ON THE SPACE Q_{p} AND ITS DYADIC COUNTERPART

SVANTE JANSON

Abstract. We define a dyadic counterpart Q_{p}^{d} of Q_{p}, and study some relations between Q_{p} and Q_{p}^{d}. For example, a function on 𝕋 belongs to Q_{p} if and only if (almost) all its translates belong to Q_{p}^{d}. Conversely, functions in Q_{p} may be obtained by averaging translates of functions in Q_{p}^{d}.

1. Introduction

The space Q_{p}, 0 < p < 1 was introduced in [1] as the Banach space of all analytic functions in the unit disc Δ satisfying

sup_{w∈Δ} ∫∫ _{Δ} |f′(z)|^{2}g(z, w)^{p} dx dy < ∞,  (1)

where z = x + iy and g(z, w) = log |(1 − w̄z)/(w − z)|. As is shown in [1], Q_{p} is a proper subspace of BMOA (which is obtained by taking p = 1 in (1)), and Q_{p_{1}} ⊊ Q_{p_{2}} if 0 < p_{1} < p_{2} < 1.

Essén and Xiao [4] showed that an analytic function f, in H^{1} say, belongs to Q_{p} if and only if its boundary values on the unit circle 𝕋 satisfy

sup_{I} |I|^{−p} ∫_{I} ∫_{I} \frac{|f(x) − f(y)|^{2}}{|x − y|^{2−p}} dx dy < ∞,  (2)

where I ranges over all intervals ⊆ 𝕋.

We will in the sequel consider functions defined on 𝕋 rather than Δ, and we redefine Q_{p} to be the space of all (measurable) functions on 𝕋 that satisfy (2), analytic or not. (Analyticity is not important in (2); in fact, it was shown by Essén and Xiao [4, Corollary 3.2] that a real function satisfies (2) if and only if it is the real part of (the boundary values of) an analytic function in Q_{p}.) We define ∥f∥_{Q_{p}} to be the square root of the supremum in (2); this is a seminorm only, since ∥f∥_{Q_{p}} = 0 if f is a.s. constant, and a proper norm is given by e.g. ∥f∥_{Q_{p}} + | ∫_{𝕋} f|.

Note that Q_{p} ⊆ BMO. (BMO can be defined by taking p = 2 in (2). In fact, it follows easily using the John–Nirenberg inequality that any p with 1 < p ≤ 2 yields BMO.)

It is well-known that BMO has a dyadic counterpart BMOd. The purpose of this note is to define a corresponding dyadic version Q_{p}^{d} of Q_{p}, and to study some relations between Q_{p} and Q_{p}^{d}.

2. Main results

We will, for convenience, identify the circle 𝕋 with the unit interval [0, 1), keeping in mind that subintervals may wrap around 0. In particular, our circle has length 1. (The ambiguity in interpreting |x − y| in (2) is harmless.)

A dyadic interval is an interval of the type [(m − 1)2^{−n}, m2^{−n}); let D = D(𝕋) be the set of all dyadic subintervals of 𝕋 (including 𝕋 itself) