AN INTERESTING CLASS OF OPERATORS WITH UNUSUAL SCHATTEN–VON NEUMANN BEHAVIOR

A.B. ALEKSANDROV, S. JANSON, V.V. PELLER AND R. ROCHBERG

We dedicate this paper to Jaak Peetre on occasion of his 65th birthday and to the memory of Tom Wolff. Both helped shape the mathematics of our time and profoundly influenced our

mathematical thoughts. Each, through his singular humanity, helped our hearts grow.

Abstract. We consider the class of integral operators Q_{φ} on L^{2}(ℝ_{+}) of the form (Q_{φ}f)(x) = ∫_{0}^{∞} φ(max{x, y})f(y)dy. We discuss necessary and sufficient conditions on φ to insure that Q_{φ} is bounded, compact, or in the Schatten–von Neumann class S_{p}, 1 < p < ∞. We also give necessary and sufficient conditions for Q_{φ} to be a finite rank operator. However, there is a kind of cut-off at p = 1, and for membership in S_{p}, 0 < p ≤ 1, the situation is more complicated. Although we give various necessary conditions and sufficient conditions relating to Q_{φ} ∈ S_{p} in that range, we do not have necessary and sufficient conditions. In the most important case p = 1, we have a necessary condition and a sufficient condition, using L^{1} and L^{2} modulus of continuity, respectively, with a rather small gap in between. A second cut-off occurs at p = 1/2: if φ is sufficiently smooth and decays reasonably fast, then Q_{φ} belongs to the weak Schatten–von Neumann class S_{1/2,∞}, but never to S_{1/2} unless φ = 0.

We also obtain results for related families of operators acting on L^{2}(ℝ) and ℓ^{2}(ℤ).

We further study operations acting on bounded linear operators on L^{2}(ℝ_{+}) related to the class of operators Q_{φ}. In particular we study Schur multipliers given by functions of the form φ (max {x, y}) and we study properties of the averaging projection (Hilbert–Schmidt projection) onto the operators of the form Q_{φ}.

Contents

1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2 2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3. Boundedness, compactness, and p > 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4. Positive operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 5. A sufficient condition, 1/2 < p ≤ 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19

Date: April 15, 2001; revised October 4, 2001. The first author is partially supported by Grant 99-01-00103 of Russian Foundation of Funda-

mental Studies and by Grant 326.53 of Integration. The third author is partially supported by NSF grant DMS 9970561. The fourth author is partially supported by NSF grant DMS 9970366.

6. p = 1, first results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 7. Schur multipliers of the form ψ(max{x, y}), x, y ∈ ℝ_{+} . . . . . . . . . . . . . . . . . 26 8. The case p = 1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 9. Sturm–Liouville theory and p = 1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 10. More on p = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .42 11. Averaging projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 12. Finite rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 13. A class of integral operators on L^{2}(ℝ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 14. Matrix representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 15. Necessary conditions for Q_{φ} ∈ S_{1} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .73 16. Dilation of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .87

1. Introduction

For a function φ ∈ L_{loc}^{1}(ℝ_{+}), which means that φ is a locally integrable function on ℝ_{+} = (0, ∞), we define the operator Q_{φ} on the set of bounded compactly supported functions f in L^{2}(ℝ_{+}) by

(Q_{φ}f)(x) = ∫_{0}^{∞} φ (max{x, y}) f(y)dy;  (1.1)

equivalently,

(Q_{φ}f)(x) = φ(x) ∫_{0}^{x} f(y)dy + ∫_{x}^{∞} φ(y)f(y)dy.  (1.2)

We are going to study when Q_{φ} is (i.e., extends to) a bounded operator in L^{2}(ℝ_{+}), and when this operator is compact, or belongs to Schatten–von Neumann classes S_{p}.

We will also consider the corresponding Volterra operators Q_{φ}^{+} and Q_{φ}^{−} defined by

(Q_{φ}^{+}f)(x) = φ(x) ∫_{0}^{x} f(y)dy,
(Q_{φ}^{−}f)(x) = ∫_{x}^{∞} φ(y)f(y)dy;

thus Q_{φ} = Q_{φ}^{+} + Q_{φ}^{−}.

It is straightforward to see (and proved more generally in Theorem 2.4) that if any of these three operators is bounded on L^{2}(ℝ_{+}), then ∫_{a}^{∞} |φ|^{2} < ∞ for any a > 0, and thus the integrals above converge for any f ∈ L^{2}(ℝ_{+}) and define all three operators on L^{2}(ℝ_{+}).

We find in §3 simple necessary and sufficient conditions for Q_{φ} to be bounded or compact, and for Q_{φ} ∈ S_{p}, 1 < p < ∞. The conditions are φ ∈ X_{∞}, φ ∈ X_{∞}^{0} and φ ∈ X_{p}, respectively, where the spaces X_{p}, X_{∞}, and X_{∞}^{0} are defined as follows.

Definition. If 0 < p < ∞, let X_{p} be the linear space of all measurable functions on ℝ_{+} that satisfy the equivalent conditions

∑_{n∈ℤ}2^{np/2} (∫_{2^{n}}^{2^{n+1}} |φ(x)|^{2}dx)^{p/2} < ∞;  (1.3)

∑_{n∈ℤ}2^{np/2} (∫_{2^{n}}^{∞} |φ(x)|^{2}dx)^{p/2} < ∞;  (1.4)

x^{1/2} (∫_{x}^{∞} |φ(y)|^{2}dy)^{1/2} ∈ L^{p}(dx/x).  (1.5)

Similarly, let X_{∞} be the linear space of all measurable functions on ℝ_{+} that satisfy the equivalent conditions

sup_{n∈ℤ}2^{n} (∫_{2^{n}}^{2^{n+1}} |φ(x)|^{2}dx) < ∞;  (1.6)

sup_{x>0} x ∫_{x}^{∞} |φ(y)|^{2}dy < ∞.  (1.7)

Let X_{∞}^{0} be the subspace of X_{∞} consisting of the functions that satisfy the equivalent conditions

lim_{n→±∞} 2^{n} (∫_{2^{n}}^{2^{n+1}} |φ(x)|^{2}dx) = 0;  (1.8)

lim_{x→0} x ∫_{x}^{∞} |φ(y)|^{2}dy = lim_{x→∞} x ∫_{x}^{∞} |φ(y)|^{2}dy = 0.  (1.9)

The equivalence of the different conditions is an exercise. For 1 ≤ p ≤ ∞, X_{p} is a Banach space with the norm

∥φ∥_{X_{p}} = ∥x^{1/2}(∫_{x}^{∞} |φ(y)|^{2}dy)^{1/2}∥_{L^{p}(dx/x)};

for 0 < p < 1, this is a quasi-norm and X_{p} is a quasi-Banach space. X_{∞}^{0} is a closed subspace of X_{∞}, and thus a Banach space too. Note that X_{p} ⊂ X_{q} if 0 < p ≤ q ≤ ∞.

Remark. It is well known [Pee] that φ ∈ X_{p} if and only if the Fourier transform Fφ belongs to the homogeneous Besov space B_{2 p}^{1/2} (here we identify φ with the function extended to ℝ by zero on ℝ_{−}).

Note that the operators Q_{φ} appear in a natural way in [MV] when studying the

boundedness problem for the Sturm–Liouville operator L from L^{∘}_{2}^{1}(ℝ_{+}) → L_{2}^{−1}(ℝ_{+}) defined by Lu = −u′′ + qu. To be more precise, Maz’ya and Verbitsky studied in [MV] the problem of identifying potentials q for which the inequality

|∫_{ℝ_{+}} |u(t)|^{2}q(t)dt| ≤ const ∫_{ℝ_{+}} |u′(t)|^{2}dt

holds for any C^{∞} compactly supported function u on (0, ∞). This inequality is in turn equivalent to the boundedness of the quadratic form

|∫_{ℝ_{+}} u(t)\overline{v(t)}q(t)dt| ≤ const ∥u′∥_{L^{2}(ℝ_{+})}∥v′∥_{L^{2}(ℝ_{+})}.  (1.10)

In [MV] under the assumption that the limit

lim_{y→∞} ∫_{x}^{y} q(t)dt = ∫_{x}^{∞} q(t)dt =^{def} φ(x)

exists for any x > 0 the problem of boundedness (compactness) of the quadratic form (1.10) was reduced to the problem of boundedness (compactness) of the operator Q_{φ} on L^{2}(ℝ_{+}). Note that in [MV] the authors also obtained boundedness and compactness criteria for the operators Q_{φ} in terms of conditions (1.7) and (1.9).

We also mention here the papers [OP] and [AO] where the authors study the

properties of the imbedding operators from L^{∘}_{p}^{l}(Ω) to L^{p}(Ω), where Ω is a domain in ℝ^{n}. There results also lead to certain estimates of s_{n}(Q_{φ}^{+}) for φ supported on [0, a], a ∈ ℝ_{+}.

The conditions in the above definition are conditions on the size of φ only, and define Banach lattices of functions on ℝ_{+}. Thus, if |ψ| ≤ |φ| and Q_{φ} is bounded, compact, or belongs to S_{p}, p > 1, then Q_{ψ} has the same property and, for example, ∥Q_{ψ}∥_{S_{p}} ≤ C_{p}∥Q_{φ}∥_{S_{p}}. Moreover, we will see that the same conditions are necessary and sufficient for these properties for the operators Q_{φ}^{+} and Q_{φ}^{−} too; thus, if one of the three operators has one these properties, then all three have it. These results for Q_{φ}^{+} are not new, see for example [ES, EEH, No, NeS, St], and the results for Q_{φ} can easily be derived. Nevertheless we give complete proofs, by another method, as a background for the case p ≤ 1.

At p = 1, there is a kind of threshold, and for S_{p}, p ≤ 1, the situation is much more complex. First, Q_{φ}^{+} and Q_{φ}^{−} never belong to S_{1}, except in the trivial case φ = 0 a.e., when the operators vanish (Theorem 6.6). Secondly, although φ ∈ X_{1} is a necessary condition for Q_{φ} ∈ S_{1}, it is not sufficient. Indeed, for p ≤ 1, we have not succeeded in finding both necessary and sufficient conditions for Q_{φ} ∈ S_{p}, and any such conditions would have to be fairly complicated. For one thing, the property Q_{φ} ∈ S_{p} does not depend on the size of φ only; although φ_{0}(x) = χ_{[0,1]}, the characteristic function of the unit interval, yields an operator Q_{φ_{0}} of rank 1, we show (see the example following Theorem 6.5) that there exists a function ψ with |ψ| = |φ_{0}| such that Q_{ψ} ∉ S_{1}. In the positive direction we show (§5 and §10) that if φ is sufficiently smooth and decays sufficiently rapidly at infinity, then Q_{φ} ∈ S_{p}, 1/2 < p ≤ 1. Conversely, we give in §15 (Theorem 15.22) a necessary condition on the L^{1} modulus of continuity for Q_{φ} ∈ S_{1}.

At p = 1/2 there is a second threshold. We prove in §8 and §9, by two different methods, that if φ is smooth (locally absolutely continuous is enough), then Q_{φ} never belongs to S_{1/2} except when φ = 0 a.e. More precisely, if φ is sufficiently smooth and decays sufficiently rapidly at infinity, and does not vanish identically, then the singular numbers s_{n}(Q_{φ}) decay asymptotically exactly like n^{−2} (Theorem 9.3).

On the other hand, Q_{φ} may belong to S_{1/2} for non-smooth functions: It is easily seen that if φ is a step function, then Q_{φ} has finite rank, and thus Q_{φ} ∈ S_{p} for every p > 0. Taking suitable infinite sums of step functions we find also other functions in S_{p}, p ≤ 1/2.

The role of smoothness is thus complicated and not well understood. It seems to be a help towards Q_{φ} ∈ S_{p} for 1/2 < p ≤ 1, but it is not necessary and it completely prevents Q_{φ} ∈ S_{p} for p ≤ 1/2. On the other hand, it is irrelevant for p > 1.

As said above, Q_{φ} has finite rank when φ is a step function. We show in Theorem 12.2 that this is the only case when Q_{φ} has finite rank.

The kernel in Definition (1.1) is symmetric, and thus Q_{φ} is self-adjoint if and only if φ is real. In §4 we show that Q_{φ} is a positive operator if and only if φ is a nonnegative non-increasing function. In this special case, for each p > 1/2, Q_{φ} ∈ S_{p} if and only if φ ∈ X_{p}. In this case we also give an even simpler necessary and sufficient conditions for boundedness, compactness and Q_{φ} ∈ S_{p}, p > 1/2 (Theorem 4.6). In particular, for positive operators we have a necessary and sufficient condition for p = 1 too.

When φ is real and thus Q_{φ} is self-adjoint, the singular values are the absolute values of the eigenvalues. In §9, we study the eigenvalues, which leads to a Sturm– Liouville problem that we study. We include one example (see §9), where the singular values can be calculated exactly by this method. We give also another example (Theorem 6.5) where the singular values are calculated within a constant factor by Fourier analysis.

In §13 and §14, we consider related families of operators acting on L^{2}(ℝ) and ℓ^{2}(ℤ); the latter operators include some given by weighted Hankel matrices.

We further study operations acting on bounded linear operators on L^{2}(ℝ_{+}) related to the class of operators Q_{φ}. We study Schur multipliers given by functions of the form φ (max {x, y}) in §7 and properties of the averaging projection onto the operators of the form Q_{φ} in §11.

We give in this paper several necessary conditions and sufficient conditions for properties such as Q_{φ} ∈ S_{p}. In all cases there are corresponding norm estimates, which follow by inspection of the proofs or by the closed graph theorem, although we usually do not state these estimates explicitly.

We denote by |I| the length of an interval I. We also use |S| for the cardinality of a finite set S; there is no danger of confusion.

We use c and C, sometimes with subscripts or superscripts, to denote various unspecified constants, not necessarily the same on different occurrences. These constants are universal unless we indicate otherwise by subscripts.

2. Preliminaries

Definition (1.1) shows that the adjoint Q_{φ}^{∗} = Qφ̄; in particular, Q_{φ} is self-adjoint if and only if φ is real. Similarly, (Q_{φ}^{+})^{∗} = Q− ; φ̄; which has the same norm and singular numbers as Q_{φ}^{−}. Hence, we will mainly consider Q_{φ}^{+}; all results obtained in this paper for Q_{φ}^{+} immediately holds for Q_{φ}^{−} too.

Schatten classes. We denote the singular numbers of a bounded operator T on a Hilbert space (or from one Hilbert space into another) by s_{n}(T), n = 0, 1, 2, …; thus s_{n}(T) =^{def} inf {∥T − R∥ : rank(R) ≤ n}. We will frequently use the simple facts

s_{m+n}(T + R) ≤ s_{m}(T) + s_{n}(R),  m, n ≥ 0,  (2.1)

and

s_{m+n}(T R) ≤ s_{m}(T)s_{n}(R),  m, n ≥ 0,  (2.2)

Recall that the Schatten–von Neumann classes S_{p}, 0 < p < ∞ are defined by

S_{p} = {T : ∑_{n≥0} s_{n}(T)^{p} < ∞} ,

and the Schatten–Lorentz classes S_{p,q} are defined by

S_{p,q} = {T : ∑_{n≥0}(s_{n}(T))^{q}(1 + n)^{q/p−1} < ∞} , 0 < p < ∞, 0 < q < ∞,
S_{p,∞} = {T : s_{n}(T) ≤ const(1 + n)^{−1/p}} ,  0 < p < ∞.

See for example [GK1] and [BS].

Other intervals. We have defined our operators for the interval ℝ_{+} = (0, ∞). More generally, for any interval I ⊆ ℝ and a function φ ∈ L_{loc}^{1}(I), we define Q_{φ}^{I} to be the integral operator on L^{2}(I) with kernel φ(max(x, y)).

It is easily seen that if I = (−∞, a) with −∞ < a ≤ ∞, then Q_{φ}^{I} is bounded only for φ = 0 a.e. By translation invariance, it remains only to consider the cases I = (0, ∞), as above, and I = (0, a) for some finite a. The latter case will be used sometimes below, but it can always be reduced to the case (0, ∞). Indeed, if we extend φ to (0, ∞) letting φ = 0 on (a, ∞), then Q_{φ}^{I} and Q_{φ} may be identified. (Formally, they are defined on different spaces, and Q_{φ}^{I} is the restriction of Q_{φ} to L^{2}(I), but the complementary restriction to L^{2}(a, ∞) vanishes. In particular, Q_{φ}^{I} and Q_{φ} have the same singular numbers.)

The case of a finite interval can also be reduced to [0,1] by the following simple homogeneity result.

Lemma 2.1. If t > 0, and φ_{t}(x) = tφ(tx), then Q_{φ} and Q_{φ_{t}} are unitarily equivalent. Similarly, for a subinterval I ⊂ (0, ∞), Q_{φ}^{I} and Q_{φ_{t}}^{t^{−1}I} are unitarily equivalent.

Proof. The mapping T_{t} : f(x) ↦ t^{1/2}f(tx) is a unitary operator in L^{2}(ℝ_{+}), and Q_{φ_{t}} = T_{t}Q_{φ}T_{t}^{−1}. ■

Note that the spaces X_{p} have the homogeneity property exhibited in this lemma: if φ ∈ X_{p}, then φ_{t} ∈ X_{p} with the same norm. Of course, it is natural to have this property for any necessary or sufficient condition for Q_{φ} ∈ S_{p}.

Distributions. We can also define the operators Q_{φ}, Q_{φ}^{+}, and Q_{φ}^{−} in the case when φ is a distribution.

For an open subset G of ℝ^{n} we denote by D(G) the space of compactly supported C^{∞} functions in G and denote by D′(G) the space of distributions on G, i.e., continuous linear functionals on D(G). We refer the reader to [Sch] for basic facts about distributions. We use the notation ⟨φ, f⟩ for φ(f), where φ ∈ D′(G) and f ∈ D(G).

Suppose now that Σ and Ω are open subsets of ℝ. In this paper we usually consider the case when Σ = Ω = ℝ_{+} or Σ = Ω = ℝ. Let Φ ∈ D′(Σ × Ω). We say that Φ determines a bounded linear operator from L^{2}(Ω) into L^{2}(Σ) if there exists a constant C such that |⟨Φ(x, y), f(y)\overline{g(x)}⟩| ≤ C∥f∥_{L^{2}}∥g∥_{L^{2}} for any f ∈ D(Ω) and any g ∈ D(Σ); the corresponding operator T then is given by ⟨T f, g⟩ = ⟨Φ(x, y), f(y)\overline{g(x)}⟩ and Φ is called the kernel of T. We say that Φ determines an operator in S_{p} if this operator is an operator from L^{2}(Ω) into L^{2}(Σ) of class S_{p}. Note that for any bounded operator T : L^{2}(Ω) → L^{2}(Σ) there exists a distribution Φ ∈ D′(Σ × Ω) which determines the operator T (a special case of Schwartz’s kernel theorem). Indeed, it is easy to see that any function in D(Ω×Σ) defines an operator from L^{2}(Σ) into L^{2}(Ω) of class S_{1}, and we have a continuous imbedding j : D(Ω × Σ) → S_{1}(L^{2}(Σ), L^{2}(Ω)). We may define the distribution Φ_{T} ∈ D′(Σ × Ω) by the following formula ⟨Φ_{T} (x, y), f(x, y)⟩ =^{def} trace(T A), where f ∈ D(Σ × Ω) and A is the integral operator with kernel function f(y, x). Clearly, Φ_{T} determines the operator T.

We also consider the space S(ℝ^{n}) of infinitely smooth functions whose derivatives of arbitrary orders decay at infinity faster than (1 + |x|)^{−n} for any n ∈ ℤ_{+} and the dual space S′(ℝ^{n}) of tempered distributions (see [Sch] for basic facts). Recall that the Fourier transform

f ↦ (Ff)(x) =^{def} ∫_{ℝ^{n}} f(t)e^{−2πi(t,x)} dt  (2.3)

where (t, x) is the scalar product of x and t in ℝ^{n}, is an isomorphism of S(ℝ^{n}) onto itself, and that it can be extended to S′(ℝ^{n}) by duality.

We need the following elementary facts.

Lemma 2.2. Suppose that a distribution Φ ∈ D′(ℝ^{2}) determines a bounded operator on L^{2}(ℝ). Then Φ is a tempered distribution.

Proof. It is easy to see that any function Φ ∈ S(ℝ^{2}) determines an operator of class S_{1} and that the corresponding imbedding of S(ℝ^{2}) into S_{1} is continuous. The result follows now by duality. ■

Lemma 2.3. Let Φ ∈ S′(ℝ^{2}). Consider the distribution Ψ on ℝ^{2} defined by Ψ(x, y) =^{def} (FΦ)(x, −y). Then Φ determines a bounded operator on L^{2}(ℝ) if and only if Ψ does. Moreover, these two operators are unitarily equivalent.

Proof. It suffices to observe that

⟨Ψ(x, y), f(y)\overline{g(x)}⟩ = ⟨Φ(x, y), F(f(−y)\overline{g(x)})⟩
= ⟨Φ(x, y), (Ff)(−y)\overline{Fg(−x)})⟩. ■

Now we are ready to define the operators Q_{φ}, Q_{φ}^{+} and Q_{φ}^{−} in the case where φ is distribution.

It is not hard to see that the operator f ↦ ∫_{0}^{x} f(x, y)dy is a continuous operator

from D(ℝ_{+} × ℝ_{+}) into D(ℝ_{+}). Hence, with any φ ∈ D′(ℝ_{+}) we can associate the distributions Λ_{φ}^{+}, Λ_{φ}^{−},and Λ_{φ} in D′(ℝ_{+} × ℝ_{+}) defined by

⟨Λ_{φ}^{+}, f(x, y)⟩ =^{def} ⟨φ, ∫_{0}^{x} f(x, y)dy⟩,  (2.4)

⟨Λ_{φ}^{−}, f(x, y)⟩ =^{def} ⟨φ, ∫_{0}^{x} f(y, x)dy⟩  (2.5)

and

Λ_{φ} =^{def} Λ_{φ}^{+} + Λ_{φ}^{−}.  (2.6)

For a distribution φ in D′(ℝ_{+}), we can consider now the operators Q_{φ}^{+}, Q_{φ}^{−}, and Q_{φ} determined by the distributions Λ_{φ}^{+}, Λ_{φ}^{−},and Λ_{φ} respectively. It is easy to see that in case φ ∈ L_{loc}^{1}(ℝ_{+}), the new definition coincides with the old one. The following theorem shows however that if one of those operators is bounded on L^{2}(ℝ_{+}), then φ must be a locally integrable function on ℝ_{+}, and so we have not enlarged the class of bounded operators of the form Q_{φ}.

Theorem 2.4. Let φ ∈ D′(ℝ_{+}). Suppose that at least one of the distributions Λ_{φ}^{+}, Λ_{φ}^{−}, or Λ_{φ} determines a bounded operator on L^{2}(ℝ_{+}). Then φ ∈ L_{loc}^{2}(ℝ_{+}).

Proof. We consider the cases of the distributions Λ_{φ}^{+} and Λ_{φ}. For Λ_{φ}^{−} the proof is the same as for Λ_{φ}^{+}. Let a ∈ ℝ_{+}. Fix a function f_{0} ∈ D(ℝ_{+}) such that supp f_{0} ⊂ [0, a] and ∫_{ℝ_{+}} f_{0}(x) dx = 1. We have

⟨Λ_{φ}^{+}, f_{0}(y)\overline{g(x)}⟩ = ⟨Λ_{φ}, f_{0}(y)\overline{g(x)}⟩ = ⟨φ, ḡ⟩

for any g ∈ D(ℝ_{+}) with supp g ⊂ [a, +∞). Therefore

|⟨φ, \overline{g}⟩| ≤ C∥f_{0}∥_{L^{2}(ℝ_{+})}∥g∥_{L^{2}(ℝ_{+})}

for any g ∈ D(a, ∞). Thus, φ|(a, ∞) ∈ L^{2}(a, ∞). ■

Triangular projection. On the class of operators on S_{p}(L^{2}(ℝ_{+})), p < ∞, we define the operator of triangular projection P as follows. Consider first the case p ≤ 2. Let T be an operator on L^{2}(ℝ_{+}) of class S_{p}, p ≤ 2. Then T is an integral operator with kernel function k_{T}:

(T f)(x) = ∫_{0}^{∞} k_{T} (x, y)f(y) dy, f ∈ L^{2}(ℝ_{+}).

Then by definition

(PT f)(x) = ∫_{0}^{x} k_{T} (x, y)f(y) dy, f ∈ L^{2}(ℝ_{+}).  (2.7)

It is well known that

∥PT ∥_{S_{p}} ≤ c_{p}∥T ∥_{S_{p}}, 1 < p ≤ 2,  (2.8)

where c_{p} depends only on p. This allows one to extend by duality the definition of P and inequality (2.8) to the case 2 ≤ p < ∞. Note also that P has weak type (1, 1), i.e.,

s_{n}(PT) ≤ const(1 + n)^{−1}∥T ∥_{S_{1}}, T ∈ S_{1}.  (2.9)

We will need these results on the triangular projection P in a more general situation. Let μ and ν be regular Borel measures on ℝ_{+}. As above we can associate with any operator T from L^{2}(μ) to L^{2}(ν) of class S_{2} the operator PT by multiplying the kernel function of T by the characteristic function of the set {(x, y) ∈ ℝ_{+}^{2} : 0 < y < x}.

Theorem 2.5. P is a bounded linear projection on S_{p}(L^{2}(μ), L^{2}(ν)) for 1 < p < ∞ and P has weak type (1, 1), i.e., P is a bounded linear operator from S_{1}(L^{2}(μ), L^{2}(ν)) to S_{1,∞}(L^{2}(μ), L^{2}(ν)).

Theorem 2.5 is well known at least when μ = ν. Let us explain how to reduce Theorem 2.5 to the case of the triangular projection onto the upper triangular matrices.

Let {K_{j}}_{j≥0} and {H_{k}}_{k≥0} be Hilbert spaces. Put K =^{def} ⨁_{j≥0} K_{j} and H =^{def} ⨁_{k≥0} H_{k}.

We identify operators A ∈ B(H, K) with its block matrix representation {A_{jk}}_{j,k≥0}, where A_{jk} ∈ B(H_{k}, K_{j}). We define the triangular projection P by (PA)_{jk} =^{def} A_{jk} for j > k and (PA)_{jk} =^{def} 0 for j ≤ k.

Lemma 2.6. Let 1 < p < ∞. Then P is bounded on S_{p}(H, K) and has weak type (1, 1). Moreover, the norms of P can be bounded independently of H and K.

In the case dim K_{j} = dim H_{k} = 1, this is the Krein–Matsaev theorem (it is equivalent to Theorems III.2.4 and III.6.2 of [GK2], see also Theorem IV.8.2 of [GK1]). In general the result can be reduced easily to this special case. Indeed, it is easy to reduce the general case to the case when dim H_{j} = dim K_{j} < ∞. Then it is easy to see that if A ∈ S_{p}, 1 ≤ p < ∞, then the diagonal part of A also belongs to S_{p}, and so we may assume without loss of generality that A_{jj} = 0, j ∈ ℤ_{+}. We can take an orthonormal basis in each H_{j} and consider the orthonormal basis of H that consists of those basis vectors of H_{j}, j ∈ ℤ_{+}. Then we can consider the matrix representation of A with respect to this orthonormal basis. We have now two triangular projections: with respect to the orthonormal basis and the projection P, the triangular with respect to the the decomposition H = ⨁_{k≥0} H_{k}.

It is not hard to check that since the diagonal part of A is zero, both triangular projections applied to A give the same result. This reduces the general case to the Krein–Matsaev theorem mentioned above. ■

Now it is easy to deduce Theorem 2.5 from Lemma 2.6.

Proof of Theorem 2.5. Let T be an integral operator with kernel function k. For ε > 0 we put k_{ε}(x, y) =^{def} k(x, y)χ_{\{(x,y)∈ℝ_{+}^{2}:[ \frac{x}{ε}]ε>y>0\}}, where [a] denote the largest integer that is less than or equal to a. Suppose that p > 1. It is sufficient to consider the case 1 < p ≤ 2 and then use duality. Let T_{ε} be the integral operator with kernel function k_{ε}. It follows easily from Lemma 2.6 that

∥T_{ε}∥_{S_{p}(L^{2}(μ),L^{2}(ν))} ≤ c_{p}∥T ∥_{S_{p}(L^{2}(μ),L^{2}(ν))}

for any ε > 0. Clearly, T_{ε} → T in the weak operator topology as ε → 0. It follows that ∥PT ∥_{S_{p}(L^{2}(μ),L^{2}(ν))} ≤ c_{p}∥T ∥_{S_{p}(L^{2}(μ),L^{2}(ν))}. The case p = 1 may be considered in the same way. ■

We have Q_{φ}^{+} = PQ_{φ}, which together with the equivalence

Q_{φ}^{+} ∈ S_{p} ⟺ Q_{φ}^{−} ∈ S_{p}

yields

Q_{φ}^{+} ∈ S_{p} ⟺ Q_{φ} ∈ S_{p}

for 1 < p < ∞. We will give a direct proof of this in Theorem 3.3.

We introduce a more general operation. Let A be a measurable subset of ℝ_{+} × ℝ_{+}. For an operator T on L^{2}(ℝ_{+}) of class S_{2} with kernel function k_{T} we consider the integral operator P_{A}T whose kernel function is χ_{A}k_{T}, where χ_{A} is the characteristic function of A. In other words,

(P_{A}T f)(x) = ∫_{0}^{∞} χ_{A}(x, y)k_{T} (x, y)f(y)dy.

If 0 < p ≤ 2 and P_{A} maps S_{p} into itself, it follows from the closed graph theorem that the linear transformation P_{A} is a bounded linear operator on S_{p}. If 1 < p ≤ 2 and P_{A} is a bounded linear operator on S_{p}, then by duality we can define in a natural way the bounded linear operator P_{A} on S_{p′}. If P_{A} is bounded on S_{1}, we can define by duality P_{A} on the space B(L^{2}(ℝ_{+})) of bounded linear operators on L^{2}(ℝ_{+}). Note that the projection P defined by (2.7) is equal to P_{A} with A = {(x, y) : x ∈ ℝ_{+}, y ≤ x}.

3. Boundedness, compactness, and p > 1

Recall the spaces X_{p} defined in the Introduction.

Theorem 3.1. Let φ ∈ L_{loc}^{2}(ℝ_{+}). The following are equivalent:

(i) Q_{φ} is bounded on L^{2}(ℝ_{+});

(ii) Q_{φ}^{+} is bounded on L^{2}(ℝ_{+});

(iii) φ ∈ X_{∞}.

Recall that the equivalence of (i) and (iii) was also established in [MV] by a different method.

Proof. Let us show that (ii) implies (i). If Q_{φ}^{+} is bounded, then the integral

operator Q_{φ}^{−} =^{def} Q_{φ} − Q_{φ}^{+} is also bounded, since its kernel function is the reflection of the kernel function of Q_{φ}^{+} with respect to the line {x = y}. Hence, Q_{φ} is bounded.

Let us deduce now (iii) from (i). For n ∈ ℤ put

A_{n} = [2^{n}, 2^{n+1}] × [2^{n−1}, 2^{n}].

Certainly, if Q_{φ} is bounded, then

sup_{n∈ℤ}∥P_{A_{n}}Q_{φ}∥ < ∞.

It is easy to see that P_{A_{n}}Q_{φ} is a rank one operator and

∥P_{A_{n}}Q_{φ}∥ = (2^{n−1} ∫_{2^{n}}^{2^{n+1}} |φ(x)|^{2}dx)^{1/2}

which implies (1.6).

It remains to prove that (1.6) implies (ii). Put

B = ⋃_{n∈ℤ}B_{n},  (3.1)

where

B_{n} = {(x, y) : 2^{n} ≤ x ≤ 2^{n+1}, 2^{n} < y < x}.  (3.2)

We also define the sets

A_{n}^{(k)} = [2^{n}, 2^{n+1}] × [2^{n−k}, 2^{n−k+1}]  (3.3)

and

A^{(k)} = ⋃_{n∈ℤ}A_{n}^{(k)}.  (3.4)

Clearly,

{(x, y) : x > 0, 0 < y < x} = B ∪ ⋃_{k≥1} A^{(k)},

and so

∥Q_{φ}^{+}∥ ≤ ∥P_{B}Q_{φ}∥ + ∑_{k≥1} ∥P_{A^{(k)}}Q_{φ}∥.  (3.5)

Since the projections of the B_{n} onto the coordinate axes are pairwise disjoint, it is straightforward to see that

∥P_{B}Q_{φ}^{+}∥ = sup_{n∈ℤ}∥P_{B_{n}}Q_{φ}^{+}∥.

Let R_{n} be the integral operator with kernel function

k_{R_{n}}(x, y) = φ(x)χ_{[2^{n},2^{n+1}]}(x)χ_{[2^{n},2^{n+1}]}(y).  (3.6)

Obviously, rank R_{n} = 1 and ∥R_{n}∥_{S_{2}} = ∥R_{n}∥ = (2^{n} ∫_{2^{n}}^{2^{n+1}} |φ(x)|^{2}dx)^{1/2}. It is also evident that PR_{n} = P_{B_{n}}Q_{φ}^{+}, and since P an orthogonal projection on S_{2}, we have

∥P_{B_{n}}Q_{φ}^{+}∥ = ∥PR_{n}∥ ≤ ∥PR_{n}∥_{S_{2}} ≤ ∥R_{n}∥_{S_{2}}
= (2^{n} ∫_{2^{n}}^{2^{n+1}} |φ(x)|^{2}dx)^{1/2} ≤ const ∥φ∥_{X_{∞}}.

Next, it is also easy to see that

∥P_{A^{(k)}}Q_{φ}∥ = sup_{n∈ℤ}∥P_{A_{n}^{(k)}}Q_{φ}∥.

Also, P_{A_{n}^{(k)}}Q_{φ} has rank one and norm (2^{n−k} ∫_{2^{n}}^{2^{n+1}} |φ(x)|^{2}dx)^{1/2}, and so

∑_{k≥1} ∥P_{A^{(k)}}Q_{φ}∥ = ∑_{k≥1} sup_{n∈ℤ}(2^{n−k} ∫_{2^{n}}^{2^{n+1}} |φ(x)|^{2}dx)^{1/2}

= ∑_{k≥1} 2^{−k/2} sup_{n∈ℤ}(2^{n} ∫_{2^{n}}^{2^{n+1}} |φ(x)|^{2}dx)^{1/2}

= const sup_{n∈ℤ}(2^{n} ∫_{2^{n}}^{2^{n+1}} |φ(x)|^{2}dx)^{1/2} .

The result follows now from (3.5). ■

Theorem 3.2. Let φ ∈ L_{loc}^{1}(ℝ_{+}). The following are equivalent:

(i) Q_{φ} is compact on L^{2}(ℝ_{+});

(ii) Q_{φ}^{+} is compact on L^{2}(ℝ_{+});

(iii) φ ∈ X_{∞}^{0}.

Recall that the equivalence of (i) and (iii) was also established in [MV] by a different method.

Proof. It is easy to see that the estimates given in the proof of Theorem 3.1 actually lead to the proof of Theorem 3.2; for the step (i) ⟹ (iii) we observe that if Q_{φ} is compact, then lim_{n→±∞} ∥P_{A_{n}}Q_{φ}∥ = 0. ■

Theorem 3.3. Let 1 < p < ∞ and let φ ∈ L_{loc}^{1}(ℝ_{+}). Then the following conditions are equivalent:

(i) Q_{φ} ∈ S_{p};

(ii) Q_{φ}^{+} ∈ S_{p};

(iii) φ ∈ X_{p}.

Note that the fact that (ii)⇔(iii) was proved in [No] by a different method, see also [NeS] and [St] for the case of more general Volterra operators.

Proof. The fact that (ii)⇒(i) can be proved exactly as in the proof of Theorem 3.1. Let us show that (i) implies (iii). Consider the sets A_{n} = A_{n}^{(1)} introduced in the proof of Theorem 3.1 (see (3.3)). Recall that A^{(1)} = ⋃_{n∈ℤ}A_{n}^{(1)}. It is easy to see

that

∥P_{A^{(1)}}Q_{φ}∥_{S_{p}} ≤ ∥Q_{φ}∥_{S_{p}}.

Clearly,

∥P_{A^{(1)}}Q_{φ}∥_{S_{p}}^{p} = ∑_{n∈ℤ}∥P_{A_{n}}Q_{φ}∥_{S_{p}}^{p},

the operator P_{A_{n}}Q_{φ} has rank one and

∥P_{A_{n}}Q_{φ}∥_{S_{p}} = (2^{n−1} ∫_{2^{n}}^{2^{n+1}} |φ(x)|^{2}dx)^{1/2} .

This implies (1.3).

Let us show that (1.3) implies (ii). Consider the sets, B_{n}, B, A_{n}^{(k)}, A^{(k)} defined in (3.2), (3.1), (3.3), and (3.4). Clearly,

∥Q_{φ}^{+}∥_{S_{p}} ≤ ∥P_{B}Q_{φ}^{+}∥_{S_{p}} + ∑_{k≥1} ∥P_{A^{(k)}}Q_{φ}^{+}∥_{S_{p}}.

Let us first estimate ∥P_{B}Q_{φ}^{+}∥_{S_{p}}. Clearly,

∥P_{B}Q_{φ}^{+}∥_{S_{p}}^{p} = ∑_{n∈ℤ}∥P_{B_{n}}Q_{φ}^{+}∥_{S_{p}}^{p}.

Consider the rank one operators R_{n} defined in (3.6). As in the proof of Theorem 3.1 we have PR_{n} = P_{B_{n}}Q_{φ}^{+} and since P is bounded on S_{p}, we obtain

∥P_{B_{n}}Q_{φ}^{+}∥_{S_{p}} = ∥PR_{n}∥_{S_{p}} ≤ const_{p} ∥R_{n}∥_{S_{p}} = const_{p} (2^{n} ∫_{2^{n}}^{2^{n+1}} |φ(x)|^{2}dx)^{1/2} , and so

∥P_{B}Q_{φ}^{+}∥_{S_{p}}^{p} ≤ const_{p} ∑_{n∈ℤ}(2^{n} ∫_{2^{n}}^{2^{n+1}} |φ(x)|^{2}dx)^{p/2} .

It is also easy to see that

∥P_{A^{(k)}}Q_{φ}^{+}∥_{S_{p}}^{p} = ∑_{n∈ℤ}∥P_{A_{n}^{(k)}}Q_{φ}^{+}∥_{S_{p}}^{p}

and, since P_{A_{n}^{(k)}}Q_{φ} has rank 1,

∥P_{A_{n}^{(k)}}Q_{φ}^{+}∥_{S_{p}} = ∥P_{A_{n}^{(k)}}Q_{φ}∥_{S_{p}} = (2^{n−k} ∫_{2^{n}}^{2^{n+1}} |φ(x)|^{2}dx)^{1/2} ,

and so

\begin{aligned}∥P_{A^{(k)}}Q_{φ}^{+}∥_{S_{p}} & = (∑_{n∈ℤ}(2^{n−k} ∫_{2^{n}}^{2^{n+1}} |φ(x)|^{2}dx)^{p/2})^{1/p} \\ & = 2^{−k/2}(∑_{n∈ℤ}(2^{n} ∫_{2^{n}}^{2^{n+1}} |φ(x)|^{2}dx)^{p/2})^{1/p} \end{aligned}  (3.7)

and

∑_{k≥1} ∥P_{A^{(k)}}Q_{φ}^{+}∥_{S_{p}} ≤ const(∑_{n∈ℤ}(2^{n} ∫_{2^{n}}^{2^{n+1}} |φ(x)|^{2}dx)^{p/2})^{1/p}  (3.8)

which completes the proof. ■

Remark. The same proof shows that for p = 1, (ii) ⇒ (i) ⇒ (iii), but the final part of it fails because the triangular projection is not bounded on S_{1}. We will see later that, indeed, none of the implications can be reversed for p = 1.

In the Hilbert–Schmidt case p = 2, the result simplifies further. Indeed, we have the equalities

∥Q_{φ}∥_{S_{2}} = (∫_{0}^{∞} ∫_{0}^{∞} |φ(max(x, y))|^{2} dx dy)^{1/2} = (2 ∫_{0}^{∞} x|φ(x)|^{2} dx)^{1/2}

and

∥Q_{φ}^{+}∥_{S_{2}} = 2^{−1/2}∥Q_{φ}∥_{S_{2}} = ∥x^{1/2}φ(x)∥_{2} = ∥φ∥_{X_{2}}.

4. Positive operators

We consider the special case when Q_{φ} is a positive operator, i.e., ⟨Q_{φ}f, f⟩ ≥ 0 for every f ∈ L^{2}(ℝ_{+}). In this case we obtain rather complete results. We first characterize the corresponding symbols φ.

Theorem 4.1. Suppose that φ ∈ L_{loc}^{1}(ℝ_{+}) is such that Q_{φ} is a bounded operator. Then Q_{φ} is a positive operator if and only if φ is a.e. equal to a non-increasing, non-negative function.

Proof. Suppose that Q_{φ} is positive. Define, for z, h > 0, f_{z,h} = h^{−1}χ_{(z,z+h)} and let Leb(φ) be the set of Lebesgue points of φ. Then, if z ∈ Leb(φ),

|⟨Q_{φ}f_{z,h}, f_{z,h}⟩ − φ(z)| = h^{−2} |∫_{z}^{z+h} ∫_{z}^{z+h} (φ{max(x, y)} − φ(z)) dx dy|

≤ h^{−2} ∫_{z}^{z+h} ∫_{z}^{z+h}(|φ(x) − φ(z)| + |φ(y) − φ(z)|) dx dy

= 2h^{−1} ∫_{z}^{z+h} |φ(x) − φ(z)| dx → 0

as h → 0. Since ⟨Q_{φ}f_{z,h}, f_{z,h}⟩ ≥ 0 for all h > 0, this implies φ(z) ≥ 0.

Moreover, if z_{1}, z_{2} ∈ Leb(φ) are two Lebesgue points with z_{1} < z_{2} and 0 < h < z_{2} − z_{1}, then, similarly,

⟨Q_{φ}f_{z_{1},h}, f_{z_{2},h}⟩ = ⟨Q_{φ}f_{z_{2},h}, f_{z_{1},h}⟩ = h^{−1} ∫_{z_{2}}^{z_{2}+h} φ(y) dy → φ(z_{2})

as h → 0, and thus, with g_{h} = f_{z_{1},h} − f_{z_{2},h},

⟨Q_{φ}g_{h}, g_{h}⟩ → φ(z_{1}) + φ(z_{2}) − 2φ(z_{2}) = φ(z_{1}) − φ(z_{2}).

Hence, φ(z_{1}) ≥ φ(z_{2}).

It follows that the function φ̃(x) = sup {φ(z) : z ≥ x, z ∈ Leb(φ)} is non-negative and non-increasing, and that φ = φ̃ a.e.

Conversely, if φ is non-negative and non-increasing, then lim_{x→∞} φ(x) = 0, since a positive lower bound is impossible by Theorem 3.1. Thus there exists a measure μ on (0, ∞) such that φ(x) = μ(x, ∞) a.e. If, say, f is bounded with compact support in (0, ∞), then by Fubini’s theorem

\begin{aligned}⟨Q_{φ}f, f⟩ & = ∫∫ φ(max\{x, y\})f(x)f̄(y) dx dy = ∫∫∫_{max\{x,y\}<z} f(x)f̄(y) dx dy dμ(z) \\ & = ∫_{0}^{∞} |∫_{0}^{z} f(x) dx|^{2} dμ ≥ 0. \end{aligned}

Hence, Q_{φ} is a positive operator. ■

We have used the fact that Q_{φ} is a sum of the Volterra operators Q_{φ}^{+} and Q_{φ}^{−}. Operators of the type Q_{φ} also appear as the composition of Volterra operators.

Theorem 4.2. Suppose that ψ_{1} and ψ_{2} are functions on ℝ_{+} such that Q_{ψ_{1}} and Q_{ψ_{2}} are bounded linear operators. Let φ be the function defined by

φ(x) = ∫_{x}^{∞} ψ_{1}(t)ψ_{2}(t)dt.  (4.1)

Then the operator Q_{φ} is bounded and admits a factorization

Q_{φ} = Q_{ψ_{1}}^{−}Q_{ψ_{2}}^{+}.

Proof. Let k_{1} be the kernel function of Q_{ψ_{1}}^{−} and k_{2} the kernel function of Q_{ψ_{2}}^{+}. We have

k_{1}(x, t) = \begin{cases}ψ_{1}(t), & t ≥ x \\ 0, & t < x \end{cases}

and

k_{2}(t, y) = \begin{cases}0, & t < y \\ ψ_{2}(t), & t ≥ y. \end{cases}

Then the kernel function k of the product Q_{ψ_{1}}^{−}Q_{ψ_{2}}^{+} is given by

k(x, y) = ∫_{ℝ_{+}} k_{1}(x, t)k_{2}(t, y)dt = ∫_{max{x,y}}^{∞} ψ_{1}(t)ψ_{2}(t)dt = φ(max{x, y})

by the hypotheses; the integrals converge by Theorem 3.1 and the Cauchy–Schwarz inequality. ■

The function φ in (4.1) is always locally absolutely continuous. In order to treat more general non-increasing φ, we define, for a positive measure μ on (0, ∞), the operator Q_{μ}^{+} : L^{2}(0, ∞) → L^{2}(μ) by Q_{μ}^{+}f(x) = ∫_{0}^{x} f(y) dy. (Thus, the operator itself does not depend on μ; only its range space does.) We have the following analogues of Theorems 3.1 and 3.3. (We leave the corresponding criterion for compactness to the reader.)

Theorem 4.3. Let μ be a positive measure on ℝ_{+}. The following are equivalent:

(i) Q_{μ}^{+} is bounded operator from L^{2}(ℝ_{+}) to L^{2}(μ);

(ii) sup_{n∈ℤ}2^{n}μ[2^{n}, 2^{n+1}) < ∞;

(iii) sup_{x>0} xμ[x, ∞) < ∞.

Theorem 4.4. Let 1 < p < ∞ and let μ be a positive measure on ℝ_{+}. The following conditions are equivalent:

(i) Q_{μ}^{+} ∈ S_{p};

(ii) ∑_{n∈ℤ}2^{np/2}(μ[2^{n}, 2^{n+1}))^{p/2} < ∞;

(iii) ∑_{n∈ℤ}2^{np/2}(μ[2^{n}, ∞))^{p/2} < ∞;

(iv) x^{1/2}(μ(x, ∞))^{1/2} ∈ L^{p}(dx/x).

The proofs of Theorems 4.3 and 4.4 are almost the same as the proofs of Theorems 3.1 and 3.3. The main difference is that we have to apply the theorem on the boundedness of the triangular projection on S_{p}, 1 < p < ∞, in the case of weighted L^{2} spaces (see Theorem 2.5). ■

Furthermore, the factorization in Theorem 4.2 extends.

Theorem 4.5. Suppose that μ is a positive measure on ℝ_{+} such that Q_{μ}^{+} is a bounded linear operator. Let φ be the function defined by φ(x) = μ(x, ∞). Then the operator Q_{φ} is bounded and Q_{φ} = (Q_{μ}^{+})^{∗}Q_{μ}^{+}.

Proof. By Theorem 4.3, 0 ≤ φ(x) ≤ C_{φ}/x, and thus Q_{φ} is bounded by Theorem 3.1.

If, say, f, g ∈ L^{2}(ℝ_{+}) are non-negative, then by Fubini’s theorem

\begin{aligned}⟨(Q_{μ}^{+})^{∗}Q_{μ}^{+}f, g⟩ & = ⟨Q_{μ}^{+}f, Q_{μ}^{+}g⟩ = ∫_{0}^{∞} ∫_{0}^{z} f(x) dx ∫_{0}^{z} ḡ(y) dy dμ(z) \\ & = ∫∫∫_{max\{x,y\}<z} f(x)ḡ(y) dx dy dμ(z) = ⟨Q_{φ}f, g⟩. ■ \end{aligned}

For positive operators Q_{φ}, we have a simple result, Theorem 4.6 below. (For (i), cf. the discussion of the Hille condition in [MV].)

Theorem 4.6. Suppose that φ is a non-negative, non-increasing function on ℝ_{+}.

(i) Q_{φ} is bounded if and only if xφ(x) is bounded.

(ii) Q_{φ} is compact if and only if xφ(x) → 0 as x → 0 and as x → ∞.

(iii) If 1/2 < p < ∞, then the following are equivalent: (a) Q_{φ} ∈ S_{p};

(b) φ ∈ X_{p};

(c) xφ(x) ∈ L^{p}(dx/x).

Proof. The equivalence of φ ∈ X_{p} and xφ(x) ∈ L^{p}(dx/x) for non-increasing, non-negative φ is elementary, using φ(2^{n})^{2} ≥ 2^{−n} ∫_{2^{n}}^{2^{n+1}} |φ|^{2} ≥ φ(2^{n+1})^{2}. Hence, (i) follows from Theorem 3.1, and (iii) for p > 1 from Theorem 3.3; furthermore, (ii) follows similarly from Theorem 3.2.

For (iii) for a general p > 1/2, we first note that any of the three conditions (a), (b) and (c) implies that xφ(x) is bounded. (This follows by (i) for (a), and by elementary estimates for (b) and (c).) We can assume without loss of generality that φ is right-continuous on (0, ∞). If we let μ be the measure on ℝ_{+} with μ(x, ∞) = φ(x), then by Theorems 4.3 and 4.5, Q_{μ}^{+} is bounded and Q_{φ} = (Q_{μ}^{+})^{∗}Q_{μ}^{+}. Hence, Q_{φ} ∈ S_{p} ⇔ Q_{μ}^{+} ∈ S_{2p}, and the result follows by Theorem 4.4. ■

We will see in the example given at the beginning of §9 that Theorem 4.6 (iii) does not extend to p ≤ 1/2.

5. A sufficient condition, 1/2 < p ≤ 1

By linearity, we immediately obtain from Theorem 4.6 a sufficient, but not necessary, condition for general symbols φ.

Definition. Y_{p} is the subspace of X_{p} spanned by non-increasing functions. I.e., φ ∈ Y_{p} if and only if Re φ and Im φ both are differences of non-increasing functions in X_{p}.

Theorem 5.1. Let p > 1/2. If φ ∈ Y_{p}, then Q_{φ} ∈ S_{p}. ■

The condition φ ∈ Y_{p} can be made more explicit and useful as follows. We denote by ∥φ∥′_{BV (I)} the total variation of a function φ over an interval I, and let

∥φ∥_{BV (I)} =^{def} ∥φ∥′_{BV (I)}+sup_{I} |φ|. Moreover, we let V_{φ}(x) denote the total variation of

a function φ over the interval [x, ∞). Note that if φ is locally absolutely continuous, then V_{φ}(x) = ∫_{x}^{∞} |φ′(y)| dy.

Lemma 5.2. Let 0 < p < ∞. If φ is non-increasing, then

φ ∈ Y_{p} ⇔ φ ∈ X_{p} ⇔ ∫_{0}^{∞} |xφ(x)|^{p} \frac{dx}{x} < ∞.

Proof. For non-increasing φ, the first equivalence follows from the definition of Y_{p}, while the second equivalence was noted in the proof of Theorem 4.6. ■

Theorem 5.3. Let φ be a function on ℝ_{+} and let 0 < p < ∞. The following are equivalent:

(i) φ ∈ Y_{p}; (ii) V_{φ} ∈ X_{p} and lim_{x→∞} φ(x) = 0; (iii) xV_{φ}(x) ∈ L^{p}(dx/x) and lim_{x→∞} φ(x) = 0; (iv) φ has locally bounded variation, lim_{x→∞} φ(x) = 0 and

∑_{n∈ℤ}2^{np} (∫_{2^{n}}^{∞} |dφ(x)|)^{p} < ∞;

(v) φ has locally bounded variation, lim_{x→∞} φ(x) = 0 and

∑_{n∈ℤ}2^{np} (∫_{2^{n}}^{2^{n+1}} |dφ(x)|)^{p} < ∞;

(vi) ∑_{n∈ℤ}2^{np}∥φ∥_{BV [2^{n},2^{n+1}]}^{p} < ∞;

(vii) ∑_{n∈ℤ}∥xφ(x)∥_{BV [2^{n},2^{n+1}]}^{p} < ∞.

Proof. To show that (i) implies (ii), it suffices to consider a non-increasing φ ∈ X_{p}; it is easily seen that then lim_{x→∞} φ(x) = 0 and V_{φ} = φ, whence (ii) follows.

Conversely, suppose that (ii) holds. By considering real and imaginary parts, we may assume that φ is real. Then φ = V_{φ} − (V_{φ} − φ), where V_{φ} and V_{φ} − φ are non-increasing functions in X_{p}; note that 0 ≤ V_{φ} − φ ≤ 2V_{φ}. Consequently (i) holds.

Since V_{φ} is non-increasing, (ii)⇔(iii) follows by Lemma 5.2.

Next, (iii)⇔(iv) follows easily because V_{φ}(x) = ∫_{x}^{∞} |dφ(y)|, and (iv)⇔(v) is easily verified.

If (iv) holds, then ∥φ∥_{BV [2^{n},2^{n+1}]} = ∫_{2^{n}}^{2^{n+1}} |dφ(x)|+ sup_{[2^{n},2^{n+1}]} |φ| ≤ 2 ∫_{2^{n}}^{∞} |dφ(x)| and

(vi) follows. Conversely, (vi) immediately implies (v).

Finally, for any functions φ and ψ on an interval I, ∥ψφ∥_{BV (I)} ≤ ∥ψ∥_{BV (I)}∥φ∥_{BV (I)}, and the equivalence (vi)⇔(vii) follows by taking ψ(x) = x and ψ(x) = 1/x. ■

We can define a norm in Y_{p} (a quasi-norm for p < 1) by

∥φ∥_{Y_{p}} =^{def} (∑_{n∈ℤ}2^{np}(∫_{2^{n}}^{2^{n+1}} |dφ(x)|)^{p})^{1/p};  (5.1)

an alternative is

(∫_{0}^{∞} |xV_{φ}(x)|^{p} \frac{dx}{x})^{1/p} ,

which yields an equivalent (quasi-)norm.

We obtain as corollaries to Theorems 5.1 and 5.3 the following simple sufficient conditions for Q_{φ} ∈ S_{p}.

Corollary 5.4. If φ is absolutely continuous on [0, ∞), lim_{x→∞} φ(x) = 0 and sup_{x>0} x^{γ}|φ′(x)| < ∞ for some γ > 2, then φ ∈ Y_{p} for every p > 0 and thus Q_{φ} ∈ S_{p} for every p > 1/2.

Proof. V_{φ}(x) is bounded and |V_{φ}(x)| ≤ const ·x^{1−γ}, and thus xV_{φ}(x) ∈ L^{p}(dx/x) for every p > 0. ■

Corollary 5.5. If φ has bounded variation and support in a finite interval, then φ ∈ Y_{p} for every p > 0 and thus Q_{φ} ∈ S_{p} for every p > 1/2. ■

6. p = 1, first results

Let us now consider the case p = 1. We know already that φ ∈ X_{1} is a necessary and φ ∈ Y_{1} a sufficient condition for Q_{φ} ∈ S_{1}. We will later see that neither condition is both necessary and sufficient (see the example following Theorem 6.5). We restate these results as follows.

Theorem 6.1. If φ has locally bounded variation, ∫_{0}^{∞} x|dφ(x)| < ∞ and lim_{x→∞} φ(x) = 0, then Q_{φ} ∈ S_{1}.

Proof. It follows from Theorem 5.3 and the calculation

∫_{0}^{∞} V_{φ}(x) dx = ∫_{0}^{∞} ∫_{x}^{∞} |dφ(y)| dx = ∫_{0}^{∞} y|dφ(y)|

that the assumption is equivalent to φ ∈ Y_{1}, so the result follows from Theorem 5.1. ■

Theorem 6.2. If Q_{φ} ∈ S_{1}, then φ ∈ X_{1}. Furthermore, φ ∈ L^{1}(0, ∞) and

trace Q_{φ} = ∫_{0}^{∞} φ(x) dx.

Proof. By the Remark at the end of §3, Q_{φ} ∈ S_{1} ⇒ φ ∈ X_{1}. Next, X_{1} ⊂ L^{1}(0, ∞), since by the Cauchy–Schwarz inequality and (1.3)

∫_{0}^{∞} |φ| ≤ ∑_{n∈ℤ}2^{n/2} (∫_{2^{n}}^{2^{n+1}} |φ(x)|^{2}dx)^{1/2} < ∞.

Finally, the trace formula follows from Theorem 6.3 below, since with k(x, y) = φ(max{x, y}) for x, y > 0,

∫_{−∞}^{∞} k(x, x + a) dx = ∫_{|a|}^{∞} φ(x) dx. ■

Remark. This theorem gives a formula for the trace of Q_{φ} if that operator has a trace; i.e., if it is in the trace class, S_{1}. The theorem also shows that if Q_{φ} is in the trace class then we must have φ ∈ X_{1}. We will see later in this section that φ ∈ X_{1} is not enough to insure that Q_{φ} is in the trace class. However, we will see later, Corollary 10.2, that φ ∈ X_{1} is sufficient to insure that Q_{φ} and related operators do have a Dixmier trace.

In the previous theorem we used following fact from [A], improving an earlier result in [B]:

Theorem 6.3. If T is an integral operator on L^{2}(ℝ) of class S_{1} with kernel function k, then the function

x ↦ k(x, x + a), x ∈ ℝ,

is in L^{1}(ℝ) for almost all a ∈ ℝ and the function

a ↦ ∫_{ℝ} k(x, x + a)dx, a ∈ ℝ,

is almost everywhere equal to the Fourier transform Fh of a function h ∈ L^{1}(ℝ), in particular, it coincides a.e. with a continuous function on ℝ. Moreover,

trace T = (Fh)(0).

Proof. It is sufficient to prove the result when k(x, y) = f(x)g(y) with f and g in L^{2}(ℝ), in which case it can be verified straightforwardly, with h(ξ) = (Ff)(ξ)(Fg)(−ξ). ■

We can reduce the estimation of ∥Q_{φ}∥_{S_{1}} to the estimation of ∥Q_{φ}^{I}∥_{S_{1}} for dyadic intervals I.

Theorem 6.4. Let φ ∈ L_{loc}^{1}(ℝ_{+}) and let I_{n} = [2^{n}, 2^{n+1}]. Then Q_{φ} ∈ S_{1} if and only if φ ∈ X_{1} and

∑_{n∈ℤ}∥Q_{φ}^{I_{n}}∥_{S_{1}} < ∞.  (6.1)

Proof. Consider the sets A^{(k)} defined by (3.4) and consider their symmetric images A^{(−k)} about the line {(x, y) : x = y}. As in (3.8) we have

∑_{k≥1} ∥P_{A^{(k)}}Q_{φ}∥_{S_{1}} ≤ const ∑_{n∈ℤ}(2^{n} ∫_{2^{n}}^{2^{n+1}} |φ(x)|^{2}dx)^{1/2} .

Similarly,

∑_{k≥1} ∥P_{A^{(−k)}}Q_{φ}∥_{S_{1}} ≤ const ∑_{n∈ℤ}(2^{n} ∫_{2^{n}}^{2^{n+1}} |φ(x)|^{2}dx)^{1/2} .

It thus follows from φ ∈ X_{1} that

P_{A}Q_{φ} ∈ S_{1} and PĂQ_{φ} ∈ S_{1}.

where

A = ⋃_{k≥1} A^{(k)} and Ă = ⋃_{k≤−1} A^{(k)}.

Consequently, using Theorem 6.2, Q_{φ} ∈ S_{1} if and only if φ ∈ X_{1} and P_{B}Q_{φ} ∈ S_{1}, where B is defined by (3.1). Since the projections of the sets B_{n} onto the coordinate axes are disjoint, and P_{B_{n}}Q_{φ} = Q_{φ}^{I_{n}}, it follows that P_{B}Q_{φ} ∈ S_{1} if and only if (6.1) holds. ■

Let n ∈ ℤ. It is easy to see that Q_{φ}^{I_{n}} ∈ S_{1} if and only if Q_{φ_{n}} ∈ S_{1}, where

φ_{n}(x) =^{def} \begin{cases}φ(x + 2^{n}), & x ∈ [0, 2^{n}] \\ 0, & otherwise. \end{cases}

Hence, the question of when Q_{φ} belongs to S_{1} reduces to the question of estimating ∥Q_{φ}∥_{S_{1}} for functions φ supported on finite intervals.

Remark. For 0 < p < 1, it can similarly be shown that if φ ∈ X_{p} and ∑_{n∈ℤ} ∥Q_{φ}^{I_{n}}∥_{S_{p}}^{p} < ∞, then Q_{φ} ∈ S_{p}. We do not know whether the converse holds.

We next show that φ ∈ X_{1} is not sufficient for Q_{φ} ∈ S_{1}.

Theorem 6.5. Let φ_{N}(x) = e^{2πiNx}χ_{[0,1]}(x) for N = 1, 2, … Then

s_{n}(Q_{φ_{N}}) ≍ min \{\frac{1}{n + 1}, \frac{N}{(n + 1)^{2}} \} ,

and so

∥Q_{φ_{N}} ∥_{S_{1}} ≍ log(N + 1).  (6.2)

Note that ≍ means that the ratio of the two sides are bounded from above and below by positive constants (not depending on n or N). Clearly,

∥φ_{N}∥_{X_{1}} = ∥χ_{[0,1]}∥_{X_{1}} = C

is independent of N. This shows, by the closed graph theorem, that

φ ∈ X_{1} ⇏ Q_{φ} ∈ S_{1};

a concrete counterexample is given in the example following the proof of Theorem 6.5. Moreover, φ_{N} ∈ X_{p} for any p > 0, again with norm independent of N, so for every p ≤ 1

φ ∈ X_{p} ⇏ Q_{φ} ∈ S_{1} and φ ∈ X_{p} ⇏ Q_{φ} ∈ S_{p}.

It also follows from Theorem 6.5 that

∥Q_{φ_{N}} ∥_{S_{p}} ≍ N^{(1−p)/p},  \frac{1}{2} < p < 1.  (6.3)

Proof of Theorem 6.5. Since multiplication by a unimodular function is a unitary operator, the singular values of Q_{φ_{N}} are the same as the singular values s_{n}(T_{N}), where T_{N} is the integral operator on L^{2}[0, 1] with kernel

e^{−πiNx}φ_{N}(max(x, y))e^{−πiNy} = exp(2πiN max(x, y) − πiNx − πiNy)
= exp(πiN|x − y|).

Let g_{N}(x) = e^{πiN|x|} for |x| ≤ 1, and extend g_{N} to a function on ℝ with period 2. Let T′_{N} be the integral operator on L^{2}[−1, 1] with kernel g_{N}(x − y). If I_{+} = [0, 1], I_{−} = [−1, 0] and A_{αβ} = I_{α} × I_{β}, α, β ∈ {+, −}, then P_{A_{++}}T′_{N} = T_{N} and thus

s_{n}(Q_{φ_{N}}) = s_{n}(T_{N}) ≤ s_{n}(T′_{N}).  (6.4)

Moreover, each P_{A_{αβ}}T′_{N} is by a translation unitarily equivalent to either T_{N} or the integral operator on L^{2}[0, 1] with kernel g_{N}(x − y − 1) = exp(πiN(1 − |x − y|)) = (−1)^{N} \overline{g_{N}}(x − y), which has the same singular values. Hence, by (2.1),

s_{4n}(T′_{N}) ≤ 4s_{n}(T_{N}) = 4s_{n}(Q_{φ_{N}}).  (6.5)

T′_{N} is a convolution operator on the circle ℝ/2ℤ, so the elements of the orthonormal basis {2^{−1/2} exp(πikx)}_{k=−∞}^{∞} in L^{2}[−1, 1] are eigenvectors with eigenvalues

ĝ_{N}(k) =^{def} ∫_{−1}^{1} g_{N}(x)e^{−πikx}dx = ∫_{−1}^{1} e^{πiN|x|−πikx} dx

= ∫_{−1}^{0} e^{−πi(N+k)x} dx + ∫_{0}^{1} e^{πi(N−k)x} dx

= \begin{cases}0, & k ≡ N (mod 2), k ≠ ±N, \\ 1, & k = ±N, \\ \frac{−2}{πi(N+k)} + \frac{−2}{πi(N−k)} = \frac{4iN}{π(N^{2}−k^{2})}, & k ≡ N + 1 (mod 2). \end{cases}

Consequently, the singular values s_{n}(T_{N}) are the absolute values |ĝ_{N}(k)|, k ∈ ℤ, arranged in decreasing order. This easily yields

s_{n}(T′_{N}) ≍ min \{\frac{1}{n + 1}, \frac{N}{(n + 1)^{2}} \} ,

and the result follows by (6.4) and (6.5). ■

Remark. A related method is used in a more general context in §14. Indeed, the estimates (6.2) and (6.3) follow easily from Theorem 14.10 (with the norm estimates implicit there).

Moreover, (6.2) and (6.3) also follow from the results in §16, obtained by a different method.

Example. For a concrete counterexample we let N_{k} ≥ 2 be integers and define

φ = ∑_{k=1}^{∞} exp(2πi2^{k}N_{k}x)χ_{(2^{−k},2^{1−k})}(x).

Then |φ| = χ_{[0,1]}; in particular, φ ∈ X_{p} for every p > 0.

On the other hand, for every k, by Lemma 2.1 and Theorem 6.5,

∥Q_{φ}∥_{S_{1}} ≥ ∥Q_{φ}^{[2^{−k},2^{1−k}]}∥_{S_{1}} = 2^{−k}∥Q_{φ_{N_{k}}} ∥_{S_{1}} ≥ c2^{−k} log N_{k}.

Choosing N_{k} = 2^{3^{k}}, we find that Q_{φ} ∉ S_{1}.

Other choices yield further interesting examples. Thus, N_{k} = 2^{k} yields a symbol φ ∈ X_{1} but φ ∉ Y_{1} such that, by Theorem 6.4, Q_{φ} ∈ S_{1}. In fact, using the Remark followed by Theorem 6.5, we can conclude that Q_{φ} ∈ S_{p} for every p > 1/2.

The choice N_{k} = 2^{k^{2}} yields a symbol φ ∈ X_{1}\Y_{1} such that Q_{φ} ∈ S_{1} but Q_{φ} ∉ S_{p} for p < 1.

Remark. Theorem 6.5 implies also that φ ∈ X_{1} does not imply Q_{φ} ∈ S_{1,q} for any Schatten–Lorentz space S_{1,q} with q < ∞.

Let us prove now that the condition Q_{φ} ∈ S_{1} does not imply that Q_{φ}^{+} ∈ S_{1}. Moreover, as previously shown by Nowak [No], we show that there are no non-zero operators Q_{φ}^{+} of class S_{1}. (A more refined result will be given in Theorem 10.1.)

Theorem 6.6. Suppose that φ ∈ L_{loc}^{1}(ℝ_{+}). If Q_{φ}^{+} ∈ S_{1}, then φ is the zero function.

Proof. Suppose that Q_{φ}^{+} ∈ S_{1}. Let k be the kernel function of Q_{φ}^{+}, extended by 0 to ℝ^{2}, i.e.

k(x, y) = \begin{cases}φ(x), & 0 < y < x \\ 0, & otherwise. \end{cases}

Let Δ ⊂ ℝ_{+} be a compact interval. Consider the operator P_{Δ}Q_{φ}^{+}P_{Δ}, where P_{Δ} is multiplication by χ_{Δ}. Clearly, P_{Δ}Q_{φ}^{+}P_{Δ} is an integral operator with kernel

function k_{Δ} =^{def} χ_{Δ×Δ}k, (recall that χ_{A} is the characteristic function of a set A) and P_{Δ}Q_{φ}^{+}P_{Δ} ∈ S_{1}. It follows from Theorem 6.3 that the function u

u(a) =^{def} ∫_{ℝ} k_{Δ}(x, x + a) dx

is a.e. equal to a continuous function ℝ. Clearly, u(a) = 0 if a > 0. And

u(a) → ∫_{Δ} φ(x) dx

when a ↑ 0. Hence,

∫_{Δ} φ(x) dx = 0, for any interval Δ ⊂ ℝ_{+}.

Consequently, φ = 0. ■

7. Schur multipliers of the form ψ(max{x, y}), x, y ∈ ℝ_{+}

Let 0 < p ≤ 2. Recall that a function ω on ℝ^{2} is called a Schur multiplier of S_{p} if the integral operator on L^{2}(ℝ) with kernel function ωk belongs to S_{p} whenever the integral operator with kernel function k does. If 2 < p < ∞, the class of Schur multipliers of S_{p} can be defined by duality: ω is a Schur multiplier of S_{p} if ω is a Schur multiplier of S_{p′}, where p′ =^{def} p/(p − 1). We say that ω is a Schur multiplier of weak type (p, p), 0 < p ≤ 2, if the integral operator with kernel function ωk belongs to S_{p,∞} whenever the integral operator with kernel function k belongs to S_{p}. Note that in a similar way one can define Schur multipliers for an arbitrary measure space (X , μ).

In this section for a function ψ ∈ L^{∞}(ℝ) we find a sufficient condition for the function (x, y) ↦ ψ(max{x, y}), (x, y) ∈ ℝ^{2}, to be a Schur multiplier of S_{p}. We also obtain a sufficient condition for this function to be a Schur multiplier of weak type (1/2, 1/2).

Theorem 7.1. Let 1 < p < ∞ and let ψ ∈ L^{∞}(ℝ). Then the function

(x, y) ↦ ψ(max{x, y}), (x, y) ∈ ℝ^{2},

is a Schur multiplier of S_{p}.

Proof. Since the triangular projection P is bounded on S_{p}, the characteristic function of the set {(x, y) : x > y} is a Schur multiplier of S_{p}. It remains to observe that

ψ(max{x, y}) = ψ(x)χ_{{(x,y):x>y}} + ψ(y)χ_{{(x,y):x<y}}. ■  (7.1)

Theorem 7.2. Let ψ ∈ L^{∞}(ℝ). Then the function

(x, y) ↦ ψ(max{x, y}), (x, y) ∈ ℝ^{2},

is a Schur multiplier of weak type (1, 1).

Proof. The result follows from (7.1) and the fact that the triangular projection P has weak type (1, 1) (see (2.9)). ■

Theorem 7.3. Let 1/2 < p < ∞ and let ψ be a function of bounded variation. Then the function (x, y) ↦ ψ(max{x, y}) on ℝ^{2} is a Schur multiplier of S_{p}.

Proof. By Theorem 7.1 we may assume that p ≤ 1.

We consider first the case when ψ is absolutely continuous, i.e.,

ψ(x) = ∫_{x}^{∞}h(t) dt + C, h ∈ L^{1}(ℝ).  (7.2)

We may assume that C = 0.

Let ξ and η be functions in L^{2} and let T be the integral operator defined by

(T f)(x) = ∫_{ℝ} ξ(x)η(y)ψ(max{x, y})f(y) dy, f ∈ L^{2}(ℝ).  (7.3)

We have to prove that

∥T ∥_{S_{p}} ≤ C(p, h)∥ξ∥_{L^{2}(ℝ)}∥η∥_{L^{2}(ℝ)},

where C(p, h) may depend only on p and h.

We can factorize the function h in the form h = uv, where u, v ∈ L^{2}(ℝ). Put

k_{1}(x, y) =^{def} \{ \begin{matrix}0, & y < x, \\ ξ(x)u(y), & y > x, \end{matrix}

and

k_{2}(x, y) =^{def} \{ \begin{matrix}η(y)v(x), & y < x, \\ 0, & y > x. \end{matrix}

Let T_{1} and T_{2} be the integral operators on L^{2}(ℝ) with kernel functions k_{1} and k_{2}. It follow from the boundedness of the triangular projection that if 1 < q < ∞, then ∥T_{1}∥_{S_{q}} ≤ C(q)∥ξ∥_{L^{2}}∥u∥_{L^{2}} and ∥T_{2}∥_{S_{q}} ≤ C(q)∥η∥_{L^{2}}∥v∥_{L^{2}}, where C(q) may depend only on q. It is also easy to verify that T = T_{1}T_{2}. It follows that

∥T ∥_{S_{p}} ≤ ∥T_{1}∥_{S_{2p}}∥T_{2}∥_{S_{2p}} ≤ (C(2p))^{2}∥ξ∥_{L^{2}}∥η∥_{L^{2}}∥u∥_{L^{2}}∥v∥_{L^{2}}.

To reduce the general case to the case of an absolutely continuous function ψ, we can consider a standard regularization process. ■

We complete this section with the following result.

Theorem 7.4. Suppose that ψ is a function of bounded variation. Then the function (x, y) ↦ ψ(max{x, y}) is a Schur multiplier of weak type (1/2, 1/2).

We need two lemmata.

Lemma 7.5. Let 0 < p < 1 and let A ∈ S_{p,∞}. Set

∥A∥_{S_{p,∞}}^{∗} =^{def} sup_{t>0} (t^{p−1} ∑_{n≥0} min{t, s_{n}(A)})^{1/p} .

Then

∥A∥_{S_{p,∞}} ≤ ∥A∥_{S_{p,∞}}^{∗} ≤ (1 − p)^{−1/p}∥A∥_{S_{p,∞}}.

Proof. Taking t = s_{n}(A) in the definition of ∥ · ∥_{S_{p,∞}}^{∗}, we obtain

(n + 1)^{\frac{1}{p}} s_{n}(A) ≤ ∥A∥_{S_{p,∞}}^{∗}.

Consequently, ∥A∥_{S_{p,∞}} ≤ ∥A∥_{S_{p,∞}}^{∗}. Next, we have

t^{p−1} ∑_{n≥0} min\{t, s_{n}(A)\} ≤ t^{p−1} ∑_{n≥0} min\{t, ∥A∥_{S_{p,∞}}(n + 1)^{− \frac{1}{p}} \}

≤ t^{p−1} ∫_{0}^{∞}min{t, ∥A∥_{S_{p,∞}}x^{−1/p}} dx = (1 − p)^{−1}∥A∥_{S_{p,∞}}^{p}.

Hence, ∥A∥_{S_{p,∞}}^{∗} ≤ (1 − p)^{−1/p}∥A∥_{S_{p,∞}}. ■

Lemma 7.6. If 0 < p < 1, then ∥ · ∥_{S_{p,∞}}^{∗} is a p-norm, i.e.,

∥A_{1} + A_{2}∥_{S_{p,∞}}^{∗p} ≤ ∥A_{1}∥_{S_{p,∞}}^{∗p} + ∥A_{2}∥_{S_{p,∞}}^{∗p}, A_{1}, A_{2} ∈ S_{p,∞}.

Proof. By Rotfeld’s theorem [R], if Φ is a concave nondecreasing function on ℝ_{+} such that Φ(0^{+}) = 0, then

∑_{j=0}^{m} Φ(s_{j}(A_{1} + A_{2})) ≤ ∑_{j=0}^{m} Φ(s_{j}(A_{1})) + ∑_{j=0}^{m} Φ(s_{j}(A_{2})), m ∈ ℤ_{+}.  (7.4)

For t > 0 we define the function Φ_{t} on ℝ_{+} by

Φ_{t}(x) = t^{p−1} min{t, x}.

Clearly,

∥A∥_{S_{p,∞}}^{∗p} = sup_{t>0} ∑_{j≥0} Φ_{t}(s_{j}(A)).

It remains to apply 7.4 for Φ_{t} and take the supremum over t > 0. ■

Note that the fact that for p < 1 the space L^{p,∞} has a p-norm that is equivalent to the initial quasi-norm is well known (see [K]).

Proof of Theorem 7.4. The proof is similar to the proof of Theorem 7.3. Again, it is sufficient to assume that ψ has the form (7.2) with C = 0.

By Lemma 7.5 and Lemma 7.6, to prove that our function is a Schur multiplier of weak type (1/2, 1/2), it is sufficient to prove that if T is defined by (7.3), then T ∈ S_{1/2,∞}. Let u, v, k_{1}, k_{2}, T_{1}, and T_{2} be as in the proof of Theorem 7.3. Let n ≥ 2 and n = m_{1} + m_{2}, where |m_{1} − n/2| ≤ 1/2 and |m_{2} − n/2| ≤ 1/2. Since the triangular projection P has weak type (1, 1) (see (2.9)), we have

∥T_{1}∥_{S_{1,∞}} ≤ const ∥ξ∥_{L^{2}}∥u∥_{L^{2}} and ∥T_{2}∥_{S_{1,∞}} ≤ const ∥η∥_{L^{2}}∥v∥_{L^{2}}.

Hence, by (2.2),

s_{n}(T) ≤ s_{m_{1}}(T_{1})s_{m_{2}}(T_{2}) ≤ const \frac{1}{m_{1}m_{2}} ∥ξ∥_{L^{2}}∥u∥_{L^{2}}∥η∥_{L^{2}}∥v∥_{L^{2}}

≤ const \frac{1}{n^{2}} ∥ξ∥_{L^{2}}∥u∥_{L^{2}}∥η∥_{L^{2}}∥v∥_{L^{2}}

which completes the proof. ■

We let 𝔐_{p} denote the space of Schur multipliers of S_{p}, and put

∥ω∥_{𝔐_{p}} =^{def} sup ∥ωk∥_{S_{p}},

where the supremum is taken over all integral operators with kernel k ∈ L^{2} such that ∥k∥_{S_{p}} = 1. Here by ∥k∥_{S_{p}} we mean the S_{p} norm (quasi-norm if p < 1) of the integral operator with kernel function k. If ω is a Schur multiplier of weak type (p, p), we put

∥ω∥_{𝔐_{p,w}} =^{def} sup ∥ωk∥_{S_{p,∞}},

where the supremum is taken over all integral operators with kernel k ∈ L^{2} such that ∥k∥_{S_{p}} = 1.

Remark. It is clear from the proofs of Theorems 7.1, 7.2, 7.3, and 7.4 that under the hypotheses of Theorem 7.1 we have

∥ψ(max{x, y})∥_{𝔐_{p}} ≤ C(p)∥ψ∥_{L^{∞}}

and

∥ψ(max{x, y})∥_{𝔐_{1,w}} ≤ C∥ψ∥_{L^{∞}}

while under the hypotheses of Theorem 7.3 we have

∥ψ(max{x, y})∥_{𝔐_{p}} ≤ C(p)∥ψ∥_{BV} ,

and

∥ψ(max{x, y})∥_{𝔐_{1/2,w}} ≤ C∥ψ∥_{BV} .

Here we use the notation

∥φ∥′_{BV} = ∫_{ℝ} |dφ| and ∥φ∥_{BV} = ∥φ∥′_{BV} + ∥φ∥_{L^{∞}}.

The following result gives us a more accurate estimate for ∥ψ(max{x, y})∥_{𝔐_{p}} in the case 1/2 < p ≤ 1.

Theorem 7.7. Let ψ be a function of bounded variation on ℝ. Then

∥ψ(max\{x, y\})∥_{𝔐_{1}} ≤ const ∥ψ∥_{L^{∞}} log (2 + \frac{∥ψ∥′_{BV}}{∥ψ∥_{L^{∞}}} )

and

∥ψ(max\{x, y\})∥_{𝔐_{p}} ≤ const ∥ψ∥_{L^{∞}}^{2−1/p}∥ψ∥_{BV}^{1/p−1}, \frac{1}{2} < p < 1.

Proof. Let ξ and η be function in L^{2} such that ∥ξ∥_{L^{2}} = ∥η∥_{L^{2}} = 1. We have to estimate the S_{p}-norm of the integral operator with kernel ψ(max{x, y})ξ(x)η(y). Let {s_{n}}_{n≥0} be the sequence of s-numbers of this integral operator. Theorem 7.2 implies that

s_{n} ≤ const \frac{∥ψ∥_{L^{∞}}}{n + 1} .

Theorem 7.4 implies that

s_{n} ≤ const \frac{∥ψ∥_{L^{∞}} + ∥ψ∥′_{BV}}{(n + 1)^{2}} .

Consequently,

s_{n} ≤ const min \{\frac{∥ψ∥_{L^{∞}}}{n + 1} , \frac{∥ψ∥_{L^{∞}} + ∥ψ∥′_{BV}}{(n + 1)^{2}} \} .

The rest of the proof is an easy exercise. ■

8. The case p = 1/2

Theorem 8.1. Let φ be a function of bounded variation on [0, 1]. Then Q_{φ}^{[0,1]} ∈ S_{1/2, ∞}.

Proof. We may extend the function φ by putting φ(t) = 0 for t ∈ ℝ_{+} \ [0, 1]. Clearly, the integral operator with kernel function χ_{[0,1]^{2}} has rank one, and so it belongs to S_{1/2}. Consequently, by Theorem 7.4, the integral operator with kernel function χ_{[0,1]^{2}}φ(max{x, y}) belongs to S_{1/2, ∞}. ■

To see that this result cannot be improved to Q_{φ}^{[0,1]} ∈ S_{1/2}, we begin with two extensions of Theorem 6.6.

Lemma 8.2. Let φ, ψ ∈ L_{loc}^{1}(ℝ). Suppose that the integral operator with kernel function k,

k(x, y) = \begin{cases}φ(x)ψ(y), & y ≤ x, \\ 0, & otherwise, \end{cases}

belongs to S_{1}. Then φψ = 0 almost everywhere.

Proof. First we assume that φ, ψ ∈ L^{2}(ℝ). By Theorem 6.3 we have

lim_{a→0−} ∫_{ℝ} k(x, x + a)dx = lim_{a→0+} ∫_{ℝ} k(x, x + a)dx = 0,

whence ∫_{ℝ} φ(x)ψ(x) dx = 0. Now let φ and ψ be arbitrary functions in L_{loc}^{1}(ℝ). Sup-

pose that f and g are functions in L^{∞}(ℝ) such that fφ ∈ L^{2}(ℝ) and gψ ∈ L^{2}(ℝ). Consider the integral operator with kernel function

(x, y) ↦ f(x)k(x, y)g(y).

Clearly, it belongs to S_{1}. It follows from what we have just proved that ∫_{ℝ} φ(x)f(x)ψ(x)g(x)dx = 0. Since f and g are arbitrary, this implies the result. ■

Lemma 8.3. Let φ ∈ L_{loc}^{1}(ℝ_{+}) and let A = (0, ∞)×Δ, where Δ is a measurable subset of (0, ∞). Suppose that P_{A}Q_{φ}^{+} ∈ S_{1}. Then φ = 0 almost everywhere on Δ.

Proof. The result follows easily from Lemma 8.2 with ψ = χ_{Δ}. ■

Lemma 8.4. Let φ be a nonincreasing locally absolutely continuous function on ℝ_{+} and let Δ be a measurable subset of ℝ_{+}. Suppose that the integral operator with kernel function

(x, y) ↦ φ(max{x, y})χ_{Δ}(x)χ_{Δ}(y)

belongs to S_{1/2}. Then φ′ = 0 almost everywhere on Δ.

Proof. By replacing Δ with Δ ∩ (a, b), we may assume that Δ ⊂ [a, b] where 0 < a < b < ∞. We may then subtract φ(b) and modify φ outside [a, b] so that φ becomes constant on (0, a] and zero on [b, ∞). Let ψ = (−φ′)^{1/2}. Since xφ(x) is bounded, we have ψ ∈ X_{∞}. Thus Q_{ψ}^{+} is bounded and by Theorem 4.2, Q_{φ} admits a factorization Q_{φ} = (Q_{ψ}^{+})^{∗}Q_{ψ}^{+}. Let M be multiplication by χ_{Δ}. It follows that MQ_{φ}M = (Q_{ψ}^{+}M)^{∗}(Q_{ψ}^{+}M), and so Q_{ψ}^{+}M ∈ S_{1}. The result follows now from Lemma 8.3. ■

Corollary 8.5. Let φ be as in Lemma 8.4 and let ψ be a function on ℝ_{+} such that Q_{ψ} ∈ S_{1/2}. Set Δ =^{def} {x ∈ ℝ_{+} : ψ(x) = φ(x)}. Suppose that ψ is differentiable almost everywhere on Δ. Then φ′ = ψ′ = 0 almost everywhere on Δ.

Proof. It suffices to apply Lemma 8.4. ■

Theorem 8.6. Let ψ be an absolutely continuous function on [0, 1]. Suppose Q_{ψ}^{[0,1]} ∈ S_{1/2}. Then ψ is constant.

Proof. Clearly, we may assume that ψ is a real function. Suppose that ψ is not constant. Then max ψ > ψ(1) or min ψ < ψ(1). To be definite, suppose that max ψ > ψ(1). We use the “sun rising method”. Let

Δ =^{def} {x ∈ [0, 1] : ψ(x) ≥ ψ(t) for all t ≥ x}.

Clearly, Δ is closed and 1 ∈ Δ. Moreover, the restriction ψ|[α, β] is constant for any interval (α, β) such that α, β ∈ Δ and Δ ∩ (α, β) = ∅. Set

φ(x) = max_{[x,1]∩Δ} ψ.

Clearly, φ is non-increasing, φ|_{Δ} = ψ|_{Δ} and φ|[α, β] is constant for any interval (α, β) such that α, β ∈ Δ and Δ ∩ (α, β) = ∅. Consequently, φ is absolutely continuous. Thus, φ′ = 0 almost everywhere on Δ by Corollary 8.5. Moreover, φ′ = 0 outside Δ because φ is locally constant outside Δ. Consequently, φ′ = 0 almost everywhere on [0, 1], and so φ(t) = φ(1) = ψ(1) for any t ∈ [0, 1] which contradicts the condition max ψ > ψ(1). ■

Corollary 8.7. Suppose that Q_{ψ} ∈ S_{1/2}. Then ψ is constant on any interval I ⊂ ℝ_{+} on which ψ is absolutely continuous. ■

Corollary 8.8. Suppose that ψ is locally absolutely continuous and Q_{ψ} ∈ S_{1/2}. Then ψ = 0 everywhere on ℝ_{+}. ■

Lemma 8.9. Let φ be a nonincreasing function on ℝ_{+} with lim_{t→+∞} φ(t) = 0 and let ε > 0. Then there exists a nonincreasing absolutely continuous function ψ on ℝ_{+} such that lim_{t→+∞} ψ(t) = 0 and m{φ ≠ ψ} < ε.

Note that here and in what follows m denotes Lebesgue measure on ℝ or normalized Lebesgue measure on the unit circle 𝕋.

Proof. We may assume that φ is right-continuous on (0, ∞) and that φ is bounded. Consider the positive measure μ on ℝ_{+} such that φ(t) = μ(t, ∞) for any t > 0. Denote by μ_{s} the singular part of μ. There exists a Borel set E ⊂ ℝ_{+} such that m(E) = 0 and μ_{s}((0, +∞) \ E) = 0. We may find an open set U such that E ⊂ U ⊂ ℝ_{+} and μ(U) < ε. Let U = ⋃_{n≥1}(a_{n}, b_{n}), where (a_{n}, b_{n}) are mutually

disjoint. Set

f(t) = \begin{cases}−φ′(t), & t ∈ ℝ_{+} \ E, \\ \frac{μ[a_{n}, b_{n})}{b_{n} − a_{n}}, & t ∈ (a_{n}, b_{n}). \end{cases}

Set ψ(t) =^{def} ∫_{t}^{∞}f(s)ds. Clearly, φ = ψ outside U. ■

Lemma 8.10. Let φ a nonincreasing function on ℝ_{+} with lim_{t→+∞} φ(t) = 0. Suppose that the integral operator with kernel function

(x, y) ↦ φ(max{x, y})χ_{Δ}(x)χ_{Δ}(y)

belongs to S_{1/2} for a measurable subset Δ of ℝ_{+}. Then φ′ = 0 almost everywhere on Δ.

Proof. The result follows from Lemmas 8.4 and 8.9. ■

Theorem 8.11. Let ψ be a function with bounded variation on [0, 1]. Suppose that Q_{ψ}^{[0,1]} ∈ S_{1/2}. Then ψ′ = 0 almost everywhere on [0, 1].

Proof. Again, we may assume that ψ is real. We may also make the assumption that that ψ is continuous at 0 and at 1, and ψ(t) = max{ψ(t^{−}), ψ(t^{+})} for any t ∈ (0, 1). With any nondegenerate closed interval I ⊂ [0, 1] we associate the function φ_{I} : I → ℝ defined by φ_{I}(x) = sup{ψ(t) : t ∈ I and t ≥ x}. Set

Δ(I) =^{def} {x ∈ I : φ_{I}(x) = ψ(x)}.

Clearly, Δ(I) is closed. By Lemma 8.10, ψ′ = φ′_{I} = 0 almost everywhere on Δ(I).

Set E_{−} = {x ∈ (0, 1) : ψ′(x) < 0}. Let a ∈ E_{−}. Clearly, a ∈ Δ(I) if I is small enough and a ∈ I. Consequently,

E_{−} ⊂ ⋃_{n=2}^{∞} (⋃_{k=1}^{n−1} Δ ([\frac{k − 1}{n}, \frac{k + 1}{n}])) .

We have shown that Lemma 8.10 implies m(Δ(I) ∩ E_{−}) = 0 for every I. Consequently, m(E_{−}) = 0. Thus, we have proved that ψ′ ≥ 0 almost everywhere. It remains to apply this result to −ψ. ■

The following fact is an immediate consequence of Theorem 8.11.

Corollary 8.12. Suppose that Q_{ψ} ∈ S_{1/2}. Then ψ′ = 0 almost everywhere on any interval I ⊂ ℝ_{+} on which ψ is of bounded variation. ■

9. Sturm–Liouville theory and p = 1/2

If φ is real, then Q_{φ} is self-adjoint, so its singular values are the absolute values of its eigenvalues. Hence, we next study the eigenvalues and eigenfunctions. For simplicity we consider only the case of symbols φ which vanish on (1, ∞); thus it does not matter whether we consider Q_{φ} on L^{2}(ℝ_{+}) or Q_{φ}^{[0,1]} on L^{2}[0, 1].

Suppose that φ ∈ C^{1}[0, 1] and that φ is real. Let λ be a non-zero eigenvalue of Q_{φ}^{[0,1]} and g ∈ L^{2}[0, 1] a corresponding eigenfunction,

λg(x) = φ(x) ∫_{0}^{x} g(y) dy + ∫_{x}^{1} φ(y)g(y) dy,  0 ≤ x ≤ 1.  (9.1)

The right hand side is a continuous function of x; hence, g ∈ C[0, 1] and (9.1) holds for every x (and not just a.e.). By (9.1) again, g ∈ C^{1}[0, 1] with

λg′(x) = φ′(x) ∫_{0}^{x} g(y) dy.  (9.2)

Define G(x) = ∫_{0}^{x} g(y) dy. Then (9.2) can be written as the system

\begin{aligned}G′(x) & = g(x) \\ g′(x) & = λ^{−1}φ′(x)G(x) \end{aligned}  (9.3)

and we have, using (9.1) with x = 1, the boundary conditions

\begin{aligned}G(0) & = 0, \\ g(1) & = λ^{−1}φ(1)G(1). \end{aligned}  (9.4)

Conversely, any solution of (9.3) with the boundary conditions (9.4) satisfies (9.2) and (9.1), so the problem of finding the singular values of Q_{φ} reduces to finding the λ ≠ 0 for which (9.3) and (9.4) have a solution. Note that (9.3) can be written as a Sturm–Liouville problem

λG′′(x) = φ′(x)G(x).  (9.5)

If g(x_{0}) = G(x_{0}) = 0 for some x_{0} ∈ [0, 1], then (9.3) shows, by the standard uniqueness theorem, that g vanishes identically, a contradiction. In particular, since G(0) = 0, we have g(0) ≠ 0, and we may normalize the eigenfunction g by g(0) = 1.

For every λ ≠ 0, (9.3) has a unique solution (G_{λ}, g_{λ}) with G_{λ}(0) = 0, g_{λ}(0) = 1. It thus follows that all non-zero eigenvalues of Q_{φ} are simple, and that λ ≠ 0 is an eigenvalue if and only if

g_{λ}(1) = λ^{−1}φ(1)G_{λ}(1).  (9.6)

Example. Let φ(x) = 1 − x, x ∈ [0, 1], and φ(x) = 0, x > 1. Then (9.3) gives G′′(x) = −λ^{−1}G(x), and we find the solutions g_{λ}(x) = cos λ^{−1/2}x, λ > 0, and g_{λ}(x) = cosh |λ|^{−1/2}x, λ < 0.

Since φ(1) = 0, condition (9.6) is simply g_{λ}(1) = 0, and the non-zero eigenvalues are given by cos λ^{−1/2} = 0 or λ^{−1/2} = (n+ \frac{1}{2})π, n = 0, 1, . . .. (λ < 0 is impossible in this case; in other words, Q_{φ} is a positive operator, as is also seen by Theorem 4.1.) Hence, the non-zero eigenvalues are \{(n + \frac{1}{2})^{−2}π^{−2}\}_{n=0}^{∞} and the singular values are s_{n} = π^{−2}(n + \frac{1}{2})^{−2}, n ≥ 0.

The behaviour s_{n}(Q_{φ}) ≍ (n + 1)^{−2} found in the above example holds for all smooth φ on [0,1] by Sturm–Liouville theory, as will be seen in Theorem 9.3. Hence, for smooth φ with compact support, we have Q_{φ} ∈ S_{1/2,∞} but nothing better.

Let BV [0, 1] denote the Banach space of functions on [0,1] with bounded variation, with the seminorm ∥φ∥′_{BV} = ∫_{0}^{1} |dφ| and the norm ∥φ∥_{BV} = ∥φ∥′_{BV} +sup |φ|.

The following result is essentially the same as Theorem 8.1. However, we use in this section a different approach based on the study of eigenvalues of Sturm– Liouville operators.

Theorem 9.1. If φ ∈ BV [0, 1], and φ = 0 on (1, ∞), then Q_{φ} ∈ S_{1/2,∞} and ∥Q_{φ}∥_{S_{1/2,∞}} ≤ C∥φ∥_{BV}. More precisely,

s_{n}(Q_{φ}) ≤ C_{1}∥φ∥_{BV} (n + 1)^{−2},  n ≥ 0,  (9.7)

and

s_{n}(Q_{φ}) ≤ C_{2}∥φ∥′_{BV} n^{−2},  n ≥ 1.  (9.8)

Proof. Note that (9.8) follows from (9.7) since a symbol φ constant on [0,1] yields a rank one operator Q_{φ}.

We use methods from Sturm–Liouville theory, and begin by making some simplifications.

(i) Replacing φ by a regularization φ_{ε} such that φ_{ε} → φ in L^{2}[0, 1] and thus Q_{φ_{ε}} → Q_{φ} in S_{2} as ε → 0, we see that we may assume φ ∈ C^{1}[0, 1].

(ii) Considering real and imaginary parts separately, we may assume that φ is

real, and thus Q_{φ} self-adjoint. (iii) Subtracting a constant times χ_{[0,1]}, which yields a rank 1 operator, we may

assume that φ(1) = 0. (iv) By homogeneity, we may assume that ∥φ∥′_{BV} = ∫_{0}^{1} |φ′| ≤ 1 and show that

then ∥Q_{φ}∥_{S_{1/2,∞}} ≤ C.

(v) Using

φ(x) = − ∫_{x}^{1} φ′(y)dy

= − ∫_{x}^{1} max{φ′(y), 0}dy − ∫_{x}^{1} min{φ′(y), 0}dy
= φ_{1}(x) − φ_{2}(x),

and the corresponding decomposition Q_{φ} = Q_{φ_{1}} −Q_{φ_{2}}, we may also assume that φ′ ≤ 0 and thus φ ≥ 0. By Theorem 4.1, Q_{φ} is then a positive operator.

(vi) Similarly, writing φ = 2φ_{1} − φ_{2} with φ_{2} = 1 − x and φ_{1} = (φ + φ_{2})/2, we

may further assume that φ′ ≤ −1/2 on [0,1]. Let λ > 0 and let, as above, (G_{λ}, g_{λ}) be the solution to (9.3) with g_{λ}(0) = 1,

G_{λ}(0) = 0. Thus λ is an eigenvalue if and only if (9.6) holds, i.e., by (iii), if and only if g_{λ}(1) = 0.

Write ω = λ^{−1/2} and express (g_{λ}, ωG_{λ}) in polar coordinates

g_{λ}(x) = R_{ω}(x) cos Θ_{ω}(x),
ωG_{λ}(x) = R_{ω}(x) sin Θ_{ω}(x), (9.9)

where R_{ω}(x) = \sqrt{g^{2} + ω^{2}G^{2}} > 0 and Θ_{ω} is continuous with Θ_{ω}(0) = 0. Note that λ is an eigenvalue if and only if cos Θ_{ω}(1) = 0, i.e. Θ_{ω}(1) = nπ + π/2 for some integer n.

Since R_{ω}(x) > 0, R_{ω} and Θ_{ω} belong to C^{1}[0, 1], and (9.9) and (9.3) yield

λ^{1/2}R_{ω}^{2}Θ′_{ω} = gG′ − Gg′ = g^{2} − λ^{−1}φ′G^{2} = R_{ω}^{2}(cos^{2} Θ_{ω} − φ′ sin^{2} Θ_{ω})

and thus

Θ′_{ω} = ω(cos^{2} Θ_{ω} − φ′ sin^{2} Θ_{ω}).  (9.10)

In particular, since φ′ < 0 by (vi), Θ′_{ω} > 0 and thus Θ_{ω}(1) > 0.

Now suppose 0 < ω < ν and consider the corresponding functions Θ_{ω} and Θ_{ν}. We claim that

Θ_{ω}(x) < Θ_{ν}(x),  0 < x ≤ 1.  (9.11)

Indeed, since Θ_{ω}(0) = 0 = Θ_{ν}(0) and, by (9.10), Θ′_{ω}(0) = ω < ν = Θ′_{ν}(0), (9.11) holds in (0, δ) for some δ > 0. Hence, if (9.11) fails, there exists some x_{1} ∈ (0, 1] such that Θ_{ω}(x) < Θ_{ν}(x) for 0 < x < x_{1} but Θ_{ω}(x_{1}) = Θ_{ν}(x_{1}). This would imply Θ′_{ω}(x_{1}) ≥ Θ′_{ν}(x_{1}); on the other hand, then

cos^{2} Θ_{ω}(x_{1}) − φ′ sin^{2} Θ_{ω}(x_{1}) = cos^{2} Θ_{ν}(x_{1}) − φ′ sin^{2} Θ_{ν}(x_{1}) > 0,

recalling (vi), and (9.10) would yield Θ′_{ω}(x_{1}) < Θ′_{ν}(x_{1}), a contradiction.

From (9.11) follows in particular that the function ω ↦ Θ_{ω}(1) is strictly increasing. Hence, there is for each integer n ≥ 0 at most one value of ω, ω_{n} say, such that Θ_{ω_{n}}(1) = nπ + π/2, and thus a corresponding eigenvalue λ_{n} = ω_{n}^{−2}. (The solution to (9.4) depends continuously on ω, with Θ_{0}(1) = 0 and Θ_{ω}(1) → ∞ as ω → ∞, so ω_{n} exists for every n ≥ 1, but we do not need that.) Integrating (9.10) we obtain by (iv)

Θ_{ω}(1) = ∫_{0}^{1} Θ′_{ω}(x) dx ≤ ω ∫_{0}^{1}(1 + |φ′(x)|) dx ≤ 2ω

and thus 2ω_{n} ≥ Θ_{ω_{n}}(1) = nπ + π/2, which yields

λ_{n} = ω_{n}^{−2} ≤ 4π^{−2}(n + \frac{1}{2})^{−2}. ■

Considering again functions on the whole half-line ℝ_{+}, we now can prove an endpoint result corresponding to Theorem 5.1.

Theorem 9.2. If φ ∈ Y_{1/2}, then Q_{φ} ∈ S_{1/2,∞}.

Proof. Define A_{n}^{(k)} and A^{(k)} as in (3.3) and (3.4), but now for all integers k. For k ≥ 1, (3.7) holds for every p, and thus φ ∈ Y_{1/2} ⊂ X_{1/2} implies that

∑_{k≥1} ∥P_{A^{(k)}}Q_{φ}∥_{S_{1/2}}^{1/2} ≤ ∑_{k≥1} const 2^{−pk/2} < ∞.

Moreover, by symmetry, ∥P_{A^{(−k)}}Q_{φ}∥_{S_{1/2}} = ∥P_{A^{(k)}}Q_{φ}∥_{S_{1/2}} so

∑_{k≤−1} ∥P_{A^{(k)}}Q_{φ}∥_{S_{1/2}}^{1/2} < ∞

too. It follows that

Q_{φ} − P_{A^{(0)}}Q_{φ} = ∑_{k≠0} P_{A^{(k)}}Q_{φ} ∈ S_{1/2}.  (9.12)

Next, P_{A^{(0)}}Q_{φ} is the direct sum of P_{A_{n}^{(0)}}Q_{φ}, n ∈ ℤ, which act in the orthogonal spaces L^{2}[2^{n}, 2^{n+1}]. By translation invariance, Lemma 2.1 and Theorem 9.1, with φ_{n}(x) = 2^{n}φ(2^{n}x + 2^{n}),

\begin{aligned}∥P_{A_{n}^{(0)}}Q_{φ}∥_{S_{1/2,∞}} & = ∥Q_{φ}^{[2^{n},2^{n+1}]}∥_{S_{1/2,∞}} = ∥Q_{φ_{n}}^{[0,1]}∥_{S_{1/2,∞}} ≤ C∥φ_{n}∥_{BV [0,1]} \\ & ≤ C′2^{n} ∫_{2^{n}}^{∞} |dφ|. \end{aligned}

By Theorem 5.3, we thus have

∑_{n∈ℤ}(∥P_{A_{n}^{(0)}}Q_{φ}∥_{S_{1/2,∞}})^{1/2} < ∞

and it follows from Lemma 7.6 (or as in the proof of Lemma 9.5 below) that P_{A^{(0)}}Q_{φ} ∈ S_{1/2,∞}. By (9.12), Q_{φ} ∈ S_{1/2,∞} too. ■

Theorem 9.2 is the best possible; for any reasonably smooth φ, the singular numbers s_{n}(Q_{φ}) decrease like n^{−2} but not faster. More precisely, we have the following very precise result. Recall that a function in Y_{1/2} has locally bounded variation and thus is a.e. differentiable.

Theorem 9.3. Let φ ∈ Y_{1/2}. Then

n^{2}s_{n}(Q_{φ}) → π^{−2}∥φ′∥_{L^{1/2}} = π^{−2} (∫_{0}^{∞} |φ′(x)|^{1/2}dx)^{2} < ∞ as n → ∞.(9.13)

Equivalently,

ε^{1/2}|{n : s_{n}(Q_{φ}) > ε}| → π^{−1} ∫_{0}^{∞} |φ′(x)|^{1/2}dx < ∞ as ε → 0. (9.14)

In particular, n^{2}s_{n}(Q_{φ}) → 0 as n → ∞ if and only if φ′ = 0 a.e.

Proof. Note first that by the Cauchy–Schwarz inequality and (5.1),

∫_{0}^{∞} |φ′(x)|^{1/2}dx ≤ ∑_{n∈ℤ}2^{n/2} (∫_{2^{n}}^{2^{n+1}} |φ′(x)| dx)^{1/2} ≤ ∥φ∥_{Y_{1/2}}^{1/2} < ∞. (9.15)

For smooth and positive symbols on a finite interval, (9.13) follows by standard Sturm–Liouville theory, see [LS, §1.2 with the transformation in §1.1]. Indeed, much more refined asymptotics of s_{n} can be given [LS, Chapter 5].

We present here another proof that applies in the general case. We prove a sequence of lemmas. The first implies that (9.13) and (9.14) are equivalent.

Lemma 9.4. For any bounded operator T on a Hilbert space,

lim sup_{ε→0} ε^{1/2}|{n : s_{n} > ε}| = (lim sup_{n→∞}(n^{2}s_{n}))^{1/2}

and similarly with lim inf instead of lim sup on both sides.

Proof. If lim sup ε^{1/2}|{n : s_{n} > ε}| < a for some a > 0, then for all small ε, |{n : s_{n} > ε}| < aε^{−1/2}. Taking ε = a^{2}(n + 1)^{−2}, we see that for large n, s_{n} ≤ ε, and thus (n + 1)^{2}s_{n} ≤ a^{2}, so lim sup_{n→∞} n^{2}s_{n} ≤ a^{2}. The converse is similar, and the second part follows similarly by reversing the inequalities. ■

Lemma 9.5. If T_{1}, …, T_{N} are bounded operators on Hilbert spaces H_{1}, …, H_{N}, then

(lim sup_{n→∞} n^{2}s_{n}(T_{1} ⊕ ⋯ ⊕ T_{N}))^{1/2} ≤ ∑_{k=1}^{N}(lim sup_{n→∞} n^{2}s_{n}(T_{k}))^{1/2}

and

(lim inf_{n→∞} n^{2}s_{n}(T_{1} ⊕ ⋯ ⊕ T_{N}))^{1/2} ≥ ∑_{k=1}^{N}(lim inf_{n→∞} n^{2}s_{n}(T_{k}))^{1/2}.

Proof. The singular numbers s_{n}(T_{1} ⊕ ⋯ ⊕ T_{N}) consist of all s_{i}(T_{j}), rearranged into a single nonincreasing sequence. Hence,

|{n : s_{n}(T_{1} ⊕ ⋯ ⊕ T_{N}) > ε}| = ∑_{j=1}^{N} |{n : s_{n}(T_{j}) > ε}|

and the result follows by Lemma 9.4. ■

For arbitrary sums we have the following estimate.

Lemma 9.6. If T and U are bounded operators in a Hilbert space, and 0 < δ < 1, then

lim sup_{n→∞} n^{2}s_{n}(T + U) ≤ (1 − δ)^{−2} lim sup_{n→∞} n^{2}s_{n}(T) + δ^{−2} lim sup_{n→∞} n^{2}s_{n}(U),(9.16)
lim inf_{n→∞} n^{2}s_{n}(T + U) ≥ (1 − δ)^{2} lim inf_{n→∞} n^{2}s_{n}(T) − δ^{−2} lim sup_{n→∞} n^{2}s_{n}(U). (9.17)

Proof. By (2.1), s_{n}(T +U) ≤ s_{[(1−δ)n]}(T)+s_{[δn]}(U), and (9.16) follows, together with

lim inf_{n→∞} n^{2}s_{n}(T + U) ≤ (1 − δ)^{−2} lim inf_{n→∞} n^{2}s_{n}(T) + δ^{−2} lim sup_{n→∞} n^{2}s_{n}(U).(9.18)

Replacing here T by T + U and U by −U, we obtain (9.17) by rearrangement. ■ Letting δ → 0 in (9.17) and (9.18), we obtain the following result by Fan [GK1].

Lemma 9.7. If T and U are bounded operators in a Hilbert space, lim_{n→∞} n^{2}s_{n}(T)

exists and n^{2}s_{n}(U) → 0 as n → ∞, then lim_{n→∞} n^{2}s_{n}(T + U) = lim_{n→∞} n^{2}s_{n}(T). ■

Lemma 9.8. The set of φ ∈ Y_{1/2} such that (9.13) holds is a closed set.

Proof. Suppose that φ_{k} → φ in Y_{1/2} and that (9.13) holds for each φ_{k}. By Lemma 9.6 and Theorem 9.2, for every k and 0 < δ < 1,

lim sup_{n→∞} n^{2}s_{n}(Q_{φ}) ≤ (1 − δ)^{−2} lim sup_{n→∞} n^{2}s_{n}(Q_{φ_{k}}) + δ^{−2} lim sup_{n→∞} n^{2}s_{n}(Q_{φ−φ_{k}})

≤ (1 − δ)^{−2}π^{−2}∥φ′_{k}∥_{L^{1/2}} + Cδ^{−2}∥φ − φ_{k}∥_{Y_{1/2}} (9.19)

and similarly

lim inf_{n→∞} n^{2}s_{n}(Q_{φ}) ≥ (1 − δ)^{2}π^{−2}∥φ′_{k}∥_{L^{1/2}} − Cδ^{−2}∥φ − φ_{k}∥_{Y_{1/2}}.  (9.20)

Moreover, by (9.15), ∥(φ − φ_{k})′∥_{L^{1/2}} ≤ ∥φ − φ_{k}∥_{Y_{1/2}} → 0 as k → ∞, and so ∥φ′_{k}∥_{L^{1/2}} → ∥φ′∥_{L^{1/2}}. Letting first k → ∞ and then δ → 0 in (9.19) and (9.20), we obtain (9.13). ■

Lemma 9.9. If φ is linear on a finite interval I, then

n^{2}s_{n}(Q_{φ}^{I}) → π^{−2} (∫_{I} |φ′(x)|^{1/2}dx)^{2} as n → ∞.

Proof. Let φ(x) = α + βx with α, β complex numbers. Suppose first that I = [0, 1]. By the example at the beginning of the section and homogeneity,

s_{n}(Q_{−β+βx}^{I}) = |β|π^{−2}(n + \frac{1}{2})^{−2}

so n^{2}s_{n}(Q_{−β+βx}^{I}) → π^{−2}|β| as n → ∞. Since Q_{α+βx}^{I} − Q_{−β+βx}^{I} = Q_{α+β}^{I} is a rank one operator, Lemma 9.7 (or, more simply, s_{n+1}(Q_{−β+βx}^{I}) ≤ s_{n}(Q_{α+βx}^{I}) ≤ s_{n−1}(Q_{−β+βx}^{I})) yields

n^{2}s_{n}(Q_{φ}^{I}) → π^{−2}|β| = π^{−2} (∫_{I} |φ′(x)|^{1/2}dx)^{2} .  (9.21)

If I = [0, a], we have by Lemma 2.1 and (9.21)

n^{2}s_{n}(Q_{φ}^{I}) = n^{2}s_{n}(Q_{φ_{a}}^{[0,1]}) → π^{−2}a^{2}|β| = π^{−2} (∫_{I} |φ′(x)|^{1/2}dx)^{2} ,

and the general case follows by translation invariance. ■

Completion of the proof of Theorem 9.3.

Step 1. φ is piecewise linear on [0, 1] and φ = 0 on (1, ∞). Let 0 = t_{0} < t_{1} < ⋯ < t_{N} = 1 be such that φ is linear on every I_{i} = [t_{i−1}, t_{i}], i = 1, …, N. Let H_{i} = L^{2}(I_{i}), so L^{2}[0, 1] = H_{1} ⊕ ⋯ ⊕ H_{N}, and let P_{i} : L^{2}[0, 1] → H_{i} denote the orthogonal projection.

Since each P_{i}Q_{φ}P_{j}, i ≠ j, has rank 1, Q_{φ} − ∑_{i=1}^{N} P_{i}Q_{φ}P_{i} has finite rank and by

Lemma 9.7 (or directly), it suffices to consider ∑_{i=1}^{N} P_{i}Q_{φ}P_{i} = Q_{φ}^{I_{1}} ⊕ ⋯ ⊕ Q_{φ}^{I_{N}}. By Lemma 9.9,

n^{2}s_{n}(Q_{φ}^{I_{i}}) → π^{−2} (∫_{t_{i−1}}^{t_{i}} |φ′(x)|^{1/2}dx)^{2} ,

and thus Lemma 9.5 yields

( lim_{n→∞} n^{2}s_{n}(∑_{i=1}^{N} P_{i}Q_{φ}P_{i}))^{1/2} = ∑_{i=1}^{N}( lim_{n→∞} n^{2}s_{n}(Q_{φ}^{I_{i}}))^{1/2} = π^{−1} ∫_{0}^{1} |φ′(x)|^{1/2}dx,

which proves (9.13).

Step 2. φ is absolutely continuous on [0, 1] and φ = 0 on (1, ∞). Approximate φ′ by step functions h_{n} such that ∥φ′ − h_{n}∥_{L^{1}[0,1]} < 1/n, and let

ψ_{n}(x) = \begin{cases}φ(0) + ∫_{0}^{x} h_{n}(y) dy, & x ≤ 1, \\ 0, & x > 1. \end{cases}

Then (9.13) holds for each ψ_{n} by Step 1, and ψ_{n} → φ in BV [0, 1] and thus in Y_{1/2}, see Corollary 5.5, so (9.13) holds by Lemma 9.8.

Step 3. φ has bounded variation on [0, 1], φ = 0 on (1, ∞) and φ is singular, i.e., φ′ = 0 almost everywhere. We may assume that φ is right-continuous. Then φ(x) = φ(0) + ∫_{0}^{x} dμ for some singular complex measure μ supported on [0, 1]. Given any ε > 0, there thus exists a sequence of intervals (I_{i})_{1}^{∞} in [0, 1] such that

∑_{i=1}^{∞} |I_{i}| < ε and |μ| ([0, 1] \ ⋃_{i=1}^{∞} I_{i}) = 0. Let N be a positive integer such that

|μ| ([0, 1] \ ⋃_{i=1}^{N} I_{i}) < ε.

We may assume that each I_{i} is closed, and by combining any two of I_{1}, …, I_{N} that overlap, we may assume that I_{1}, …, I_{N} are disjoint. The complement

[0, 1] \ ⋃_{i=1}^{N} I_{i} is also a finite disjoint union of intervals, say ⋃_{j=1}^{M} J_{j}.

For each interval I, Theorem 9.1 and Lemma 2.1 yield

sup_{n≥1} n^{2}s_{n}(Q_{φ}^{I}) ≤ C|I|∥φ∥′_{BV (I)} ≤ C|I| |μ|(I).  (9.22)

Moreover, as in Step 1 of the proof,

Q_{φ} = Q_{φ}^{I_{1}} ⊕ ⋯ ⊕ Q_{φ}^{I_{N}} ⊕ Q_{φ}^{J_{1}} ⊕ ⋯ ⊕ Q_{φ}^{J_{M}} + R,

where R has finite rank. Hence, by Lemma 9.5, (9.22) and the Cauchy–Schwarz inequality,

(lim sup_{n→∞} n^{2}s_{n}(Q_{φ}))^{1/2} ≤ ∑_{i=1}^{N}(lim sup_{n→∞} n^{2}s_{n}(Q_{φ}^{I_{i}}))^{1/2} + ∑_{j=1}^{M}(lim sup_{n→∞} n^{2}s_{n}(Q_{φ}^{J_{j}}))^{1/2}

≤ C ∑_{i=1}^{N}(|I_{i}| |μ|(I_{i}))^{1/2} + C ∑_{j=1}^{M}(|J_{j}| |μ|(J_{j}))^{1/2}

≤ C(∑_{i=1}^{N} |I_{i}|)^{1/2}(∑_{i=1}^{N} |μ|(I_{i}))^{1/2}

+ C(∑_{j=1}^{M} |J_{j}|)^{1/2}(∑_{j=1}^{M} |μ|(J_{j}))^{1/2}

≤ Cε^{1/2}(|μ|[0, 1])^{1/2} + C · 1 · ε^{1/2}.

The result n^{2}s_{n}(Q_{φ}) → 0 follows by letting ε → 0.

Step 4. φ has bounded variation on (0, a) and φ = 0 on (a, ∞) for some a > 0. By Lemma 2.1, it suffices to consider the case a = 1. We can decompose φ = φ_{a}+φ_{s} on [0, 1], with φ_{a} absolutely continuous and φ_{s} singular; let φ_{a} = φ_{s} = 0 on (1, ∞). By Steps 2 and 3,

n^{2}s_{n}(Q_{φ_{a}}) → π^{−2}∥φ′_{a}∥_{1/2} = π^{−2}∥φ′∥_{1/2}

and n^{2}s_{n}(Q_{φ_{s}}) → 0, and the result follows by Lemma 9.7.

Step 5. φ ∈ Y_{1/2} is arbitrary. Define, for N ≥ 1,

φ_{N}(x) = \begin{cases}φ(1/N) − φ(N), & 0 < x ≤ 2^{−N}, \\ φ(x) − φ(N), & 2^{−N} < x ≤ 2^{N}, \\ 0, & 2^{N} < x. \end{cases}

It is easily seen that each φ_{N} has bounded variation and that ∥φ − φ_{N}∥_{Y_{1/2}} → 0 as N → ∞, cf. (5.1). Thus the result follows by Step 4 and Lemma 9.8. ■

As corollaries, we obtain new proofs of some results from §8.

Corollary 9.10. If I is a finite interval and φ has bounded variation on I, then

n^{2}s_{n}(Q_{φ}^{I}) → π^{−2}(∫_{I} |φ′(x)|^{1/2}dx)^{2} as n → ∞.

Proof. By translation invariance, we may assume I = [0, a]. Then, defining φ = 0 outside I, we have φ ∈ Y_{1/2} by Corollary 5.5, and the result follows by Theorem 9.3. ■

Corollary 9.11. If φ has locally bounded variation and Q_{φ} ∈ S_{1/2}, then φ′ = 0 a.e.

Proof. If 0 < a < b < ∞, then φ has bounded variation on [a, b], and since

n^{2}s_{n} (Q_{φ}^{[a,b]}) ≤ n^{2}s_{n}(Q_{φ}) → 0,

Corollary 9.10 yields ∫_{a}^{b} |φ′|^{1/2} = 0. Hence, φ′ = 0 a.e. ■

Corollary 9.12. If φ is locally absolutely continuous and Q_{φ} ∈ S_{1/2}, then φ = 0. ■

Remark. More generally, in the last two corollaries, S_{1/2} can be replaced by any Schatten–Lorentz space S_{1/2,q} with q < ∞.

10. More on p = 1

Although φ ∈ X_{1} does not imply Q_{φ}, Q_{φ}^{+} ∈ S_{1}, the corresponding weak results holds. There is, however, a striking difference between Q_{φ} and Q_{φ}^{+}; as is shown in (ii) and (iii) below, for every φ ∈ X_{1} not a.e. equal to 0, ns_{n} → 0 for Q_{φ} but not for Q_{φ}^{+}. (Note that Theorem 6.5 implies that nothing can be said about the rate of convergence of ns_{n}(Q_{φ}) to 0. In particular, if q < ∞, then φ ∈ X_{1} does not imply that Q_{φ} ∈ S_{1,q}.)

Theorem 10.1. If φ ∈ X_{1} then the following hold:

(i) Q_{φ}, Q_{φ}^{+}, Q_{φ}^{−} ∈ S_{1,∞}.

(ii) ns_{n}(Q_{φ}) → 0 as n → ∞.

(iii) ns_{n}(Q_{φ}^{+}) = ns_{n}(Q_{φ}^{−}) → π^{−1} ∫_{0}^{∞} |φ(x)| dx as n → ∞.

Proof. Since X_{1} ⊂ X_{∞}, Q_{φ}^{+} is bounded by Theorem 3.1. By Theorem 4.2,

(Q_{φ}^{+})^{∗}Q_{φ}^{+} = Q_{Φ}

where Φ(x) = ∫_{x}^{∞} |φ(y)|^{2} dy. By (1.5), x^{1/2}Φ(x)^{1/2} ∈ L^{1}(dx/x), so Φ ∈ Y_{1/2} by Theorem 5.3, and hence, Q_{Φ} ∈ S_{1/2,∞} by Theorem 9.2. Consequently, Q_{φ}^{+} ∈ S_{1,∞}. The same holds for Q_{φ}^{−} = (Q_{φ}^{+})^{∗} and Q_{φ} = Q_{φ}^{+} + Q_{φ}^{−}.

Moreover, s_{n}(Q_{Φ}) = s_{n}(Q_{φ}^{+})^{2}, and thus (iii) follows from Theorem 9.3 applied to Φ.

For (ii), we observe that φ ↦ Q_{φ} thus is a bounded linear map X_{1} → S_{1,∞}, and that the set of C^{1} functions with compact support is dense in X_{1} and mapped (by Theorem 9.2) into the closed subspace S_{1,∞}^{0} = {T ∈ S_{1,∞} : ns_{n}(T) → 0 as n → ∞} of S_{1,∞}. Hence, Q_{φ} ∈ S_{1,∞}^{0} for every φ ∈ X_{1}. ■

Remark. Note that (i) and (iii) were earlier obtained in [EEH] for a more general class of operators. Moreover, in [EEH] the authors also consider the same operators on L^{p}(ℝ_{+}) and obtain similar results for approximation numbers. See also related results in [NaS]. We also mention here [GK1, Remark IV.8.3] and [GK2, Theorem III.2.4], where similar asymptotic formulas are given for abstract Volterra operators with trace class imaginary parts.

Remark. For the related operators Q_{μ}^{+}, we similarly obtain that if xμ(x, ∞) ∈ L^{1/2}(dx/x), then Q_{μ}^{+} ∈ S_{1,∞}, and

ns_{n}(Q_{μ}^{+}) → π^{−1} ∫_{0}^{∞} (\frac{dμ}{dx})^{1/2} dx as n → ∞,

where \frac{dμ}{dx} is the Radon–Nikodym derivative of the absolutely continuous component of μ. In particular, for such μ, ns_{n}(Q_{μ}^{+}) → 0 if and only if μ is singular.

We saw earlier that φ ∈ X_{1} is not enough to insure that Q_{φ} is in the trace class. Furthermore the previous theorem shows that if φ ∈ X_{1} then neither Q_{φ}^{+} nor Q_{φ}^{−} will be in the trace class. However, the combination of size and regularity results for singular numbers given in the previous theorem does insure that these operators have a well defined Dixmier trace. Because of the recent interest in the Dixmier trace we digress briefly to record this observation. For more about the Dixmier trace and its uses we refer to IV.2.β of [C].

Let ℓ^{∞} be the space of bounded sequences indexed by non-negative integers and let c_{1} be the closed subspace consisting of sequences {a_{n}} for which lim_{n} a_{n} exists. It follows from the Hahn–Banach theorem that the functional lim(·) which is defined on c_{1} has a positive continuous extension, lim_{ω}(·), to all of ℓ^{∞}. By saying lim_{ω}(·) is positive we mean that if a_{n} ≥ 0 for n = 0, 1, 2, … then lim_{ω} ({a_{n}}) ≥ 0. This extension is not unique and we are using the subscript ω to denote the particular choice. It was noted by Dixmier in [D] that lim_{ω}(·) can also be selected to have the following scaling property:

lim_{ω} (a_{0}, a_{0}, a_{1}, a_{1}, a_{2}, a_{2}, …) = lim_{ω} (a_{0}, a_{1}, a_{2}, …) .

A simple proof is in [C]. (Although the scaling is important for the general theory it has no role in our discussion.)

Consider now the operator ideal S_{Ω} ⊃ S_{1,∞} that consists of the operators T on Hilbert space such that

∥T ∥_{S_{Ω}} =^{def} sup_{n≥0} \frac{∑_{k=0}^{n} s_{k}(T)}{∑_{k=0}^{n} \frac{1}{k+1}} < ∞.  (10.1)

Suppose that T ∈ S_{Ω}. For a fixed choice of lim_{ω}(·) we define a Dixmier trace, trace_{ω}(·), as follows. For positive T ∈ S_{Ω} set

trace_{ω}(T) = lim_{ω} (\{\frac{1}{log(n + 2)} ∑_{k=0}^{n} λ_{k}(T)\}) .

Here the λ_{k} are the (necessarily non-negative) eigenvalues of the positive operator T arranged in decreasing order. Although perhaps not obvious at first glance, it is not difficult to see that, in fact, if T_{1} and T_{2} are two positive operators in S_{Ω} then trace_{ω}(T_{1} + T_{2}) = trace_{ω}(T_{1}) + trace_{ω}(T_{2}). A proof of this is also in [C]. Using this fact, the functional trace_{ω}(·) can be extended uniquely by linearity to all of T ∈ S_{Ω}. For T ∈ S_{Ω} the value of trace_{ω}(T) need not be independent of ω. However, there are certain operators for which trace_{ω}(T) is independent of ω. Such operators are defined to be measurable. In this case we will write trace_{D}(T) for this common value and refer to it as the Dixmier trace of T.

Corollary 10.2.

(i) If φ ∈ X_{1}, then the operators |Q_{φ}^{+}| and |Q_{φ}^{−}| are measurable and

trace_{D}(|Q_{φ}^{+}|) = trace_{D}(|Q_{φ}^{−}|) = \frac{1}{π} ∫_{0}^{∞} |φ(x)|dx.

(ii) If φ ∈ Y_{1/2}, then |Q_{φ}|^{1/2} is measurable and

trace_{D}(|Q_{φ}|^{1/2}) = \frac{1}{π} ∫_{0}^{∞} |φ′(x)|^{1/2}dx.

(iii) If φ ∈ X_{1}, then Q_{φ} is measurable and

trace_{D}(Q_{φ}) = 0.

(iv) If φ ∈ X_{1}, then Q_{φ}^{+} and Q_{φ}^{−} are measurable and

trace_{D}(Q_{φ}^{+}) = trace_{D}(Q_{φ}^{−}) = 0.

Proof. We start with (i). From the very definitions s_{n}(|Q_{φ}^{+}|) = s_{n}(Q_{φ}^{+}) and hence, the previous theorem gives the asymptotic behavior of {s_{n}(|Q_{φ}^{+}|)}_{n≥0}. Those asymptotics, together with the fact that |Q_{φ}^{+}| is a positive operator, insure that |Q_{φ}^{+}| is measurable and has the indicated Dixmier trace. A similar argument applies to |Q_{φ}^{−}| and, after noting that s_{n} (|Q_{φ}|^{1/2}) = s_{n}(Q_{φ})^{1/2} and taking note of Theorem 9.3, to part (ii).

We now consider (iii). By Theorem 10.1, we have lim_{n→∞} ns_{n}(Q_{φ}) = 0. Also s_{n}(Q_{φ}) = s_{n}(Q_{φ}^{∗}). Thus by (2.1), lim_{n→∞} ns_{n}((Q_{φ} + Q_{φ}^{∗})/2) = 0. We now use the

spectral projection to write \frac{1}{2}(Q_{φ} + Q_{φ}^{∗}) as a difference of two positive operators \frac{1}{2}(Q_{φ} + Q_{φ}^{∗})_{±} and note that we will have

lim_{n→∞} ns_{n} (\frac{1}{2}(Q_{φ} + Q_{φ}^{∗})_{±}) = 0.

Arguing similarly with the skew-adjoint part of Q_{φ}, we realize Q_{φ} as a linear combination of four positive operators each of which have singular numbers which tend to zero more rapidly than n^{−1}. Those positive operators are certainly measurable and have Dixmier trace zero. The result we want now follows by the linearity of trace_{D}(·).

For (iv) we first pick and fix a choice trace_{ω}(·). Assume for the moment that φ is real, supported in [0, 1] and in L^{2}. R_{φ} =^{def} Q_{φ}^{+} − Q_{φ}^{+∗} = Q_{φ}^{+} − Q_{φ}^{−} has real anti-symmetric kernel. Thus it is normal and its eigenvalues are imaginary and symmetric. Hence, iR_{φ} is symmetric and its positive and negative parts, (iR_{φ})_{±} are unitarily equivalent. Thus

trace_{ω}(R_{φ}) = −i trace_{ω}((iR_{φ})_{+}) + i trace_{ω}((iR_{φ})_{−}) = 0.

Taking note of the fact that lim_{ω}(·) is continuous on ℓ^{∞} and of the norm estimates implicit in the previous theorem we see that we can extend this result by linearity and continuity and conclude that trace_{ω}(R_{φ}) = 0 for all φ ∈ X_{1}. Now we use the fact that ω was arbitrary to conclude trace_{D}(R_{φ}) = 0. By linearity this result together with the result in (iii) yields (iv). ■

For a function φ defined on a finite or infinite interval I, we define the standard and L^{p} moduli of continuity by

ω_{φ}^{(∞)}(h; I) =^{def} sup {|φ(x) − φ(y)| : x, y ∈ I, |x − y| ≤ h},
ω_{φ}^{(p)}(h; I) =^{def} sup_{0≤s≤h}(∫_{I∩(I−s)} |φ(x + s) − φ(x)|^{p} dx)^{1/p}, 1 ≤ p < ∞,

(10.2)

where 0 < h ≤ |I| and I − s = {x − s : x ∈ I} = {x : x + s ∈ I}. It follows easily from Minkowski’s inequality that

ω_{φ}^{(p)}(h; I) ≤ 2ω_{φ}^{(p)}(h/2; I),  1 ≤ p ≤ ∞.  (10.3)

Note further that for a finite interval I,

ω_{φ}^{(p)}(h; I) ≤ |I|^{1/p−1/q}ω_{φ}^{(q)}(h; I),  p < q ≤ ∞.  (10.4)

We often omit I from the notation.

An alternative L^{p} modulus of continuity is defined by

ω̃_{φ}^{(p)}(h; I) =^{def} ((2h)^{−1} ∫∫_{{x,y∈I \atop |x−y|<h}} |φ(x) − φ(y)|^{p} dx dy)^{1/p} .

This is equivalent to ω_{φ}^{(p)}(h; I) defined above by the following lemma, which probably is well-known to some experts.

Lemma 10.3. Let 1 ≤ p < ∞. Then, with C_{p} depending on p only,

ω̃_{φ}^{(p)}(h; I) ≤ ω_{φ}^{(p)}(h; I) ≤ C_{p}ω̃_{φ}^{(p)}(h; I).

Proof. The left hand inequality follows by

\begin{aligned}(ω̃_{φ}^{(p)}(h; I))^{p} & = \frac{1}{h} ∫∫_{{x,y∈I \atop 0<y−x<h}} |φ(y) − φ(x)|^{p} dx dy \\ & = \frac{1}{h} ∫_{0}^{h} ∫_{x∈I∩(I−s)} |φ(x + s) − φ(x)|^{p} dx dy ≤ (ω_{φ}^{(p)}(h; I))^{p}. \end{aligned}

For the converse, we assume for convenience that I = [0, 1]. The result then follows for every finite I by a linear change of variables, and for infinite I by considering I∩[−n, n] and letting n → ∞. Thus I = [0, 1] and I∩(I−s) = [0, 1−s].

Let φ_{s}(x) = φ(x + s). Assume first that h ≤ 1/2. Then, for 0 ≤ s, t ≤ h, by Minkowski’s inequality,

∥φ − φ_{s}∥_{L^{p}[0,1/2]}^{p} ≤ C_{p}∥φ − φ_{t}∥_{L^{p}[0,1/2]}^{p} + C_{p}∥φ_{s} − φ_{t}∥_{L^{p}[0,1/2]}^{p}.

Averaging over t ∈ [0, h] we find

\begin{aligned}∥φ − φ_{s}∥_{L^{p}[0,1/2]}^{p} & ≤ \frac{C_{p}}{h} ∫_{0}^{h} ∫_{0}^{1/2}(|φ(x) − φ(x + t)|^{p} + |φ(x + s) − φ(x + t)|^{p})dx dt \\ & ≤ \frac{C_{p}}{h} ∫∫_{{x,y∈[0,1] \atop 0<y−x<h}} |φ(y) − φ(x)|^{p} dx dy = C_{p}(ω̃_{φ}^{(p)}(h; [0, 1]))^{p}. \end{aligned}

A similar argument, now taking s − h ≤ t ≤ s, yields the same estimate for ∥φ − φ_{s}∥_{L^{p}[1/2−s,1−s]}^{p}, and summing we find

∥φ − φ_{s}∥_{L^{p}[0,1−s]} ≤ C_{p}ω̃_{φ}^{(p)}(h; [0, 1])  (10.5)

for every 0 ≤ s ≤ h, which proves the result for h ≤ 1/2.

If 1/2 < h ≤ 1, the result follows from the case h ≤ 1/2 and (10.3). ■

For simplicity, we state the following lemma for I = [0, 1] only.

Lemma 10.4. Let 1 ≤ p < ∞. If φ ∈ L^{p}[0, 1] and 0 < t ≤ 1, there exists a decomposition φ = φ_{0} + φ_{1} with

∥φ_{0}∥_{L^{p}[0,1]} ≤ C_{p}ω_{φ}^{(p)}(t)  and  ∥φ_{1}∥′_{BV [0,1]} ≤ C_{p}t^{−1}ω_{φ}^{(p)}(t).

In other words, the Peetre K-functional (see [BL]), can be estimated by

K(t, φ; L^{p}[0, 1], BV′[0, 1]) ≤ C_{p}ω_{φ}^{(p)}(t), 0 < t ≤ 1.

Proof. Take φ_{1}(x) = \frac{1}{t} ∫_{(1−t)x}^{(1−t)x+t} φ(y) dy and φ_{0} = φ−φ_{1}. Then φ_{1} is absolutely continuous, and thus

∥φ_{1}∥′_{BV} = ∫_{0}^{1} |φ′_{1}(x)| dx = \frac{1 − t}{t} ∫_{0}^{1}|φ((1 − t)x + t) − φ((1 − t)x)| dx

= \frac{1}{t} ∫_{0}^{1−t} |φ(y + t) − φ(y)| dy

≤ \frac{1}{t}(∫_{0}^{1−t} |φ(y + t) − φ(y)|^{p} dy)^{1/p} ≤ \frac{1}{t}ω_{φ}^{(p)}(t).

Moreover, using Hölder’s inequality again,

|φ_{0}(x)|^{p} = |\frac{1}{t} ∫_{(1−t)x}^{(1−t)x+t}(φ(x) − φ(y)) dy|^{p} ≤ \frac{1}{t} ∫_{(1−t)x}^{(1−t)x+t} |φ(x) − φ(y)|^{p} dy

and thus

∫_{0}^{1} |φ_{0}(x)|^{p} ≤ \frac{1}{t} ∫∫_{{x,y∈[0,1] \atop |y−x|<t}} |φ(y) − φ(x)|^{p} dx dy = 2(ω_{φ}^{(p)}(t))^{p}. ■

Theorem 10.5. If I is a finite interval and φ ∈ L^{2}(I), then

s_{n}(Q_{φ}^{I}) ≤ C \frac{|I|^{1/2}}{n} ω_{φ}^{(2)}(\frac{|I|}{n}) ≤ C \frac{|I|}{n}ω_{φ}^{(∞)}(\frac{|I|}{n}) , n ≥ 1.

Proof. By a linear change of variables, we may assume that I = [0, 1], cf. Lemma 2.1. Then, using the decomposition given by Lemma 10.4 with t = 1/n, (2.1), Theorem 10.1 and Theorem 9.1, we find, for n ≥ 1,

s_{2n−1}(Q_{φ}) ≤ s_{n−1}(Q_{φ_{0}})+s_{n}(Q_{φ_{1}}) ≤ Cn^{−1}∥φ_{0}∥_{L^{2}} +Cn^{−2}∥φ_{1}∥′_{BV} ≤ Cn^{−1}ω_{φ}^{(2)}(1/n),

and the result follows, using (10.3) and (10.4). ■

In particular, we see that a Dini condition implies Q_{φ} ∈ S_{1}.

Corollary 10.6. If φ ∈ L^{2}[0, 1] is such that ∫_{0}^{1} ω_{φ}^{(2)}(t)\frac{dt}{t} < ∞, in particular if ∫_{0}^{1} ω_{φ}^{(∞)}(t)\frac{dt}{t} < ∞, then Q_{φ}^{[0,1]} ∈ S_{1}.

Proof. Theorem 3.1 shows that Q_{φ} is bounded, and Theorem 10.5 yields

∑_{n=2}^{∞} s_{n}(Q_{φ}) ≤ C ∑_{n=2}^{∞} \frac{1}{n}ω_{φ}^{(2)}(\frac{1}{n}) ≤ C ∫_{0}^{1} ω_{φ}^{(∞)}(t)\frac{dt}{t}. ■

By a simple change of variables, Corollary 10.6 applies to other finite intervals too. Moreover, for functions φ on ℝ_{+}, we have the following corresponding sufficient conditions for Q_{φ} ∈ S_{1}.

Theorem 10.7. If φ ∈ X_{1} and

∑_{n=−∞}^{∞} 2^{n/2} ∫_{0}^{2^{n}} ω_{φ}^{(2)}(t; [2^{n}, 2^{n+1}])\frac{dt}{t} < ∞

then Q_{φ} ∈ S_{1}.

Proof. Let I_{n} = [2^{n}, 2^{n+1}]. Then Theorem 10.5 yields

\begin{aligned}∥Q_{φ}^{I_{n}}∥_{S_{1}} & = ∑_{k=0}^{∞} s_{k}(Q_{φ}^{I_{n}}) ≤ 2∥Q_{φ}^{I_{n}}∥_{S_{2}} + C|I_{n}|^{1/2} ∑_{k=2}^{∞} \frac{1}{k}ω_{φ}^{(2)}(\frac{2^{n}}{k}; I_{n}) \\ & ≤ C2^{n/2}∥φ∥_{L^{2}(I_{n})} + C2^{n/2} ∫_{0}^{2^{n}} ω_{φ}^{(2)}(t; I_{n})\frac{dt}{t} \end{aligned}

and the result follows by Theorem 6.4 and (1.3). ■

Corollary 10.8. If φ ∈ X_{1} and

∑_{n=−∞}^{∞} 2^{n} ∫_{0}^{2^{n}} ω_{φ}^{(∞)}(t; [2^{n}, 2^{n+1}])\frac{dt}{t} < ∞

then Q_{φ} ∈ S_{1}. ■

Note that for the functions φ_{N} considered in Theorem 6.5, the estimate of the singular numbers in Theorem 10.5 is sharp (within a constant factor) and the estimates of the S_{1} norm implicit in Corollary 10.6, Theorem 10.7 and Corollary 10.8 are of the right order.

We do not know whether the condition in Theorem 10.7 is necessary, but we will give a related necessary condition using the L^{1} modulus of continuity in §15.

We have in these applications of Theorem 10.5 considered S_{1} only, but the same arguments apply to S_{p} for other p too. In particular, Theorem 10.7 extends as follows (see the remark after Theorem 6.4).

Theorem 10.9. Let 1/2 < p ≤ 1. If φ ∈ X_{p} and

∑_{n=−∞}^{∞} 2^{n(1−p/2)} ∫_{0}^{2^{n}}(ω_{φ}^{(2)}(t; [2^{n}, 2^{n+1}]))^{p}t^{p−2} dt < ∞,

then Q_{φ} ∈ S_{p}. ■

Note also the following immediate consequence of Theorem 10.5.

Corollary 10.10. If I is a finite interval and φ satisfies a Hölder (Lipschitz) condition |f(x) − f(y)| ≤ C|x − y|^{α} for x, y ∈ I, where 0 < α ≤ 1, then Q_{φ} ∈ S_{1/(1+α),∞} and thus Q_{φ} ∈ S_{p} for every p > 1/(1 + α). ■

11. Averaging projection

In this section we study properties of the averaging projection onto the set of operators of the form Q_{ψ}. Let us first define the averaging projection on S_{2}. Let T be an operator on L^{2}(ℝ_{+}) of class S_{2} with kernel function k = k_{T} ∈ L^{2}((ℝ_{+})^{2}). We define the function φ on ℝ_{+} by

φ(x) = \frac{1}{2x} (∫_{0}^{x} k(x, t)dt + ∫_{0}^{x} k(s, x)ds) , x > 0.  (11.1)

We define the averaging projection Q on S_{2} by

QT =^{def} Q_{φ}.

It is not hard to see that if Q_{ψ} ∈ S_{2}, then QQ_{ψ} = Q_{ψ}. It is also easy to see that ∥QT ∥_{S_{2}} ≤ ∥T ∥_{S_{2}} for any T ∈ S_{2}, and so Q is the orthogonal projection of S_{2} onto the set of operators of the form Q_{ψ}.

We show in this section that Q is a bounded linear operator on S_{p} for 1 < p ≤ 2. This allows us to define by duality the projection Q on the classes S_{p} for 2 ≤ p < ∞. We also show that Q is unbounded on S_{1} but it has weak type (1,1), i.e., s_{n}(QT)(1 + n) ≤ const ∥T ∥_{S_{1}}. Finally, we use this result to show that Q maps the Matsaev ideal into the set of compact operators.

Theorem 11.1. Let 1 < p ≤ 2. Then Q is a bounded projection on S_{p}.

Proof. Let T be an integral operator in S_{p} with kernel function k and let φ be defined by (11.1). We have to show that φ ∈ X_{p} (see the definition in the Introduction). We can identify in a natural way the dual space X_{p}^{∗} with the space Z_{p′} of functions f on ℝ_{+} such that

∑_{n∈ℤ}2^{−np′/2} (∫_{2^{n}}^{2^{n+1}} |f(x)|^{2}dx)^{p′/2} < ∞

with respect to the pairing

(φ, f) = ∫_{0}^{∞} φ(x)f(x)dx.  (11.2)

Here p′ = p/(p − 1). Suppose that f is a function on (0, ∞). Define the function ψ on ℝ_{+} by ψ(x) = \frac{f(x)}{2x}, x > 0. It is straightforward to see from the definition of the X_{p} spaces that f ∈ X_{p}^{∗} if and only if ψ ∈ X_{p′} and the norm of f in X_{p}^{∗} and the norm of ψ in X_{p′} are equivalent. It is also easy to see that for 1 < p < ∞ the space X_{p} is reflexive.

Let us show that if f is a bounded function in X_{p}^{∗} with compact support in (0, ∞), then

(φ, f) = trace T Q_{ψ}.  (11.3)

We have

trace T Q_{ψ} = ∫∫_{ℝ_{+}^{2}} k(x, y)ψ(max{x, y}) dxdy

= ∫_{0}^{∞}ψ(x) (∫_{0}^{x} k(x, t)dt + ∫_{0}^{x} k(s, x)ds) dx

= ∫_{0}^{∞}2xψ(x)φ(x) dx = ∫_{0}^{∞}φ(x)f(x)dx.

It follows that

sup{|(φ, f)| : f ∈ X_{p}^{∗}, ∥f∥_{X_{p}^{∗}} ≤ 1} ≤ const ∥T ∥_{S_{p}} sup{∥Q_{ψ}∥_{S_{p′}} : ∥ψ∥_{X_{p′}} ≤ 1}
≤ const ∥T ∥_{S_{p}}

by Theorem 3.3. It follows that φ ∈ X_{p}, and again by Theorem 3.3, QT ∈ S_{p}. ■

Theorem 11.1 allows us to define for 1 < p < 2 the adjoint operator Q^{∗} on S_{p′}. Since Q is an orthogonal projection on S_{2}, Q is a self-adjoint operator on S_{2}. We denote the adjoint operator Q^{∗} on S_{p′} by the same symbol Q.

Thus we can consider the projection Q on any class S_{p} with 1 < p < ∞. It is easy to show that if T is an integral operator with kernel function k of class S_{p}, 2 < p < ∞, then QT = Q_{φ}, where φ is defined by (11.1) and Q_{φ} ∈ S_{p}. We are going to prove that for any T ∈ S_{p}, 2 < p < ∞, the operator QT has the form Q_{φ} for a function φ ∈ X_{p}.

Theorem 11.2. Let T be an operator of class S_{p}, 2 < p < ∞. Then there exists a function φ ∈ X_{p} such that QT = Q_{φ}.

Proof. Let 𝔛_{p} be the space of operators of the form Q_{φ} with φ ∈ X_{p}. Clearly, 𝔛_{p} is a Banach space with norm

∥Q_{φ}∥_{𝔛_{p}} = ∥φ∥_{X_{p}}.

It follows from Theorems 11.1 and 3.3 that for T ∈ S_{2}

∥QT ∥_{𝔛_{p}} ≤ const ∥QT ∥_{S_{p}} ≤ const ∥T ∥_{S_{p}}.

Since S_{2} dense in S_{p}, it follows that QT ∈ 𝔛_{p} for any T ∈ S_{p}. ■

We consider now the behavior of Q on S_{1}. It follows from Theorem 11.1 that if T ∈ S_{1}, then QT ∈ S_{p} for any p > 1. The next result shows that QT does not have to be in S_{1} but it has to be in S_{1,∞}.

Theorem 11.3. (i) There exists an operator T in S_{1} such that QT ∉ S_{1}. (ii) Q has weak type (1, 1), i.e., Q maps S_{1} into S_{1,∞}, i.e.,

s_{n}(QT) ≤ const(1 + n)^{−1}∥T ∥_{S_{1}}, T ∈ S_{1}.

Lemma 11.4.

QS_{1} = 𝔛_{1} =^{def} {Q_{φ} : φ ∈ X_{1}}.

Let us first deduce Theorem 11.3 from Lemma 11.4.

Proof of Theorem 11.3. (i) is an immediate consequence of Lemma 11.4 and the Example following Theorem 6.5. (ii) also follows immediately from Lemma 11.4 and Theorem 10.1. ■

Proof of Lemma 11.4. Let us first show that QS_{1} ⊂ 𝔛_{1}. Let T ∈ S_{1} and QT = Q_{φ}. We have to prove that φ ∈ X_{1}. Consider the space Z_{∞}^{0} that consists of functions f on ℝ_{+} such that

lim_{n→±∞} (2^{−n} ∫_{2^{n}}^{2^{n+1}} |f(x)|^{2}dx) = 0.

It is not difficult to see that (Z_{∞}^{0})^{∗} = X_{1} with respect to the pairing (11.2). As in the proof of Theorem 11.1 we define the function ψ by ψ(x) = \frac{f(x)}{2x}, x > 0. It follows from (11.3) that

|(φ, f)| ≤ const ∥T ∥_{S_{1}}∥Q_{ψ}∥ ≤ const ∥T ∥_{S_{1}}∥ψ∥_{X_{∞}^{0}} ≤ const ∥T ∥_{S_{1}}∥f∥_{Z_{∞}^{0}},

and so φ determines a continuous linear functional on Z_{∞}^{0}. Hence, φ ∈ X_{1}.

To prove that QS_{1} = 𝔛_{1}, we consider the operator A : S_{1} → X_{1} defined by AT = φ, where φ is the function on ℝ_{+} such that QT = Q_{φ}. We have to show that A maps S_{1} onto X_{1}. Consider the conjugate operator A^{∗} : X_{1}^{∗} → B(L^{2}(ℝ_{+})).

It is easy to see that with respect to the pairing (11.2) the space X_{1}^{∗} can be identified with the space Z_{∞} that consists of functions f on ℝ_{+} such that

sup_{n∈ℤ}2^{−n/2} (∫_{2^{n}}^{2^{n+1}} |f(x)|^{2}dx)^{1/2} < ∞.

Consider the operator J : X_{1}^{∗} → X_{∞} defined by (Jf)(x) = \frac{f(x)}{2x}, x > 0. It is easy to see that J maps isomorphically X_{1}^{∗} onto X_{∞}.

It can easily be verified that A^{∗}f = Q_{Jf}. It follows from Theorem 3.1 that ∥A^{∗}f∥ ≥ const ∥f∥_{X_{1}^{∗}}. It follows that A maps S_{1} onto X_{1}. ■

Remark. In [Pel3] metric properties of the averaging projection P onto the space of Hankel matrices were studied. In particular, it was shown in [Pel3] that PS_{1} ⊂ S_{1,2}. However, it turns out that the averaging projection Q onto the operators Q_{φ} has different properties. Theorem 11.3 shows that QS_{1} ⊂ S_{1,∞} but it follows from Lemma 11.4 and the remark preceding Theorem 6.6 that QS_{1} ⊄ S_{1,q} for any q < ∞.

Recall that the Matsaev ideal S_{ω} consists of the operators T on Hilbert space such that

∥T ∥_{S_{ω}} =^{def} ∑_{n≥0} \frac{s_{n}(T)}{1 + n} < ∞.

It is easy to see that S_{p} ⊂ S_{ω} for any p < ∞.

Consider now the operator ideal S_{Ω} defined by (10.1). It is easy to see that S_{1,∞} ⊂ S_{Ω}. It is well known (see [GK1]) that S_{ω}^{∗} = S_{Ω} with respect to the pairing

{T, R} = trace T R, T ∈ S_{ω}, R ∈ S_{Ω}.  (11.4)

Theorem 11.5. The averaging projection Q defined on S_{2} extends to a bounded linear operator from S_{ω} to the space of compact operators. If T ∈ S_{ω}, then QT = Q_{φ} for a function φ in X_{∞}^{0}.

Proof. Let us prove that Q extends to a bounded operator from S_{ω} to the space of compact operators. The proof of the second part of the theorem is the same as the proof of Theorem 11.2. Since the finite rank operators are dense in S_{ω}, it is sufficient to show that Q extends to a bounded operator from S_{ω} to B(L^{2}(ℝ_{+})).

Let T ∈ S_{2} and R ∈ S_{1}. By Theorem 11.3, QR ∈ S_{1,∞} ⊂ S_{Ω}. We have

{QT, R} = {T, QR},

and so

|{QT, R}| ≤ const ∥T ∥_{S_{ω}}∥QR∥_{S_{Ω}}
≤ const ∥T ∥_{S_{ω}}∥QR∥_{S_{1,∞}} ≤ const ∥T ∥_{S_{ω}}∥R∥_{S_{1}}

by Theorem 11.3. Since S_{1}^{∗} = B(L^{2}(ℝ_{+})) with respect to the pairing (11.4), it follows that ∥QT ∥ ≤ const ∥T ∥_{S_{ω}}, and so Q extends to a bounded linear operator from S_{ω} to B(L^{2}(ℝ_{+})). ■

12. Finite rank

We say that φ is a step function if there exist finitely many numbers 0 = x_{0} < x_{1} < ⋯ < x_{N} < ∞ such that φ is a.e. constant on each interval (x_{i−1}, x_{i}), and zero on (x_{N}, ∞). The number of steps of φ then is the smallest possible N in this definition.

There is a natural correspondence between operators Q_{φ} where the symbol φ is a step function, with given x_{1} < ⋯ < x_{N}, and matrices of the form {a_{max{i,j}}}. We need a simple result for such matrices, but will not pursue their study further.

Lemma 12.1. If a_{1}, …, a_{n} are complex numbers, then the matrix

{a_{max{i,j}}}_{1≤i,j≤n} has determinant a_{n} ∏_{i=1}^{n−1}(a_{i} − a_{i+1}).

Proof. Denote this determinant by D(a_{1}, …, a_{n}). Subtracting the last row from all others, we see that D(a_{1}, …, a_{n}) = a_{n}D(a_{1} − a_{n}, …, a_{n−1} − a_{n}), and the result follows by induction. ■

Theorem 12.2. Q_{φ} has finite rank if and only if φ is a step function. In this case, the rank of Q_{φ} equals the number of steps of φ.

Proof. If φ is a step function with N steps, we have φ = ∑_{i=1}^{N} a_{i}χ_{(0,x_{i})} for some

a_{i} and x_{i} > 0, and thus Q_{φ} is a linear combination of N rank one operators.

Conversely, suppose that rank(Q_{φ}) = M < ∞. Suppose that n > M and that z_{1} < ⋯ < z_{n} are Lebesgue points of φ. If h > 0 and f_{z,h} = h^{−1}χ_{(z,z+h)}, then the matrix (⟨Q_{φ}f_{z_{i},h}, f_{z_{j},h}⟩)_{ij} has rank at most M < n and thus its determinant vanishes. As h → 0, as shown in the proof of Theorem 4.1, ⟨Q_{φ}f_{z_{i},h}, f_{z_{j},h}⟩ → φ(max{z_{i}, z_{j}}) = φ(z_{max{i,j}}), and thus the determinant of (φ(z_{max{i,j}}))_{ij} vanishes too. By Lemma 12.1, this implies that either φ(z_{i}) = φ(z_{i+1}) for some i < n or φ(z_{n}) = 0.

Consequently, if z_{1} < ⋯ < z_{n} are Lebesgue points of φ such that φ(z_{i}) ≠ φ(z_{i+1}) for i < n and φ(z_{n}) ≠ 0, then n ≤ M. Choose such a sequence z_{1} < ⋯ < z_{n} with n maximal. If z ∈ (z_{i}, z_{i+1}) ∩ Leb(φ) for some i < n, then either φ(z) = φ(z_{i}) or φ(z) = φ(z_{i+1}), since n is maximal. Moreover, for the same reason, if φ(z) = φ(z_{i}), then φ(z′) = φ(z_{i}) for every z′ ∈ (z_{i}, z) ∩ Leb(φ), and if φ(z) = φ(z_{i+1}), then φ(z′) = φ(z_{i+1}) for every z′ ∈ (z, z_{i+1}) ∩ Leb(φ). Together with similar arguments for the intervals (0, z_{1}) and (z_{n}, ∞), which we leave to the reader, this easily shows that φ is a step function with at most n steps. ■

13. A class of integral operators on L^{2}(ℝ)

In this section we associate with the operator Q_{φ} on L^{2}(ℝ_{+}) an integral operator on L^{2}(ℝ) and we study these operators.

For a function φ ∈ L_{loc}^{2}(ℝ_{+}) we define the function φ^{♡} on ℝ by

φ^{♡}(t) =^{def} 2φ(e^{2t})e^{2t}, t ∈ ℝ;  (13.1)

this defines a one-to-one correspondence between L_{loc}^{2}(ℝ_{+}) and L_{loc}^{2}(ℝ). With a function ψ on L^{2}(ℝ) we associate the function ψ̆ on ℝ × ℝ defined by

ψ̆(s, t) = ψ(max{s, t})e^{−|s−t|}, s, t ∈ ℝ,  (13.2)

and denote by K_{ψ} the integral operator on L^{2}(ℝ) with kernel function ψ̆ (if it makes sense):

(K_{ψ}f)(s) = ∫_{ℝ} ψ(max{s, t})e^{−|s−t|}f(t)dt.

Theorem 13.1. Let φ ∈ L_{loc}^{2}(ℝ_{+}). Then the operators Q_{φ} and K_{φ^{♡}} are unitarily equivalent.

Theorem 13.1 certainly means that the boundedness of one of the operators implies the boundedness of the other one.

Proof. Consider the unitary operator U : L^{2}(ℝ_{+}) → L^{2}(ℝ) defined as follows (Uf)(t) = \sqrt{2}f(e^{2t})e^{t}. It remains to observe that K_{φ^{♡}}U = UQ_{φ}. ■

We can identify L^{2}(ℝ_{+}) with the subspace of L^{2}(ℝ) which consists of the functions vanishing on (−∞, 0). We can now extend in a natural way the operator of triangular projection P to act on the space of operators on L^{2}(ℝ) by defining it in the same way as it has been done in §2. We keep the same notation, P, for this extension. We put K_{ψ}^{+} =^{def} PK_{ψ} and K_{ψ}^{−} =^{def} K_{ψ} − PK_{ψ}^{+}.

It is easily seen from the proof of Theorem 13.1 that the operators Q_{φ}^{+} and Q_{φ}^{−} are unitarily equivalent to the operators K_{φ^{♡}}^{+} and K_{φ^{♡}}^{−} respectively.

It is easy to see that, for any p > 0,

φ ∈ X_{p} ⇔ ∑_{n=−∞}^{∞} ∥φ^{♡}∥_{L^{2}[n,n+1]}^{p} < ∞  (13.3)

(and correspondingly for X_{∞} and X_{∞}^{0}) and, using Theorem 5.1 (vii),

φ ∈ Y_{p} ⇔ ∑_{n=−∞}^{∞} ∥φ^{♡}∥_{BV [n,n+1]}^{p} < ∞.  (13.4)

(In (13.3) and (13.4), the intervals [n, n + 1] can be replaced by any partition of ℝ into intervals of the same length.) We can thus translate results from the preceding sections to K_{ψ}, for example as follows.

Theorem 13.2. Let ψ ∈ L_{loc}^{2}(ℝ). The following are equivalent:

(i) K_{ψ} is bounded on L^{2}(ℝ); (ii) K_{ψ}^{+} is bounded on L^{2}(ℝ);

(iii) sup_{n∈ℤ}∫_{n}^{n+1} |ψ(x)|^{2}dx < ∞.

Proof. The theorem is a direct consequence of Theorems 3.1 and 13.1. ■ Similarly we find from Theorems 3.3, 6.2, and 5.1, respectively, the following

three theorems.

Theorem 13.3. Let ψ ∈ L_{loc}^{2}(ℝ). If 1 < p < ∞, the following are equivalent:

(i) K_{ψ} ∈ S_{p}; (ii) K_{ψ}^{+} ∈ S_{p};

(iii) ∑_{n∈ℤ}∥ψ∥_{L^{2}[n,n+1]}^{p} < ∞. ■

Theorem 13.4. If K_{ψ} ∈ S_{1}, then ∑_{n∈ℤ}∥ψ∥_{L^{2}[n,n+1]} < ∞ and thus ψ ∈ L^{1}(ℝ).

Moreover, then trace K_{ψ} = ∫_{ℝ} ψ(x) dx. ■

Theorem 13.5. If 1/2 < p ≤ 1 and ∑_{n∈ℤ}∥ψ∥_{BV [n,n+1]}^{p} < ∞, then K_{ψ} ∈ S_{p}. ■

The following two results involving the modulus of continuity also can be obtained by changes of variables in the corresponding Theorems 10.5 and 10.9, using (10.3) and Lemma 10.3, but the details are involved and we prefer to imitate the proofs.

Theorem 13.6. If ψ ∈ L^{2}(ℝ) has support on [0, 1], then

s_{n}(K_{ψ}) ≤ C \frac{1}{n}ω_{ψ}^{(2)}(\frac{1}{n}) + C \frac{1}{n^{2}} ∥ψ∥_{L^{2}}, n ≥ 1.

Proof. We interpolate using Lemma 10.4 as in the proof of Theorem 10.5. ■

Theorem 13.7. Let 1/2 < p ≤ 1. If ∑_{n∈ℤ}∥ψ∥_{L^{2}[n,n+1]}^{p} < ∞ and

∑_{n=−∞}^{∞} ∫_{0}^{1}(ω_{ψ}^{(2)}(t; [n, n + 1]))^{p}t^{p−2} dt < ∞,

then K_{ψ} ∈ S_{p}.

Proof. We argue as in the proofs of Theorems 10.7 and 10.9, using Theorem 13.6. ■

We denote by F the Fourier transformation on L^{2}(ℝ^{n}), which is a unitary operator defined by (2.3) for f ∈ L^{2}(ℝ^{n}) ⋂ L^{1}(ℝ^{n}). Let T be the integral operator with kernel function k ∈ L^{2}(ℝ^{2}). Denote by R the integral operator with kernel function Fk. The following lemma has a straightforward verification.

Lemma 13.8. FT F = R. ■

Note that Lemma 13.8 is similar to Lemma 2.3.

Corollary 13.9. Let p > 0. Then ∥T ∥_{S_{p}} = ∥R∥_{S_{p}}. ■

Denote by Z be the integral operator with kernel function (x, y) ↦ (Fk)(x, −y). By Lemma 2.3, T is unitarily equivalent to Z. Indeed, the equality FT F = R implies FT F^{−1} = Z.

Lemma 13.10. If ψ ∈ L^{1}(ℝ), then ψ̆ ∈ L^{1}(ℝ^{2}), and

(Fψ̆)(x, y) = (Fψ)(x + y) (\frac{1}{1 − 2πix} + \frac{1}{1 − 2πiy} ) .  (13.5)

Proof. The inclusion ψ̆ ∈ L^{1}(ℝ^{2}) is obvious. We have

(Fψ̆)(x, y) = ∫_{ℝ^{2}} ψ(max{s, t})e^{−|s−t|}e^{−2πisx−2πity}dsdt

= ∫_{ℝ} (ψ(s)e^{−s}e^{−2πisx} ∫_{−∞}^{s} e^{t−2πity}dt) ds

+ ∫_{ℝ} (ψ(t)e^{−t}e^{−2πity} ∫_{−∞}^{t} e^{s−2πisx}ds) dt

= ∫_{ℝ} \frac{ψ(s)e^{−2πis(x+y)} ds}{1 − 2πiy} + ∫_{ℝ} \frac{ψ(t)e^{−2πit(x+y)} dt}{1 − 2πix}

= (\frac{1}{1 − 2πix} + \frac{1}{1 − 2πiy} ) (Fψ)(x + y). ■

Consider the functions

ψ̆_{+}(s, t) =^{def} χ_{{(s,t):s>t}}ψ̆(s, t) and ψ̆_{−}(s, t) =^{def} χ_{{(s,t):s<t}}ψ̆(s, t).  (13.6)

It can easily be seen from the proof of Lemma 13.10 that

(Fψ̆_{+})(x, y) = \frac{(Fψ)(x + y)}{1 − 2πiy} and (Fψ̆_{−})(x, y) = \frac{(Fψ)(x + y)}{1 − 2πix}.  (13.7)

It is easy to verify that if ψ is a tempered distribution on ℝ (see §2), we can define tempered distributions ψ̆, ψ̆_{+}, and ψ̆_{−} by (13.2) and (13.6); the formal definitions are by duality and analogous to (2.4)–(2.6). It is easy to check that formulas (13.5) and (13.7) also hold for tempered distributions ψ.

As a corollary to Theorem 13.2, we have the following lemma.

Lemma 13.11. Let ψ ∈ L_{loc}^{2}(ℝ). Suppose that the operator K_{ψ} is bounded on L^{2}(ℝ). Then ψ determines a tempered distribution. ■

Theorem 13.12. Let 0 < p < ∞. Suppose that φ ∈ L_{loc}^{2}(ℝ_{+}) and φ^{♡} is defined by (13.1). The following are equivalent:

(i) Q_{φ} ∈ S_{p};

(ii) K_{φ^{♡}} ∈ S_{p};

(iii) the integral operator on L^{2}(ℝ) with kernel

(x, y) ↦ (Fφ^{♡})(x + y) (\frac{1}{1 − 2πix} + \frac{1}{1 − 2πiy} )

belongs to S_{p}.

Proof. The theorem is a consequence of Theorem 13.1, Lemma 13.8 and Lemma 13.10. ■

Note that if for p > 2 we have a tempered distribution in (iii) rather than a function, by the integral operator we mean the operator determined by this tempered distribution (see §2).

In the same way one can prove the following result.

Theorem 13.13. Let 0 < p < ∞ and let φ ∈ L_{loc}^{2}(ℝ_{+}). The following are equivalent:

(i) Q_{φ}^{+} ∈ S_{p};

(ii) Q_{φ}^{−} ∈ S_{p};

(iii) K_{φ^{♡}}^{+} ∈ S_{p};

(iv) K_{φ^{♡}}^{−} ∈ S_{p};

(v) the integral operator on L^{2}(ℝ) with the kernel

(x, y) ↦ \frac{(Fφ^{♡})(x + y)}{1 − 2πix}

belongs to S_{p}. ■

It is straightforward to show that if p > 0 and the integral operator on L^{2}(ℝ) with kernel function \frac{h(x+y)}{x+α} belongs to S_{p} for some α ∈ ℂ \ ℝ, then it belongs to S_{p} for any α ∈ ℂ \ ℝ. Let us show that such an integral operator can belong to S_{p} for p ≤ 1 only if it is zero.

Consider the operator E : D(ℝ^{2}) → D(ℝ_{+} × ℝ_{+}) defined by the following formula (Ef)(s, t) =^{def} \frac{1}{2}(st)^{−1/2}f(\frac{1}{2} log s, \frac{1}{2} log t). Clearly, E is an isomorphism. Consequently, the conjugate operator E′ is an isomorphism from D′(ℝ_{+} × ℝ_{+}) onto D′(ℝ^{2}). Clearly, (E′Φ)(x, y) = 2Φ(e^{2x}, e^{2y})e^{x}e^{y} if Φ ∈ L_{loc}^{1}(ℝ_{+} × ℝ_{+}). Put

2Φ(e^{2x}, e^{2y})e^{x}e^{y} =^{def} (E′Φ)(x, y) for Φ ∈ D′(ℝ_{+} × ℝ_{+}).

Theorem 13.14. Let Φ ∈ D′(ℝ_{+} × ℝ_{+}). Then Φ determines a bounded operator on L^{2}(ℝ_{+}) if and only if the distribution 2Φ(e^{2x}, e^{2y})e^{x}e^{y} determines a bounded operator on L^{2}(ℝ). Moreover, these two operators are unitarily equivalent operators.

Proof. It suffices to note that

⟨2Φ(e^{2x}, e^{2y})e^{x}e^{y}, \sqrt{2}e^{y}f(e^{2y})\overline{\sqrt{2}e^{x}g(e^{2x})}⟩ = ⟨Φ(s, t), f(t)\overline{g(s)}⟩

for any f, g ∈ D(ℝ_{+} × ℝ_{+}), and the map h ↦ \sqrt{2}e^{x}h(e^{2x}) is a unitary operator from L^{2}(ℝ_{+}) onto L^{2}(ℝ). ■

Theorem 13.15. Let h ∈ D′(ℝ). Suppose that the distribution \frac{h(x+y)}{1−2πix} determines an operator on L^{2}(ℝ) of class S_{1}. Then h = 0.

Proof. By Lemma 2.2, \frac{h(x+y)}{1−2πix} ∈ S′(ℝ^{2}). Consequently, h(x + y) ∈ S′(ℝ^{2}), whence h ∈ S′(ℝ). Thus, there exists a distribution φ ∈ D′(ℝ_{+}) such that φ^{♡} ∈ S′(ℝ) and Fφ^{♡} = h (the operation φ ↦ φ^{♡} defined in (13.1) extends in an obvious way to distributions φ). Lemma 2.3 and formula (13.7) imply that Q_{φ}^{−} and thus also Q_{φ}^{+} belongs to S_{1}. By Theorem 2.4, φ ∈ L_{loc}^{2}(ℝ_{+}). Thus φ = 0 by Theorem 6.6. ■

We are going to prove now that for p > 1/2 if the integral operator with kernel function h(x + y)(\frac{1}{x+α} + \frac{1}{y+β} ) belongs to S_{p} for some α, β ∈ ℂ \ ℝ with α + β ∉ ℝ, then it belongs to S_{p} for any α, β ∈ ℂ \ ℝ. We will also show that this is not true for p ≤ 1/2.

Lemma 13.16. The function

(x, y) ↦ \frac{1}{x + y + i} χ_{[0,1]}(x), x, y ∈ ℝ,

is a Schur multiplier of S_{p} for any p > 0.

Proof. First we prove that the function

(x, y) ↦ \frac{1}{x + y + i} χ_{[0,1]}(x)χ_{ℝ\[−2,2]}(y), x, y ∈ ℝ,  (13.8)

is a Schur multiplier of S_{p}. We have

\frac{1}{x + y + i} χ_{[0,1]}(x)χ_{ℝ\[−2,2]}(y) = ∑_{n≥0}(−1)^{n} \frac{(x + i)^{n}}{y^{n+1}} χ_{[0,1]}(x)χ_{ℝ\[−2,2]}(y).

Clearly, the p-multiplier norm of the n-th summand is bounded by 2^{−n/2}. Consequently, the function (13.8) is a Schur multiplier of S_{p}. It remains to prove that the function

(x, y) ↦ \frac{1}{x + y + i} χ_{[0,1]}(x)χ_{[−2,2]}(y), x, y ∈ ℝ,

is a Schur multiplier of S_{p}. For any (ξ, η) ∈ [0, 1] × [−2, 2] we can expand the function \frac{1}{x+y+i} in a Taylor series in a neighborhood of (ξ, η). It follows easily that for a sufficiently small ε > 0 the function

(x, y) ↦ \frac{1}{x + y + i} χ_{[0,1]∩[ξ−ε,ξ+ε]}(x)χ_{[−2,2]∩[η−ε,η+ε]}(y), x, y ∈ ℝ,

is a Schur multiplier of S_{p}. It remains to choose a finite subcover of [0, 1] × [−2, 2] that consists of rectangles of the form [ξ − ε, ξ + ε] × [η − ε, η + ε]. ■

Remark. In the same way we can prove that the function

(x, y) ↦ \frac{1}{x + y + α} χ_{[ξ,η]}(x), x, y ∈ ℝ,

is a Schur multiplier of S_{p} for any p > 0 for any α ∈ ℂ \ ℝ and for any ξ, η ∈ ℝ.

Corollary 13.17. Let α, β, γ ∈ ℂ and γ ∉ ℝ. Then the function

(x, y) ↦ \frac{(x + α)(y + β)}{x + y + γ}χ_{[0,1]}(x), x, y ∈ ℝ,

is a Schur multiplier of S_{p} for any p > 0.

Proof. We have

\frac{(x + α)(y + β)}{x + y + γ} = (x + α) (1 + \frac{β − γ − x}{x + y + γ}) .

It remains to note that

χ_{[0,1]}(x), (x + α)χ_{[0,1]}(x), (β − γ − x)χ_{[0,1]}(x), and \frac{1}{x + y + γ} χ_{[0,1]}(x)

are Schur multipliers of S_{p}. ■

Corollary 13.18. Let p > 0 and let α, β ∈ ℂ \ ℝ such that α + β ∉ ℝ. Suppose that the integral operator on L^{2}(ℝ) with kernel function

(x, y) ↦ h(x + y) (\frac{1}{x + α} + \frac{1}{y + β} ) , x, y ∈ ℝ,

belongs to S_{p}. Then the integral operator with kernel function

(x, y) ↦ h(x + y)χ_{[0,1]}(x), x, y ∈ ℝ,

belongs to S_{p}. ■

Theorem 13.19. Let p > 1/2 and let α_{0}, β_{0} ∈ ℂ \ ℝ such that α_{0} + β_{0} ∉ ℝ. Suppose that the integral operator on L^{2}(ℝ) with kernel function

(x, y) ↦ h(x + y) (\frac{1}{x + α_{0}} + \frac{1}{y + β_{0}} ) , x, y ∈ ℝ,

belongs to S_{p}. Then the integral operator with kernel function

(x, y) ↦ h(x + y) (\frac{1}{x + α} + \frac{1}{y + β} ) , x, y ∈ ℝ,

also belongs to S_{p} for any α, β ∈ ℂ \ ℝ.

Proof. By Corollary 13.18, the integral operator with kernel h(x + y)χ_{[0,1]}(x) belongs to S_{p}. Obviously, for any n ∈ ℤ,

∥h(x + y)χ_{[0,1]}(x)∥_{S_{p}} = ∥h(x + y)χ_{[n,n+1]}(x)∥_{S_{p}}

(as usual we write ∥k∥_{S_{p}} for the S_{p} norm (or quasi-norm) of the integral operator with kernel k). Consequently,

∥h(x + y) (\frac{1}{x + α} − \frac{1}{x + α_{0}} ) χ_{[n,n+1]}(x)∥_{S_{p}} ≤ const(1 + |n|)^{−2}.

It is now clear that the integral operator with kernel function

(x, y) ↦ h(x + y) (\frac{1}{x + α} − \frac{1}{x + α_{0}} ) , x, y ∈ ℝ,

belongs to S_{p} for p > 1/2. Similarly, we prove that the integral operator with kernel function

(x, y) ↦ h(x + y) (\frac{1}{y + β} − \frac{1}{y + β_{0}} ) , x, y ∈ ℝ,

belongs to S_{p} for p > 1/2. ■

Theorem 13.20. Let φ ∈ L_{loc}^{2}(ℝ_{+}), a ∈ ℝ \ {0}, and p > 1/2. Put

φ_{[a]}(t) =^{def} φ(t^{a})t^{a−1}.

Then Q_{φ} ∈ S_{p} if and only if Q_{φ_{[a]}} ∈ S_{p}.

Proof. Recall that φ^{♡}(t) = 2φ(e^{2t})e^{2t} and φ_{[a]}^{♡}(t) = 2φ_{[a]}(e^{2t})e^{2t} = 2φ(e^{2at})e^{2at}. Consequently, φ_{[a]}^{♡}(t) = φ^{♡}(at). By Theorem 13.12, Q_{φ_{[a]}} ∈ S_{p} if and only if the integral operator on L^{2}(ℝ) with kernel

(x, y) ↦ (Fφ^{♡}) (\frac{x + y}{a} ) (\frac{1}{1 − 2πix} + \frac{1}{1 − 2πiy} ) , x, y ∈ ℝ,

belongs to S_{p}, and thus if and only if the integral operator on L^{2}(ℝ) with kernel

(x, y) ↦ (Fφ^{♡})(x + y) (\frac{1}{1 − 2πiax} + \frac{1}{1 − 2πiay} ) , x, y ∈ ℝ,

belongs to S_{p}. By Theorem 13.19, this holds if and only if the integral operator on L^{2}(ℝ) with kernel

(x, y) ↦ (Fφ^{♡})(x + y) (\frac{1}{1 − 2πix} + \frac{1}{1 − 2πiy} ) , x, y ∈ ℝ,

belongs to S_{p}. It remains to apply Theorem 13.12 once more. ■

Corollary 13.21. Let ψ ∈ L_{loc}^{2}(ℝ), a ∈ ℝ \ {0}, and p > 1/2. Define ψ_{a}(t) =^{def} ψ(at). Then K_{ψ} ∈ S_{p} if and only if K_{ψ_{a}} ∈ S_{p}. ■

Remark. Theorem 13.20 and its corollary do not generalize to the case p ≤ 1/2. Indeed, if φ is the characteristic function of an interval, then Q_{φ} ∈ S_{p} for any p > 0 but if a ≠ 1, then Q_{φ_{[a]}} ∉ S_{p} by Corollary 8.12. It follows from the proof above that Theorem 13.19 too does not extend to p ≤ 1/2.

Remark. Note that if φ ∈ X_{p}, then φ_{[a]}(t) ∈ X_{p} for any a ∈ ℝ \ {0} and any p > 0. Moreover, if φ ∈ Y_{p}, then φ_{[a]}(t) ∈ Y_{p} for any a ∈ ℝ \ {0} and any p > 0. Indeed, let A > 1. It is easy to see that (1.3) is equivalent to the condition

∑_{n∈ℤ}A^{np/2} (∫_{A^{n}}^{A^{n+1}} |φ(x)|^{2}dx)^{p/2} < ∞,

while the condition in Theorem 5.3 (vii) is equivalent to

∑_{n∈ℤ}∥xφ(x)∥_{BV [A^{n},A^{n+1}]}^{p} < ∞,

which easily implies the above assertions.

Theorem 13.22. Let α, β ∈ ℂ \ ℝ and let p > 0. Then the integral operator on L^{2}(ℝ) with kernel function

(x, y) ↦ h(x + y) (\frac{1}{x + α} + \frac{1}{y + β} ) , x, y ∈ ℝ,

belongs to S_{p} if and only if convolution with the function h(x)(x + α + β) is an operator from L^{2}(ℝ, (1 + x^{2}) dx) to L^{2}(ℝ, (1 + x^{2})^{−1} dx) of class S_{p}.

Proof. Clearly, the integral operator on L^{2}(ℝ) with kernel function

(x, y) ↦ h(x + y) (\frac{1}{x + α} + \frac{1}{y + β} ) , x, y ∈ ℝ,

belongs to S_{p} if and only if so does the integral operator with kernel function

(x, y) ↦ h(x − y)\frac{x − y + α + β}{(x + α)(y − β)}, x, y ∈ ℝ.

To complete the proof, it remains to observe that multiplication by (x − β)^{−1} is an isomorphism from L^{2}(ℝ) onto L^{2}(ℝ, (1 + x^{2}) dx) and multiplication by (x + α)^{−1} is an isomorphism from L^{2}(ℝ, (1 + x^{2})^{−1} dx) onto L^{2}(ℝ). ■

Corollary 13.23. Let α, β, γ ∈ ℂ \ ℝ such that α + β ∉ ℝ and let p > 1/2. Then the integral operator with kernel function

(x, y) ↦ h(x + y) (\frac{1}{x + α} + \frac{1}{y + β} ) , x, y ∈ ℝ,

belongs to S_{p} if and only if convolution with the function h(x)(x+γ) is an operator from L^{2}(ℝ, (1 + x^{2}) dx) to L^{2}(ℝ, (1 + x^{2})^{−1} dx) of class S_{p}.

Proof. It suffices to apply Theorem 13.19. ■

Remark. In the same way we can prove that the following statements are equivalent for any α ∈ ℂ \ ℝ and for any p > 0:

(i) the integral operator on L^{2}(ℝ) with kernel function

(x, y) ↦ k(x + y)(x + α)^{−1}, x, y ∈ ℝ,

belongs to S_{p};

(ii) the integral operator on L^{2}(ℝ) with kernel function

(x, y) ↦ k(x + y)(y + α)^{−1}, x, y ∈ ℝ,

belongs to S_{p};

(iii) convolution with k is an operator from L^{2}(ℝ) to L^{2}(ℝ, (1 + x^{2})^{−1} dx) of class S_{p};

(iv) convolution with k is an operator from L^{2}(ℝ, (1 + x^{2}) dx) to L^{2}(ℝ) of class

S_{p}. Let us repeat that Theorem 13.15 implies that the integral operator with kernel

function k(x + y)(x + α)^{−1} can be a nonzero operator in S_{p} only if p > 1.

14. Matrix representation

Let φ be a function in L_{loc}^{2}(ℝ) such that φ(x + 1) = φ(x), x ∈ ℝ. Consider the operators Q_{φ}^{[0,1]} and Q_{φ}^{[0,2]} on L^{2}[0, 1] and L^{2}[0, 2] respectively. Obviously,

∥Q_{φ}^{[0,1]}∥_{S_{p}} ≤ ∥Q_{φ}^{[0,2]}∥_{S_{p}} , 0 < p < ∞.  (14.1)

Obviously,

∥Q_{φ}^{[0,2]}∥ ≥ ∥P_{[1,2]×[0,1]}Q_{φ}^{[0,2]}∥ = ∥φ∥_{L^{2}[0,1]}.  (14.2)

It is also easy to see that

∥Q_{φ}^{[0,2]}∥_{S_{p}} ≤ C(p) (∥Q_{φ}^{[0,1]}∥_{S_{p}} + ∥φ∥_{L^{2}[0,1]}) , 0 < p < ∞,  (14.3)

where C(p) is a constant that may depend only on p.

Theorem 14.1. Let 0 < p < ∞. Suppose that φ is a function in L_{loc}^{2}(ℝ) such that φ(x) = φ(x + 1) and ψ(x) =^{def} φ(1 − x). Then

C_{1}(p) (∥Q_{φ}^{[0,1]}∥_{S_{p}} + ∥Q_{ψ}^{[0,1]}∥_{S_{p}}) ≤ ∥Q_{φ}^{[0,2]}∥_{S_{p}} ≤ C_{2}(p) (∥Q_{φ}^{[0,1]}∥_{S_{p}} + ∥Q_{ψ}^{[0,1]}∥_{S_{p}}) .

Proof. To prove the left inequality, consider the integral operator K on L^{2}[0, 1] with kernel function

(x, y) ↦ φ(min{x, y}).

Clearly, the operators K and Q_{ψ}^{[0,1]} are unitarily equivalent, and so ∥K∥_{S_{p}} = ∥Q_{ψ}^{[0,1]}∥_{S_{p}} for any p > 0. Note that

φ(min{x, y}) + φ(max{x, y}) = φ(x) + φ(y).

Hence, K +Q_{φ}^{[0,1]} is the integral operator with kernel function (x, y) ↦ φ(x)+φ(y). Thus ∥Q_{φ}^{[0,1]} + K∥_{S_{p}} ≤ C(p)∥φ∥_{L^{2}}. Now the left inequality is obvious. To prove the right inequality, we have to show that

∥φ∥_{L^{2}[0,1]} ≤ C(p) (∥Q_{φ}^{[0,1]}∥_{S_{p}} + ∥Q_{ψ}^{[0,1]}∥_{S_{p}}) .

Clearly,

(K + Q_{φ}^{[0,1]}) 1 = φ + ∫_{0}^{1} φ(t) dt.

It follows that

∥(K + Q_{φ}^{[0,1]}) 1∥_{L^{2}[0,1]}^{2} = ∥φ∥_{L^{2}[0,1]}^{2} + 3 |∫_{0}^{1} φ(t) dt|^{2} ≥ ∥φ∥_{L^{2}[0,1]}^{2}.

Thus,

∥K + Q_{φ}^{[0,1]}∥_{S_{p}} ≥ ∥K + Q_{φ}^{[0,1]}∥ ≥ ∥φ∥_{L^{2}[0,1]},

and so

∥φ∥_{L^{2}[0,1]} ≤ ∥(K + Q_{φ}^{[0,1]})∥_{S_{p}}
≤ C(p) (∥K∥_{S_{p}} + ∥Q_{φ}^{[0,1]}∥_{S_{p}})

= C(p) (∥Q_{φ}^{[0,1]}∥_{S_{p}} + ∥Q_{ψ}^{[0,1]}∥_{S_{p}}) . ■

Corollary 14.2. Under the hypotheses of Theorem 14.1

∥Q_{ψ}^{[0,2]}∥_{S_{p}} ≤ C(p) ∥Q_{φ}^{[0,2]}∥_{S_{p}} . ■

Theorem 14.3. Let p > 0 and a ∈ ℝ. If φ is a function satisfying the hypotheses of Theorem 14.1 and ψ(x) =^{def} φ(x − a), then

∥Q_{ψ}^{[0,2]}∥_{S_{p}} ≤ C(p) ∥Q_{φ}^{[0,2]}∥_{S_{p}} .

Proof. Clearly, it is sufficient to consider the case a ∈ (0, 1). Then Q_{ψ}^{[0,1]} is by a translation unitarily equivalent to Q_{φ}^{[1−a,2−a]}, and thus

∥Q_{ψ}^{[0,1]}∥_{S_{p}} = ∥Q_{φ}^{[1−a,2−a]}∥_{S_{p}} ≤ ∥Q_{φ}^{[0,2]}∥_{S_{p}} .

The result follows by (14.3) and (14.2). ■

Let ϕ be a function on the unit circle 𝕋. Put 𝕋_{+} =^{def} {ζ ∈ 𝕋 : Im ζ ≥ 0},

𝕋_{−} =^{def} {ζ ∈ 𝕋 : Im ζ < 0} and

k_{ϕ}(ζ, τ) =^{def} \{ \begin{matrix}ϕ(ζ^{2}), & ζ\overline{τ} ∈ 𝕋_{+}, \\ ϕ(τ^{2}), & ζ\overline{τ} ∈ 𝕋_{−}, \end{matrix} (ζ, τ) ∈ 𝕋^{2}.  (14.4)

It is easy to see that the functions ϕ and k_{ϕ} are equimeasurable. In particular, ∥ϕ∥_{L^{2}(𝕋)} = ∥k_{ϕ}∥_{L^{2}(𝕋^{2})}. Note also that if ϕ is continuous on 𝕋, then k_{ϕ} is continuous on 𝕋^{2}. Let ϕ ∈ L^{2}(𝕋). Denote by K_{ϕ} the integral operator on L^{2}(𝕋) with kernel function k_{ϕ}.

Theorem 14.4. Let p > 0. Suppose that ϕ ∈ L^{2}(𝕋) and φ(t) =^{def} ϕ(e^{2πit}), t ∈ ℝ. Then

C_{1}(p) ∥Q_{φ}^{[0,2]}∥_{S_{p}} ≤ ∥K_{ϕ}∥_{S_{p}} ≤ C_{2}(p) ∥Q_{φ}^{[0,2]}∥_{S_{p}} ,

where C_{1}(p) and C_{2}(p) may depend only on p.

Proof. Consider the integral operator K on L^{2}[0, 1] with kernel function k ∈ L^{2}([0, 1]^{2}) defined by

k(x, y) = \{ \begin{matrix}φ(2 max\{x, y\}), & |x − y| ≤ 1/2, \\ φ(2 min\{x, y\}), & |x − y| > 1/2. \end{matrix}

It is easy to see that K is unitarily equivalent to K_{ϕ}. For α, β = 0, 1 we consider the integral operator K^{(α,β)} with kernel function

(x, y) ↦ k(x, y)χ_{[α/2,(1+α)/2]}(x)χ_{[β/2,(1+β)/2]}(y).

Using the substitution (x, y) ↦ (2x, 2y), we find that 2∥K^{(0,0)}∥_{S_{p}} = ∥Q_{φ}^{[0,1]}∥_{S_{p}}. In

a similar way we can obtain 2∥K^{(1,1)}∥_{S_{p}} = ∥Q_{φ}^{[0,1]}∥_{S_{p}}. Let ψ(t) =^{def} φ(1 − t). It is

also easy to see that 2∥K^{(0,1)}∥_{S_{p}} = 2∥K^{(1,0)}∥_{S_{p}} = ∥Q_{ψ}^{[0,1]}∥_{S_{p}}. Hence,

\frac{1}{4} (∥Q_{φ}^{[0,1]}∥_{S_{p}} + ∥Q_{ψ}^{[0,1]}∥_{S_{p}}) = \frac{1}{4} ∑_{α=0}^{1} ∑_{β=0}^{1} ∥K^{(α,β)}∥_{S_{p}}

≤ ∥K_{ϕ}∥_{S_{p}} ≤ C(p) ∑_{α=0}^{1} ∑_{β=0}^{1} ∥K^{(α,β)}∥_{S_{p}}

= C(p) (∥Q_{φ}^{[0,1]}∥_{S_{p}} + ∥Q_{ψ}^{[0,1]}∥_{S_{p}}) .

It remains to apply Theorem 14.1. ■

We denote by f̂(n) denote the nth Fourier coefficient of a function f in L^{1}(𝕋). For convenience we put

f̂(n + 1/2) =^{def} 0, n ∈ ℤ.

For a function k in L^{1}(𝕋^{2}) we denote by {k̂(m, n)}_{(m,n)∈ℤ^{2}} the sequence of its Fourier coefficients.

Let ϕ be a function on 𝕋. Put

k_{ϕ}^{+}(ζ, τ) =^{def} \{ \begin{matrix}ϕ(ζ^{2}), & ζ\overline{τ} ∈ 𝕋_{+}, \\ 0, & ζ\overline{τ} ∈ 𝕋_{−}, \end{matrix}

and

k_{ϕ}^{−}(ζ, τ) =^{def} \{ \begin{matrix}0, & ζ\overline{τ} ∈ 𝕋_{+}, \\ ϕ(τ^{2}), & ζ\overline{τ} ∈ 𝕋_{−}. \end{matrix}

Clearly, k_{ϕ}^{+}(ζ, τ) = k_{ϕ}^{−}(τ, ζ) and k_{ϕ}(ζ, τ) = k_{ϕ}^{+}(ζ, τ) + k_{ϕ}^{−}(ζ, τ), where k_{ϕ} is defined by (14.4).

Lemma 14.5. Let ϕ ∈ L^{1}(𝕋). Then for (m, n) ∈ ℤ^{2}

k̂_{ϕ}^{+}(m, n) = \{ \begin{matrix}\frac{1}{2} ϕ̂(\frac{m}{2}), & n = 0, \\ \frac{i}{πn} ϕ̂(\frac{m+n}{2}), & m, n are odd, \\ 0, & otherwise. \end{matrix}

Proof. Let us first observe that for any ζ ∈ 𝕋 and n ∈ ℤ we have

∫_{\{τ∈𝕋:ζ\overline{τ}∈𝕋_{+}\}} τ^{−n} dm(τ) = \{ \begin{matrix}\frac{1}{2}, & n = 0, \\ \frac{i}{πn}ζ^{−n}, & n is odd, \\ 0, & otherwise. \end{matrix}

It follows that

∫_{𝕋} k_{ϕ}^{+}(ζ, τ)ζ^{−m}τ^{−n} dm(ζ) = ∫_{\{τ∈𝕋:ζ\overline{τ}∈𝕋_{+}\}} ϕ(ζ^{2})ζ^{−m}τ^{−n} dm(τ)

= \{ \begin{matrix}\frac{1}{2}ϕ(ζ^{2})ζ^{−m}, & n = 0, \\ \frac{i}{πn}ϕ(ζ^{2})ζ^{−m−n}, & n is odd, \\ 0, & otherwise. \end{matrix}

It remains to integrate the last identity in ζ. ■ Corollary 14.6. Let ϕ ∈ L^{1}(𝕋). Then

k̂_{ϕ}^{−}(m, n) = \{ \begin{matrix}\frac{1}{2} ϕ̂(\frac{n}{2}), & m = 0, \\ \frac{i}{πm} ϕ̂(\frac{m+n}{2}), & m, n are odd, \\ 0, & otherwise. \end{matrix}

Proof. It suffices to observe that k_{ϕ}^{−}(ζ, τ) = k_{ϕ}^{+}(τ, ζ). ■ Corollary 14.7. Let ϕ ∈ L^{1}(𝕋). Then

k̂_{ϕ}(m, n) = \{ \begin{matrix}\frac{1}{2} ϕ̂ (\frac{m}{2}) , & n = 0, \\ \frac{1}{2} ϕ̂(\frac{n}{2}), & m = 0, \\ ϕ̂(0), & m = n = 0, \\ \frac{i}{π} (\frac{1}{m} + \frac{1}{n}) ϕ̂ (\frac{m+n}{2}) , & m, n are odd, \\ 0, & otherwise. \end{matrix}

Proof. It suffices to observe that k_{ϕ}(ζ, τ) = k_{ϕ}^{+}(ζ, τ) + k_{ϕ}^{+}(τ, ζ). ■ Theorem 14.8. Let p > 0. Suppose that φ(t) = ∑_{k∈ℤ}a_{k}e^{2πikt} and ∑_{k∈ℤ}|a_{k}|^{2} < ∞.

Then Q_{φ}^{[0,2]} ∈ S_{p} if and only if the matrix

\{a_{m+n+1} (\frac{1}{m + \frac{1}{2}} + \frac{1}{n + \frac{1}{2}} )\}_{m,n∈ℤ}  (14.5)

belongs to S_{p}.

Here we identify operators on ℓ^{2}(ℤ) with their matrices with respect to the standard orthonormal basis of ℓ^{2}(ℤ).

Proof. Consider the function ϕ on 𝕋 defined by ϕ(z) = ∑_{n∈ℤ}a_{n}z^{n}. By Theorem

14.4, Q_{φ}^{[0,2]} ∈ S_{p} if and only if K_{ϕ} ∈ S_{p} . It easy to see that the operator K_{ϕ} belongs to S_{p} if and only if the matrix {k̂_{ϕ}(m, n)}_{m,n∈ℤ} belongs to S_{p}. Corollary 14.7 implies that k̂_{ϕ}(m, n) ≠ 0 only if mn is odd or mn = 0. Hence, it is easy to check that {k̂_{ϕ}(m, n)}_{m,n∈ℤ} ∈ S_{p} if and only if {k̂_{ϕ}(2m + 1, 2n + 1)}_{m,n∈ℤ} ∈ S_{p}. It remains to note that

k̂_{ϕ}(2m + 1, 2n + 1) = \frac{i}{π} (\frac{1}{2m + 1} + \frac{1}{2n + 1}) a_{m+n+1}

by Corollary 14.7. ■

Clearly, the same reasoning shows that Q_{φ}^{[0,2]} is bounded on L^{2}[0, 1] if and only if the matrix (14.5) is bounded. The following result shows that the boundedness of (14.5) is equivalent to its membership of S_{p}, p > 1.

Theorem 14.9. Let {a_{k}}_{k∈ℤ} be a two-sided sequence of complex numbers and let

A =^{def} \{a_{m+n+1} (\frac{1}{m + \frac{1}{2}} + \frac{1}{n + \frac{1}{2}} )\}_{m,n∈ℤ} .

Suppose that p > 1. The following are equivalent:

(i) A is a bounded operator on ℓ^{2}(ℤ); (ii) A ∈ S_{p}; (iii) {a_{k}}_{k∈ℤ} ∈ ℓ^{2}(ℤ).

Proof. Suppose that A is bounded. Then the sequence

\{a_{n+1} (2 + \frac{1}{n + \frac{1}{2}} )\}_{n∈ℤ}

belongs to ℓ^{2}(ℤ) which implies (iii). Clearly, (iii) is equivalent to the fact that φ ∈ L^{2}[0, 1]. By Theorem 3.3, Q_{φ}^{[0,2]} ∈ S_{p}, and so by Theorem 14.8, A ∈ S_{p}. The implication (ii)⇒(i) is trivial. ■

Related results, that matrices of a roughly similar sort are bounded if and only if they are in certain S_{p} can be found in [W1].

Remark. Note that the following identities hold:

\frac{i}{π} ∑_{m,n∈ℤ}a_{m+n+1} (\frac{1}{m + \frac{1}{2}} + \frac{1}{n + \frac{1}{2}} ) ζ^{m}τ^{n} = \frac{ϕ(ζ) − ϕ(τ)}{\sqrt{ζτ}},

\frac{i}{π} ∑_{m,n∈ℤ} \frac{a_{m+n+1}}{n + \frac{1}{2}} ζ^{m}τ^{n} = \frac{ϕ(ζ)}{\sqrt{ζτ}},

\frac{i}{π} ∑_{m,n∈ℤ} \frac{a_{m+n+1}}{m + \frac{1}{2}} ζ^{m}τ^{n} = −\frac{ϕ(τ)}{\sqrt{ζτ}},

where \sqrt{ζτ} is chosen so that \overline{τ}\sqrt{ζτ} ∈ 𝕋_{+} (the series converge in L^{2}(𝕋^{2})). Indeed, it suffices to note that by Corollary 14.7,

\frac{i}{π} ∑_{m,n∈ℤ}a_{m+n+1} (\frac{1}{2m + 1} + \frac{1}{2n + 1}) ζ^{2m+1}τ^{2n+1} = \frac{1}{2}(k_{ϕ}(ζ, τ) − k_{ϕ}(ζ, −τ)),

by Lemma 14.5,

\frac{i}{π} ∑_{m,n∈ℤ} \frac{a_{m+n+1}}{2n + 1} ζ^{2m+1}τ^{2n+1} = \frac{1}{2}(k_{ϕ}^{+}(ζ, τ) − k_{ϕ}^{+}(ζ, −τ)),

and by Corollary 14.6,

\frac{i}{π} ∑_{m,n∈ℤ} \frac{a_{m+n+1}}{2m + 1}ζ^{2m+1}τ^{2n+1} = \frac{1}{2}(k_{ϕ}^{−}(ζ, τ) − k_{ϕ}^{−}(ζ, −τ)).

Remark. Note that if p > 1/2, then

A =^{def} \{a_{m+n+1} (\frac{1}{m + \frac{1}{2}} + \frac{1}{n + \frac{1}{2}} )\}_{m,n∈ℤ} ∈ S_{p}

if and only if

B =^{def} \{a_{m+n} (\frac{1}{m + \frac{1}{2}} + \frac{1}{n + \frac{1}{2}} )\}_{m,n∈ℤ} ∈ S_{p}.  (14.6)

Indeed, put ψ(t) =^{def} e^{2πit}φ(t). By Theorem 14.8, it suffices to prove that Q_{φ}^{[0,2]} ∈ S_{p} implies Q_{ψ}^{[0,2]} ∈ S_{p}. This follows from Theorem 7.3.

Note however that for p ≤ 1/2 this is not true. Indeed, if a_{0} = 1 and a_{n} = 0 for n ≠ 0 (in other words, φ(t) = 1, t ∈ ℝ), then it is easy to see that A is the zero matrix, and so it belongs to S_{p} for any p > 0. On the other hand, the matrix B has nonzero entries −(n^{2} − 1/4)^{−1} for m = −n, and so it belongs to S_{p} only for p > 1/2. The situation is similar in the case where the restriction φ|[0, 1) is the characteristic function of an interval in which case A ∈ S_{p} for any p > 0 but B ∈ S_{p} only for p > 1/2, see Theorem 8.11 and Corollary 9.11.

Suppose now that p > 1/2 and consider the following submatrices of the matrix B defined by (14.6):

B_{1} = \{a_{m+n} (\frac{1}{m + \frac{1}{2}} + \frac{1}{n + \frac{1}{2}} )\}_{m,n≥0} ,

B_{2} = \{a_{m+n} (\frac{1}{m + \frac{1}{2}} + \frac{1}{n + \frac{1}{2}} )\}_{m≥0, n<0} ,

B_{3} = \{a_{m+n} (\frac{1}{m + \frac{1}{2}} + \frac{1}{n + \frac{1}{2}} )\}_{m<0, n≥0} ,

and

B_{4} = \{a_{m+n} (\frac{1}{m + \frac{1}{2}} + \frac{1}{n + \frac{1}{2}} )\}_{m,n<0} .

Clearly, B ∈ S_{p} if and only if all matrices B_{j}, 1 ≤ j ≤ 4, belong to S_{p}.

It is direct that B_{1} ∈ S_{p} if and only if the matrix

\{a_{j+k} \frac{j + k + 1}{(j + 1)(k + 1)}\}_{j,k≥0}  (14.7)

belongs to S_{p}. Matrices of the form

{a_{j+k}(1 + j)^{α}(1 + k)^{β}}_{j,k≥0}  (14.8)

are called weighted Hankel matrices. It was proved in [Pel2] that if α > −1/2, β > −1/2, and 0 < p ≤ 1, the matrix (14.8) belongs to S_{p} if and only if the function ∑_{n≥0} a_{n}z^{n} belongs to the Besov class B_{p p}^{1/p+α+β} of functions on the unit

circle 𝕋. More recent results on Schatten class properties of weighted Hankel matrices are in [RW] and [W2]. However, in the case of interest, the weighted Hankel matrix (14.7) for α = β = −1, no characterization of such matrices of class S_{p} is known. In the next section we obtain some necessary conditions for the matrix (14.7) to belong to S_{1}.

It is also easy to see that B_{4} ∈ S_{p} if and only if the weighted Hankel matrix

\{a_{−(j+k+2)} \frac{j + k + 1}{(j + 1)(k + 1)}\}_{j,k≥0}  (14.9)

belongs to S_{p}.

It can also be easily shown that B_{2} ∈ S_{p} if and only if B_{3} ∈ S_{p} and this is equivalent to the fact that the weighted Toeplitz matrix

\{a_{j−k−1} \frac{j − k}{(1 + j)(1 + k)}\}_{j,k≥0}  (14.10)

belongs to S_{p}.

Summarizing the above, we can state the following result.

Theorem 14.10. Let 1/2 < p ≤ 1 and let φ be a function in L^{2}[0, 1] of the form φ(t) = ∑_{n∈ℤ}a_{n}e^{2πint}, t ∈ [0, 1]. Then Q_{φ}^{[0,1]} ∈ S_{p} if and only if the matrices

(14.7), (14.9), and (14.10) belong to S_{p}. ■

In the next section we use the results above to obtain necessary conditions for the nuclearity of operators Q_{φ}.

Let us consider now the family of functions {F_{λ}}_{λ∈ℂ} on [0, 1)^{2} defined for a function φ ∈ L^{1}[0, 1] by

F_{λ}(t, s) =^{def} φ (max{s, t}) e^{−2πλi|s−t|} + φ (min{s, t}) e^{2πλi|s−t|−2πλi}.

Clearly, F_{λ} ∈ L^{1}([0, 1)^{2}). We identify [0, 1)^{2} with 𝕋^{2} via the map (s, t) ↦ (e^{2πis}, e^{2πit}) and we can consider the Fourier coefficients of functions on [0, 1)^{2}.

Theorem 14.11. Suppose that λ ∉ ℤ. Then

F̂_{λ}(m, n) = \frac{1 − e^{−2πiλ}}{2πi} (\frac{1}{λ − m} + \frac{1}{λ − n}) φ̂(m + n).  (14.11)

Proof. We have

F̂_{λ}(m, n) =^{def} ∫∫_{[0,1)×[0,1)} F_{λ}(s, t)e^{−2πims−2πint} dsdt = ∫∫_{t≥s} + ∫∫_{t≤s} .

Let us compute the first integral:

∫∫_{t≥s} = ∫_{0}^{1} (∫_{0}^{t} φ(t)e^{2πiλ(s−t)}e^{−2πimt−2πins} ds) dt

+ ∫_{0}^{1} (∫_{s}^{1} φ(s)e^{2πiλ(t−s)−2πiλ}e^{−2πimt−2πins} dt) ds

= ∫_{0}^{1} φ(t)e^{−2πiλt−2πimt} (∫_{0}^{t} e^{2πis(λ−n)} ds) dt

+ ∫_{0}^{1} φ(s)e^{−2πiλs−2πiλ−2πins} (∫_{s}^{1} e^{2πit(λ−m)} dt) ds

= ∫_{0}^{1} φ(t)e^{−2πiλt−2πimt} \frac{e^{2πit(λ−n)} − 1}{2πi(λ − n)}dt

+ ∫_{0}^{1} φ(s)e^{−2πiλs−2πiλ−2πins} \frac{e^{2πiλ} − e^{2πis(λ−m)}}{2πi(λ − m)} ds

= \frac{φ̂(m + n)}{2πi(λ − n)} − \frac{1}{2πi(λ − n)} ∫_{0}^{1} φ(t)e^{−2πiλt−2πimt} dt

+ \frac{1}{2πi(λ − m)} ∫_{0}^{1} φ(s)e^{−2πiλs−2πins} ds − \frac{e^{−2πiλ}φ̂(m + n)}{2πi(λ − m)}.

Similarly,

∫∫_{t≤s} = \frac{φ̂(m + n)}{2πi(λ − m)} − \frac{1}{2πi(λ − m)} ∫_{0}^{1} φ(s)e^{−2πiλs−2πins} ds

+ \frac{1}{2πi(λ − n)} ∫_{0}^{1} φ(t)e^{−2πiλt−2πimt} dt − \frac{e^{−2πiλ}φ̂(m + n)}{2πi(λ − n)}

which implies (14.11). ■

Theorem 14.12. Let λ ∈ ℂ. The integral operator with kernel function F_{λ} is bounded on L^{2}[0, 1] if and only if φ ∈ L^{2}([0, 1)).

Proof. Clearly, the integral operator with kernel function F_{λ} belongs to S_{2} if φ ∈ L^{2}([0, 1)). Suppose now that the integral operator with kernel function F_{λ} is bounded. If λ ∈ ℤ, then

F_{λ}(s, t) = (φ(max{s, t})e^{−4πiλ max{s,t}} + φ(min{s, t})e^{−4πiλ min{s,t}}) e^{2πiλ(s+t)}
= (φ(s)e^{−4πiλs} + φ(t)e^{−4πiλt}) e^{2πiλ(s+t)}.

Consequently, the boundedness of this operator implies that φ ∈ L^{2}[0, 1]. If λ ∉ ℤ, then the boundedness of the integral operator with kernel function F_{λ} implies that

∑_{m∈ℤ} |F̂_{λ}(m, 0)|^{2} < +∞

and by Theorem 14.11 we obtain

∑_{m∈ℤ}|φ̂(m)|^{2} |\frac{2λ − m}{λ − m}|^{2} < +∞,

whence φ ∈ L^{2}[0, 1]. ■

Lemma 14.13. Let w, a ∈ ℂ and w ≠ 1. Let p > \frac{1}{2}. Then the function

(s, t) ↦ (w − e^{a|s−t|})^{−1}χ_{Δ}(s)χ_{Δ}(t), s, t ∈ ℝ,

is a Schur multiplier of S_{p} if Δ is an interval of sufficiently small length.

Proof. Clearly, it suffices to consider the case p < 1. Note that w − e^{a|s−t|} is a Schur multiplier of S_{p}(L^{2}(Δ)) by Theorem 7.3, since

w − e^{a|s−t|} = w − e^{a max{s,t}}e^{−a min{s,t}}.

We have to prove that this multiplier is an isomorphism of S_{p}(L^{2}(Δ)) if Δ has sufficiently small length. For ω ∈ L^{∞}(Δ^{2}) we put

∥ω∥_{𝔐_{p}(Δ)} =^{def} sup ∥ωk∥_{S_{p}(L^{2}(Δ))},

where the supremum is taken over all integral operators with kernel k ∈ L^{2}(Δ^{2}) such that ∥k∥_{S_{p}} = 1. Here by ∥k∥_{S_{p}} we mean the S_{p} norm (quasi-norm if p < 1) of the integral operator with kernel function k. Obviously, it suffices to prove the inequality

∥e^{a max{s,t}}e^{−a min{s,t}} − 1∥_{𝔐_{p}(Δ)} < |w − 1|

provided the length of Δ is sufficiently small. Theorem 7.7 implies that, for any x_{0} ∈ Δ,

lim_{|Δ|→0} ∥e^{a(max{s,t}−x_{0})} − 1∥_{𝔐_{p}(Δ)} = 0 and lim_{|Δ|→0} ∥e^{−a(min{s,t}−x_{0})} − 1∥_{𝔐_{p}(Δ)} = 0.

Hence, the desired inequality is obvious. ■

Theorem 14.14. Suppose that λ ∉ ℤ and p > 1/2. Then Q_{φ}^{[0,2]} ∈ S_{p} if and only if the integral operator with kernel function F_{λ} belongs to S_{p}.

Proof. Suppose that Q_{φ}^{[0,2]} ∈ S_{p}. Then the integral operators with kernel functions φ(max{s, t}) and φ(min{s, t}) belong to S_{p}(L^{2}[0, 1]) (see Theorem 14.1). Note that e^{2πiλ|s−t|} = e^{2πiλ(2 max{s,t}−s−t)}. It follows now from Theorem 7.3 that the integral operator with kernel function F_{λ} belongs to S_{p}.

Suppose now that the integral operator with kernel function F_{λ} belongs to S_{p}. We have to prove that Q_{φ}^{[0,2]} ∈ S_{p}. By Theorem 14.12, φ ∈ L^{2}[0, 1], and so it suffices to show that Q_{φ}^{[0,1]} ∈ S_{p}. By Lemma 14.13, we can choose a positive number δ such that the function (e^{2πiλ} − e^{4πiλ|s−t|})^{−1} belongs to 𝔐_{p}(Δ) for any interval Δ of length less than δ. We can represent the interval [0, 1) in the form ⋃_{j=1}^{N} Δ_{j}, where

the Δ_{j} are pairwise disjoint intervals with lengths less than δ. Clearly,

F_{λ}(s, t) = φ(max{s, t})e^{−2πλi|s−t|} + (φ(s) + φ(t) − φ(max{s, t})e^{2πλi|s−t|−2πλi}.

Let s, t ∈ Δ_{j}. Then

φ(max\{s, t\}) = \frac{F_{λ}(s, t) − (φ(s) + φ(t))e^{2πλi|s−t|−2πλi}}{e^{−2πλi|s−t|} − e^{2πλi|s−t|−2πλi}} =

\frac{(F_{λ}(s, t) − (φ(s) + φ(t))e^{2πλi(max\{s,t\}−min\{s,t\}−1})}{e^{2πλi} − e^{4πλi|s−t|}}e^{2πλi(1+max\{s,t\}−min\{s,t\})}.

Theorem 7.3 and Lemma 14.13 imply that the integral operator with kernel function

(s, t) ↦ φ(max{t, s})χ_{Δ_{j}}(t)χ_{Δ_{j}}(s), s, τ ∈ ℝ,

belongs to S_{p}. To complete the proof, it remains to observe that the kernel function

(s, t) ↦ φ(max{s, t}) − ∑_{j=1}^{N} φ(max{s, t})χ_{Δ_{j}}(s)χ_{Δ_{j}}(t)

determines a finite rank operator. ■

Theorem 14.14 implies that if p > 1/2, λ_{1}, λ_{2} ∉ ℤ, and F_{λ_{1}} ∈ S_{p}, then F_{λ_{2}} ∈ S_{p}. This can also be easily deduced from the following elementary fact: if x ∈ ℓ^{2}(ℤ) and y ∈ ℓ^{p}(ℤ) with p ≤ 2, then {x_{m+n} y_{n}}_{m,n∈ℤ} ∈ S_{p}.

15. Necessary conditions for Q_{φ} ∈ S_{1}

In this section we obtain various necessary conditions for Q_{φ} ∈ S_{1}.

Theorem 15.1. Let {a_{n}}_{n≥0} be a sequence in ℓ^{2}. If the matrix

Γ = \{a_{j+k} (\frac{1}{j + \frac{1}{2}} + \frac{1}{k + \frac{1}{2}} )\}_{j,k≥0}

belongs to S_{1}, then the function ∑_{n≥0} log(2 + n)a_{n}z^{n} belongs to the Hardy class H^{1}.

We need the following well-known lemma (see e.g., [Pel1]).

Lemma 15.2. Suppose that the matrix {a_{jk}}_{j,k≥0} belongs to S_{1}. Then the func-

tion ∑_{n≥0} (∑_{j=0}^{n} a_{j n−j}) z^{n} belongs to the Hardy class H^{1}.

Proof. It is sufficient to prove this when the matrix has rank one in which case the result is an immediate consequence of the fact that H^{2} · H^{2} ⊂ H^{1}. ■

Lemma 15.3. Let m ∈ ℤ_{+} and let

β_{n} =^{def} ∑_{j=0}^{n} \frac{1}{j + \frac{1}{2}} .  (15.1)

Then there exists d ∈ ℝ such that

|β_{n} − log(2 + n) − d| ≤ const \frac{1}{1 + n}.

Proof. We use the following well known fact (see, for example, [Z, Ch. I, (8.9)])

|∑_{j=1}^{n} \frac{1}{j} − log n − γ| ≤ const ·n^{−1},  (15.2)

where γ is the Euler constant. We have

∑_{j=0}^{n} \frac{1}{j + \frac{1}{2}} = 2 ∑_{j=0}^{n} \frac{1}{2j + 1} = 2 ∑_{j=1}^{2n+1} \frac{1}{j} − ∑_{j=1}^{n} \frac{1}{j},

and so by (15.2),

|∑_{j=0}^{n} \frac{1}{j + \frac{1}{2}} − 2 log(2n + 1) + log n − γ| ≤ const ·n^{−1}

which implies the result. ■

Proof of Theorem 15.1. By Lemma 15.2, we have

∑_{m≥0} a_{m} (∑_{j=0}^{m} (\frac{1}{j + \frac{1}{2}} + \frac{1}{(m − j + \frac{1}{2})})) z^{m} = 2 ∑_{m≥0} a_{m} (∑_{j=0}^{m} \frac{1}{j + \frac{1}{2}} ) z^{m} ∈ H^{1}.

Since {a_{n}}_{n≥0} ∈ ℓ^{2}, it is not hard to check that Lemma 15.3 implies that

∑_{n≥0} log(2 + n)a_{n}z^{n} ∈ H^{1}. ■

Theorem 15.4. Let φ ∈ L^{2}[0, b] and φ(x) = ∑_{n∈ℤ}a_{n}e^{2πinx/b}. If Q_{φ}^{[0,b]} ∈ S_{1}, then

the functions ∑_{n≥0} a_{n} log(2 + n)z^{n} and ∑_{n≥0} a_{−n} log(2 + n)z^{n} in the unit disc 𝔻 belong

to the Hardy class H^{1}.

Proof. Without loss of generality we may assume that b = 1. By Theorem 14.8, the matrices

Γ = \{a_{j+k+1} (\frac{1}{j + \frac{1}{2}} + \frac{1}{k + \frac{1}{2}} )\}_{j,k≥0}

74

and

Γ = \{a_{−(j+k+1)} (\frac{1}{j + \frac{1}{2}} + \frac{1}{k + \frac{1}{2}} )\}_{j,k≥0}

belong to S_{1}. The result follows now from Theorem 15.1. ■

Theorem 15.5. Let I be a compact interval in (0, ∞) and let φ be a function in L_{loc}^{1}(ℝ_{+}) such that Q_{φ} ∈ S_{1}. If

a_{n} = ∫_{I} φ(x)e^{−2πinx/|I|}dx, n ∈ ℤ,

then the functions ∑_{n≥0} a_{n} log(2 + n)z^{n} and ∑_{n≥0} a_{−n} log(2 + n)z^{n} belong to the Hardy

class H^{1}.

Proof. Since I is separated away from 0, it follows that φ|I ∈ L^{2}(I). We can now apply a translation and reduce the result to Theorem 15.4. ■

Corollary 15.6. Under the hypotheses of either Theorem 15.4 or 15.5 the following holds:

(i) |a_{n}| ≤ const(log(2 + |n|))^{−1}, n ∈ ℤ;

(ii) suppose that {n_{k}}_{k≥0} is an Hadamard lacunary sequence of positive integers,

i.e.,

inf_{k≥0} \frac{n_{k+1}}{n_{k}} > 1,

then

∑_{k≥0} |a_{n_{k}}|^{2}(log(1 + n_{k}^{2}))^{2} < ∞ and ∑_{k≥0} |a_{−n_{k}}|^{2}(log(1 + n_{k}^{2}))^{2} < ∞.

Proof. (i) follows immediately from Theorem 15.5 and the obvious fact that the Fourier coefficients of an H^{1} function are bounded. Finally, (ii) is an immediate consequence of Theorem 15.5 and Paley’s inequality (see [Z, v. 2, Ch. XII, (7.8)]). ■

Note that if I is a compact interval in (0, ∞), the restrictions of function in X_{1} to I fill the space L^{2}(I), and so the sequence of Fourier coefficients {a_{n}}_{n∈ℤ} can be an arbitrary sequence in ℓ^{2}. Thus Corollary 15.6 also shows that the condition φ ∈ X_{1} is not sufficient for Q_{φ} ∈ S_{1}.

Now we are going to use Theorem 13.12 to obtain another necessary condition for Q_{φ} ∈ S_{1}. We denote by ℌ^{1} the Stein–Weiss space of functions f in L^{1}(ℝ) such that F^{−1}(χ_{ℝ_{+}}Ff) ∈ L^{1}(ℝ), where F is Fourier transformation.

Theorem 15.7. Let h ∈ L_{loc}^{2}(ℝ). Suppose that the integral operator on L^{2}(ℝ) with kernel function

(x, y) ↦ h^{♠}(x, y) =^{def} h(x + y) (\frac{1}{x + i} + \frac{1}{y − i}) , x, y ∈ ℝ,

belongs to S_{1}. Then the Fourier transform of the function h(x) log(1 + x^{2}) belongs to the Stein–Weiss space ℌ^{1}.

Proof. Clearly, the integral operator with kernel function h^{♠}χ_{[0,+∞)^{2}} belongs to S_{1}. Put

g(x) =^{def} ∫_{ℝ} h^{♠}(t, x − t)χ_{[0,+∞)^{2}}(t, x − t)dt.

We have

g(x) = \{ \begin{matrix}h(x) ∫_{0}^{x} (\frac{1}{t+i} + \frac{1}{x−t−i}) dt = h(x) log(1 + x^{2}), & x > 0, \\ 0, & x < 0. \end{matrix}

It follows from Theorem 6.3 that Fg ∈ L^{1}(ℝ). In the same way it can be shown that the Fourier transform of the function h(x) log(1+x^{2})χ_{ℝ_{−}}(x) belongs to L^{1}(ℝ). This implies the result. ■

Corollary 15.8. Let h ∈ L_{loc}^{2}(ℝ) and let a, b ∈ ℂ \ ℝ such that a + b ∉ ℝ. Suppose that the integral operator on L^{2}(ℝ) with kernel function

h_{a,b}^{♠}(x, y) =^{def} h(x + y) (\frac{1}{x + a} + \frac{1}{y + b})

belongs to S_{1}. Then the Fourier transform of the function h(x + c) log(1 + x^{2}) belongs to ℌ^{1} for any c ∈ ℝ.

Proof. Clearly, the integral operators on L^{2}(ℝ) with kernels functions h(x + y + c) (\frac{1}{x+a+c} + \frac{1}{y+b}) belong to S_{1}. Consequently, by Theorem 13.19, the

integral operator on L^{2}(ℝ) with kernel function h(x + y + c) (\frac{1}{x+i} + \frac{1}{y−i}) belongs to S_{1}. It remains to apply Theorem 15.7. ■

Corollary 15.9. Suppose that h, a and b satisfy the hypotheses of Corollary 15.8. Then h(x) log |x| → 0 as |x| → ∞. ■

Now Corollary 15.9 and Theorem 13.12 imply the following theorem

Theorem 15.10. Let φ ∈ L_{loc}^{2}(ℝ_{+}). Suppose that Q_{φ} ∈ S_{1}. Then

log |x| ∫_{ℝ_{+}} φ(t)t^{xi}dt → 0 as |x| → ∞. ■  (15.3)

Note that it follows from Theorem 6.2 that φ ∈ L^{1}(ℝ_{+}), and so the integral in (15.3) is well defined. It is easy to see that if φ is an arbitrary L^{2} function supported on a compact subset of (0, ∞), then φ ∈ X_{1}. However, φ does not have to satisfy (15.3), and so Theorem 15.10 also implies that the condition φ ∈ X_{1} is not sufficient for Q_{φ} ∈ S_{1}.

We conclude this section with necessary conditions on the L^{1} modulus of continuity of symbols. If f is a function on 𝕋, then its L^{1} modulus of continuity ω_{f}^{(1)} is defined by, in analogy with (10.2),

ω_{f}^{(1)}(t) =^{def} sup_{ζ∈𝕋, |1−ζ|<t} ∫_{𝕋} |f(ζτ) − f(τ)| dm(τ), t > 0.

The following result is possibly known to experts. We were not able to find a reference, and we prove it here.

Theorem 15.11. Let f ∈ L^{1}(𝕋) and let

g(z) = ∑_{n∈ℤ} \frac{f̂(n)}{log(|n| + 2)}z^{n}.

Then g ∈ L^{1}(𝕋) and

lim_{t→0} ω_{g}^{(1)}(t) log \frac{1}{t} = 0.  (15.4)

Consider the function h on 𝕋 defined by

h(z) =^{def} ∑_{n∈ℤ}(log(|n| + 2))^{−1} z^{n}.

It is well-known (see, for example, [Z, Ch. V, (1.5)]) that the series converges for z ∈ 𝕋 \ {1}, h ≥ 0 and h ∈ L^{1}(𝕋). We define the function h on ℝ by

h(x) =^{def} h(e^{ix}) = (log 2)^{−1} + 2 ∑_{n≥1} (log(n + 2))^{−1} cos nx.  (15.5)

Then h is continuously differentiable on ℝ \ 2πℤ, see [Z, Ch. V, Miscellaneous theorems and examples, 7].

We use the following notation. Let φ and ψ be nonvanishing functions on an interval (0, α). We write

φ ∼^{0} ψ, if lim_{x→0} \frac{φ(x)}{ψ(x)} = 1.

Lemma 15.12. Let h be the function defined by (15.5). Then

h(x) ∼^{0} \frac{π}{x(log x)^{2}}  (15.6)

and

h′(x) ∼^{0} −\frac{π}{x^{2}(log x)^{2}} .  (15.7)

Proof. (15.6) is proved in [Z, Ch. V, (2.17)]. Let us prove (15.7). Using Abel’s transformation, we obtain

h(x) = ∑_{n≥0} ((log(n + 2))^{−1} − (log(n + 3))^{−1}) \frac{sin(n + \frac{x}{2})}{sin \frac{x}{2}}

= cot \frac{x}{2} (∑_{n≥0} ((log(n + 2))^{−1} − (log(n + 3))^{−1}) sin nx)

+ ∑_{n≥0} ((log(n + 2))^{−1} − (log(n + 3))^{−1}) cos nx.

Consequently,

h′(x) = −\frac{1}{2 sin^{2} \frac{x}{2}} (∑_{n≥0} ((log(n + 2))^{−1} − (log(n + 3))^{−1}) sin nx)

+ cot \frac{x}{2} (∑_{n≥0} ((log(n + 2))^{−1} − (log(n + 3))^{−1}) n cos nx)

− ∑_{n≥0} ((log(n + 2))^{−1} − (log(n + 3))^{−1}) n sin nx =^{def} Σ_{1} + Σ_{2} + Σ_{3}.

It remains to observe that

Σ_{1} ∼^{0} −\frac{π}{x^{2}(log x)^{2}}

by [Z, Ch. V, (2.13)], while

Σ_{2} ∼^{0} −\frac{2π}{x^{2}(log x)^{3}} ,

and

Σ_{3} ∼^{0} −\frac{1}{x(log x)^{2}}

by [Z, Ch. V, (2.18)]. ■

Corollary 15.13. The following inequality holds

∫_{𝕋} |h(ζτ) − h(τ)|dm(τ) ≤ C (log \frac{3}{|ζ − 1|})^{−1}

for any ζ ∈ 𝕋.

Proof. It suffices to prove that

∫_{−π}^{π} |h(x + t) − h(x)| dx ≤ C (log \frac{1}{t})^{−1}

for sufficiently small positive t. We have

∫_{−π}^{π} |h(x + t) − h(x)| dx = ∫_{|x|≤2t} |h(x + t) − h(x)| dx

+ ∫_{2t≤|x|≤π} |h(x + t) − h(x)| dx

≤ 2 ∫_{|x|≤3t} |h(x)|dx + ∫_{2t≤|x|≤π} |h(x + t) − h(x)| dx

=^{def} 2I_{1} + I_{2}.

Using Lemma 15.12, we obtain

I_{1} ≤ C ∫_{0}^{3t} \frac{1}{x(log x)^{2}} dx ≤ C (log \frac{1}{t})^{−1}

and

I_{2} ≤ Ct ∫_{2t}^{π} \frac{dx}{x^{2}(log \frac{x}{10})^{2}} ≤ C (log \frac{1}{t})^{−2} ,

if t > 0 is sufficiently small. ■

Proof of Theorem 15.11. Note that g = f ∗ h. Consequently, g ∈ L^{1}(𝕋). It follows easily from Corollary 15.13 that

ω_{g}^{(1)}(t) ≤ const ∥f∥_{L^{1}} (log \frac{3}{t})^{−1} , 0 < t < 2.

The result follows now from the obvious fact that (15.4) holds for trigonometric polynomials f. ■

For a function f ∈ L^{1}(ℝ), we defined the L^{1} modulus of continuity ω_{f}^{(1)} in (10.2):

ω_{f}^{(1)}(t) = sup_{|s|≤t} ∫_{ℝ} |f(x + s) − f(x)|dx, t > 0.

In fact this definition can be extended to functions f not necessarily in L^{1}(ℝ). It is sufficient to assume that

∫_{ℝ} |f(x + s) − f(x)|dx < ∞, s ∈ ℝ.

In a similar way we can prove the following analog of Theorem 15.11.

Theorem 15.14. Let f ∈ L^{1}(ℝ). Then there exists a function g ∈ L^{1}(ℝ) such that

(Ff)(x) = (Fg(x)) log(|x| + 2), x ∈ ℝ,

and

lim_{t→0} ω_{g}^{(1)}(t) log \frac{1}{t} = 0

Proof. Indeed, let

𝔥(x) =^{def} ∫_{ℝ} (log(2 + |t|))^{−1} e^{−2πitx} dt = 2 ∫_{0}^{∞}(log(2 + t))^{−1} cos(2πtx) dt.

Then 𝔥 is an even positive continuously differentiable function on ℝ \ {0}. We can repeat the above reasoning to prove that

𝔥(x) ∼^{0} \frac{1}{2x(log x)^{2}}

and

𝔥′(x) ∼^{0} −\frac{1}{2x^{2}(log x)^{2}} .

Moreover, |𝔥(x)| ≤ const · x^{−2} and |𝔥′(x)| ≤ const · x^{−2} everywhere. These estimates allow us to obtain the inequality

∫_{ℝ} |𝔥(x + t) − 𝔥(x)| dx ≤ const (log \frac{1}{t})^{−1}

for t ∈ (0, \frac{1}{2}) and repeat the reasoning in the proof of Theorem 15.11. ■

Let us introduce some more notation. Set ℂ_{+} =^{def} {z ∈ ℂ : Im z > 0} and ℂ_{−} =^{def} {z ∈ ℂ : Im z < 0}. Let f be a function in L_{loc}^{1}(ℝ) such that

∫_{ℝ} \frac{|f(t)|}{1 + |t|} dt < +∞.

Consider the Cauchy transform of f defined by

(Cf)(ζ) =^{def} \frac{1}{2πi} ∫_{ℝ} \frac{f(t) dt}{t − ζ}, Im ζ ≠ 0.

It is well known that (Cf)|ℂ_{+} and (Cf)|ℂ_{−} are holomorphic functions of bounded characteristic in ℂ_{+} and ℂ_{−} respectively, and so they have finite angular boundary values almost everywhere on ℝ. Set

f_{+}(x) =^{def} lim_{y→0+}(Cf)(x + iy) and f_{−}(x) =^{def} − lim_{y→0−}(Cf)(x + iy).

By the Privalov theorem (see [Pr] for the case of a rectifiable Jordan curve), f = f_{+} + f_{−} almost everywhere on ℝ. If f ∈ L^{1}(ℝ) + L^{2}(ℝ), then

(Cf)(z) = ∫_{0}^{∞}(Ff)(t)e^{2πitz} dt, z ∈ ℂ_{+},

and

(Cf)(z) = − ∫_{−∞}^{0} (Ff)(t)e^{2πitz} dt, z ∈ ℂ_{−}.

Note that f_{+} does not have to be in L^{1}(ℝ) for an arbitrary function f ∈ L^{1}(ℝ). In fact, if f ∈ L^{1}(ℝ), then f_{+} ∈ L^{1}(ℝ) if and only if f belongs to the Stein–Weiss space ℌ^{1}.

Theorem 15.15. Let f ∈ ℌ^{1}. Suppose that there exists a function g ∈ L^{1}(ℝ) such that (Ff)(x) = (Fg)(x) log(1 + x^{2}) for all x ∈ ℝ. Then

lim_{t→0} ω_{g_{+}}^{(1)}(t) log \frac{1}{t} = 0

and

lim_{t→0} ω_{g_{−}}^{(1)}(t) log \frac{1}{t} = 0.

Note, however, that the assumptions of Theorem 15.15 do not imply that g_{+} ∈ L^{1}(ℝ) or g_{−} ∈ L^{1}(ℝ).

We need some auxiliary facts. Let M(ℝ) be the space of finite Borel measures on ℝ.

Lemma 15.16. Let f ∈ L^{1}(ℝ). Suppose that f′′ ∈ M(ℝ) (in the distributional sense). Then Ff ∈ L^{1}(ℝ) and

∥Ff∥_{L^{1}(ℝ)} ≤ C\sqrt{∥f∥_{L^{1}(ℝ)}∥f′′∥_{M(ℝ)}}.

Proof. The result follows from the obvious inequality:

|(Ff)(x)| ≤ min \{∥f∥_{L^{1}}, \frac{∥f′′∥_{M(ℝ)}}{4π^{2}x^{2}} \} , x ∈ ℝ. ■

Corollary 15.17. Let f ∈ L^{1}(ℝ). Suppose that supp Ff is bounded above. Then

ω_{f_{+}}^{(1)}(t) ≤ const · t, t > 0.

Proof. It suffices to construct a function g_{s} ∈ L^{1}(ℝ) such that ∥g_{s}∥_{L^{1}(ℝ)} ≤ C|s| and f_{+}(x+s)−f_{+}(x) = (f ∗g_{s})(x) for all x ∈ ℝ. Suppose that supp f ⊂ (−∞, M], where M > 0. We may take a function g_{s} such that

(Fg_{s})(t) = \begin{cases}0, & t ≤ 0, \\ e^{2πist} − 1, & t ∈ [0, M], \\ e^{2πis(2M−t)} − 1, & t ∈ [M, 2M], \\ 0, & t ≥ 2M. \end{cases}

Clearly, f_{+}(x+s)−f_{+}(x) = (f∗g_{s})(x) for all x ∈ ℝ. The inequality ∥g_{s}∥_{L^{1}(ℝ)} ≤ C|s| follows from Lemma 15.16 (with C depending on M). ■

Lemma 15.18. Set ρ(t) =^{def} 2 log(2 + |t|) − log(1 + t^{2}). Then Fρ ∈ L^{1}.

Proof. It suffices to observe that ρ is even, lim_{t→∞} ρ(t) = 0, ρ has two continuous derivatives on (0, ∞), and

∫_{0}^{∞}t|ρ′′(t)|dt < ∞;

this implies that ρ(x) = − ∫_{|x|}^{∞} ρ′(t) dt and

ρ(x) = ∫_{|x|}^{∞}(t − |x|)ρ′′(t) dt = ∫_{0}^{∞} ρ′′(t)(t − |x|)_{+} dt = ∫_{0}^{∞} ρ′′(t)tFK_{t}(x) dt,

where K_{t} is the Fejér kernel with ∥K_{t}∥_{L^{1}} = 1. ■

Lemma 15.19. Let φ be an even positive function in C^{2}(ℝ) such that φ(x) = log(1 + x^{2}) for sufficiently large |x|. Then F(φ^{−1}) ∈ L^{1}.

Proof. See the proof of the previous lemma. ■

Proof of Theorem 15.15. We prove the first equality (the proof of the second one is the same). Let ψ be a function in L^{1}(ℝ) such that supp Fψ is compact and supp Fψ = 1 in a neighborhood of 0. Then f = f ∗ ψ + (f − f ∗ ψ). The Fourier transform of the first summand has a compact support while the support of the Fourier transform of the second summand does not contain 0. Thus it is sufficient to consider two cases.

Case 1, supp Ff is compact. The result follows from Corollary 15.17.

Case 2, 0 ∉ supp Ff. Clearly,

(Ff_{+})(x) = (Fg_{+})(x) log(1 + x^{2}), x ∈ ℝ.

Let δ > 0 be such that Ff = 0 on [−δ, δ], and let φ be an even positive function in C^{2}(ℝ) such that φ(x) = log(1 + x^{2}) for |x| ≥ δ. By Lemma 15.19, 1/φ = FΦ for some Φ ∈ L^{1}(ℝ). Hence,

Fg_{+} = Ff_{+}/φ = F(f_{+} ∗ Φ),

which implies that g_{+} ∈ L^{1}(ℝ). Moreover, if ρ is as in Lemma 15.18, and thus ρ = FF with F ∈ L^{1}(ℝ), then

2(Fg_{+})(x) log(2 + |x|) = (Fg_{+})(x)ρ(x) + (Ff_{+})(x) = F(g_{+} ∗ F + f_{+})(x),

so (Fg_{+})(x) log(2 + |x|) is the Fourier transform of an L^{1}-function. It remains to apply Theorem 15.14. ■

Theorem 15.20. Let φ be a function in L^{2}[0, b] such that Q_{φ}^{[0,b]} ∈ S_{1} and let

φ(x) = ∑_{n∈ℤ}a_{n}e^{2πinx/b}. If

ϕ_{+}(ζ) =^{def} ∑_{n≥0} a_{n}ζ^{n} and ϕ_{−}(ζ) =^{def} ∑_{n<0} a_{n}ζ^{n},  (15.8)

then

lim_{t→0} ω_{ϕ_{+}}^{(1)}(t) log \frac{1}{t} = 0 and lim_{t→0} ω_{ϕ_{−}}^{(1)}(t) log \frac{1}{t} = 0.  (15.9)

Proof. The result follows immediately from Theorems 15.4 and 15.11. ■

Theorem 15.21. Let I be a compact interval in (0, ∞) and let φ be a function in L_{loc}^{1}(ℝ_{+}) such that Q_{φ} ∈ S_{1}. If

a_{n} = ∫_{I} φ(x)e^{−2πinx/|I|}dx, n ∈ ℤ,

and ϕ_{+} and ϕ_{−} are defined by (15.8), then (15.9) holds.

Proof. The result is an immediate consequence of Theorem 15.20. ■

Recall that for a function φ ∈ L_{loc}^{2}(ℝ_{+}) the function φ^{♡} is defined by (13.1). Note that if Q_{φ} ∈ S_{1}, then by Theorem 6.2, φ ∈ L^{1}(ℝ_{+}) and thus φ^{♡} ∈ L^{1}(ℝ).

Theorem 15.22. Let φ be a function in L_{loc}^{2}(ℝ_{+}) such that Q_{φ} ∈ S_{1}. Then

lim_{t→0} ω_{(φ^{♡})_{+}}^{(1)}(t) log \frac{1}{t} = 0  and  lim_{t→0} ω_{(φ^{♡})_{−}}^{(1)}(t) log \frac{1}{t} = 0.

In particular,

lim_{t→0} ω_{φ^{♡}}^{(1)}(t) log \frac{1}{t} = 0.

Proof. The result follows from Theorem 13.12, Corollary 15.8, and Theorem 15.15. ■

This result should be compared to Theorems 10.7 and 13.7. In particular, if φ has compact support in ℝ_{+}, we see that a Dini condition on the L^{2} modulus of continuity is sufficient for Q_{φ} ∈ S_{1}, while the slightly weaker condition lim_{t→0} ω_{φ}^{(1)}(t) log \frac{1}{t} = 0 on the L^{1} modulus of continuity is necessary.

Theorem 15.23. Let φ be a function in L_{loc}^{2}(ℝ_{+}) such that Q_{φ} ∈ S_{1}. Then

lim_{a→1} ∫_{ℝ_{+}} |φ(ax) − φ(x)| dx · log \frac{1}{|a − 1|} = 0.

Proof. By Theorem 15.22, we have

lim_{t→0} ∫_{ℝ} |φ(e^{2s+2t})e^{2s+2t} − φ(e^{2s})e^{2s}| ds · log \frac{1}{t} = 0.

Substituting e^{2s} = x and e^{2t} = a, we obtain

lim_{a→1} ∫_{ℝ_{+}} |aφ(ax) − φ(x)| dx · log \frac{1}{|a − 1|} = 0.

It remains to observe that by Theorem 6.2, φ ∈ L^{1}(ℝ_{+}) and obviously,

lim_{a→1} |a − 1| · log \frac{1}{|a − 1|} = 0. ■

16. Dilation of Symbols

Let φ be a function in L_{loc}^{2}(ℝ) such that φ(x + 1) = φ(x), x ∈ ℝ. For a > 0 we define the function φ_{a} on [0, 1] by φ_{a}(x) =^{def} φ(ax) for x ∈ [0, 1). We are going to obtain in this section upper and lower estimates for ∥Q_{φ_{a}}^{[0,1]}∥_{S_{p}}.

Note that we can extend φ_{a} to ℝ as a 1-periodic function on ℝ. Using an obvious estimate, see (14.1)–(14.3),

C_{1} (∥Q_{φ_{a}}^{[0,1]}∥_{S_{p}} + ∥φ_{a}∥_{L^{2}[0,1]}) ≤ ∥Q_{φ_{a}}^{[0,2]}∥_{S_{p}} ≤ C_{2} (∥Q_{φ_{a}}^{[0,1]}∥_{S_{p}} + ∥φ_{a}∥_{L^{2}[0,1]}) ,(16.1)

we can reduce the estimation of ∥Q_{φ_{a}}^{[0,1]}∥_{S_{p}} to that of ∥Q_{φ_{a}}^{[0,2]}∥_{S_{p}}. We can consider the Fourier coefficients of φ_{a} defined by

φ̂_{a}(n) =^{def} ∫_{0}^{1} φ_{a}(t)e^{−2πint}dt, n ∈ ℤ.

Theorem 16.1. Let φ be a 1-periodic function in L_{loc}^{2}(ℝ) and let a > 0. Suppose that φ has bounded variation on [0, 1]. Then

∥Q_{φ_{a}}^{[0,2]}∥_{S_{1}} ≤ C(φ) log(2 + a)

and

∥Q_{φ_{a}}^{[0,2]}∥_{S_{p}} ≤ C(φ)(1 + a)^{1/p−1}, 1/2 < p < 1.

Proof. The result follows from Theorem 7.7. ■

Theorem 16.2. Let φ be a nonconstant 1-periodic function in L_{loc}^{2}(ℝ). Then for a ≥ 1

∥Q_{φ_{a}}^{[0,1]}∥_{S_{1}} ≥ C(φ) log(2 + a).

Proof. It follows from (16.1) that it is sufficient to prove that

∥Q_{φ_{a}}^{[0,2]}∥_{S_{1}} ≥ C(φ) log(2 + a).  (16.2)

First we consider the case where a is an integer. There exists an integer k ∈ ℤ\{0} such that φ̂(k) ≠ 0. By Corollary 15.6,

|φ̂_{a}(l)| ≤ C(log(2 + |l|))^{−1}∥Q_{φ_{a}}∥_{S_{1}}.

Substituting l = ak, we obtain (16.2), since φ̂_{a}(ak) = φ̂(k).

Let now a be an arbitrary number in (1, ∞). For any σ ∈ [1, 2] there exists k_{σ} ∈ ℤ \ {0} such that φ̂_{σ}(k_{σ}) ≠ 0. Consequently, there exists a neighborhood U_{σ} of σ such that φ̂_{τ}(k_{σ}) ≠ 0 for any τ ∈ U_{σ}. The first part of the proof allows us to obtain the required estimate for any a > 1 such that a/N ∈ U_{σ} for some positive integer N. To complete the proof, we can choose a finite subcover U_{σ_{j}} of [1, 2]. ■

Theorem 16.3. Let φ be a nonconstant 1-periodic function in L_{loc}^{2}(ℝ) and let 0 < p < 1. Then for a ≥ 1

∥Q_{φ_{a}}^{[0,1]}∥_{S_{p}} ≥ C(φ)a^{1/p−1}.

Proof. It suffices to consider the case when a is an even integer. The general case may be reduced to this special case in the same way as in the proof of Theorem 16.2. With any kernel k on the square [0, 1)^{2} and any integer n ≥ 1 we associate the kernel k^{[n]} defined by

k^{[n]}(x, y) = n^{2} ∫_{\frac{j}{n}}^{\frac{j+1}{n}} ∫_{\frac{l}{n}}^{\frac{l+1}{n}}k(t, s) dtds, if x ∈ [\frac{j}{n}, \frac{j + 1}{n}) and y ∈ [\frac{l}{n}, \frac{l + 1}{n}) .

Clearly, ∥k^{[n]}∥_{S_{p}} ≤ ∥k∥_{S_{p}} for any positive p (recall that ∥k∥_{S_{p}} means the S_{p}-norm (or quasinorm) of the integral operator with kernel k).

Suppose that

∫_{0}^{1} φ(t) dt ≠ 2 ∫_{0}^{1} tφ(t) dt.

Put k_{n}(x, y) =^{def} φ (n max{x, y}) for x, y ∈ [0, 1). Clearly,

∥k_{2n}^{[n]} − k_{2n}^{[2n]}∥_{S_{p}} ≤ 2^{1/p}∥k_{2n}∥_{S_{p}}.

It is not hard to see that on [\frac{j}{n}, \frac{j+1}{n}) × [\frac{l}{n}, \frac{l+1}{n})

k_{2n}^{[n]} = \{ \begin{matrix}∫_{0}^{1} φ(t) dt, & j ≠ l, \\ \frac{1}{2} ∫_{0}^{1} φ(t) dt + ∫_{0}^{1} tφ(t) dt, & j = l. \end{matrix}

Next, on [\frac{j}{2n}, \frac{j+1}{2n}) × [\frac{l}{2n}, \frac{l+1}{2n})

k_{2n}^{[2n]} = \{ \begin{matrix}∫_{0}^{1} φ(t) dt, & j ≠ l, \\ 2 ∫_{0}^{1} tφ(t) dt, & j = l. \end{matrix}

Thus, the kernel k_{2n}^{[n]} − k_{2n}^{[2n]} vanishes outside the “diagonal” ⋃_{j=0}^{n−1} [\frac{j}{n}, \frac{j+1}{n}) × [\frac{j}{n}, \frac{j+1}{n}). Clearly, for x, y ∈ [0, \frac{n−1}{n}) × [0, \frac{n−1}{n}) we have

k_{2n}^{[n]}(x, y) − k_{2n}^{[2n]}(x, y) = k_{2n}^{[n]} (x + \frac{1}{n}, y + \frac{1}{n}) − k_{2n}^{[2n]} (x + \frac{1}{n}, y + \frac{1}{n}) .

Consequently,

∥k_{2n}^{[n]} − k_{2n}^{[2n]}∥_{S_{p}} = n^{1/p} ∥(k_{2n}^{[n]} − k_{2n}^{[2n]}) χ_{[0, \frac{1}{n})×[0, \frac{1}{n})}∥_{S_{p}} .

We have

k_{2n}^{[n]} − k_{2n}^{[2n]} = \frac{1}{2} ∫_{0}^{1} φ(t) dt − ∫_{0}^{1} tφ(t) dt

on ([0, \frac{1}{2n}) × [0, \frac{1}{2n})) ∪ ([\frac{1}{2n}, \frac{1}{n}) × [\frac{1}{2n}, \frac{1}{n})) and

k_{2n}^{[n]} − k_{2n}^{[2n]} = ∫_{0}^{1} tφ(t) dt − \frac{1}{2} ∫_{0}^{1} φ(t) dt

on ([0, \frac{1}{2n}) × [\frac{1}{2n}, \frac{1}{n})) ∪ ([\frac{1}{2n}, \frac{1}{n}) × [0, \frac{1}{2n})). Now it is easy to see that ∥k_{2n}^{[n]} − k_{2n}^{[2n]}∥_{S_{p}} = cn^{1/p−1}

for some nonzero c, since ∫_{0}^{1} φ(t) dt ≠ 2 ∫_{0}^{1} tφ(t) dt.

Suppose now that φ is an arbitrary nonconstant 1-periodic function. It suffices to prove that there exists b ∈ ℝ such that ∫_{0}^{1} φ(t − b) dt ≠ 2 ∫_{0}^{1} tφ(t − b) dt. Suppose that

∫_{0}^{1} φ(t − b) dt = 2 ∫_{0}^{1} tφ(t − b) dt, b ∈ ℝ.  (16.3)

Let h be the 1-periodic function such that h(t) = 2t − 1 for t ∈ [0, 1). Clearly, ĥ(n) ≠ 0 for n ≠ 0. Thus it follows from (16.3) that φ is constant. ■

Acknowledgement

This research was begun by the first and third authors. The two other authors joined them during the Conference on Function Spaces, Interpolation Theory, and related topics in honour of Jaak Peetre on his 65th birthday at Lund University, Sweden, August 17–22, 2000, where three of the authors were present and several of the results were proved. We thank Jonathan Arazy and Vladimir Maz’ya for helpful comments.

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Alexei B. Aleksandrov, St-Petersburg Branch, Steklov Institute of Mathematics, Fontanka 27, 191011 St-Petersburg, Russia

E-mail address: alex@pdmi.ras.ru

Svante Janson, Department of Mathematics, Uppsala University, 751 06 Uppsala, Sweden

E-mail address: svante@math.uu.se

Vladimir V. Peller, Department of Mathematics, Kansas State University, Manhattan, Kansas 66506, USA

E-mail address: peller@math.ksu.edu

Richard Rochberg, Department of Mathematics, Washington University, St. Louis, MO 63130, USA

E-mail address: rr@math.wustl.edu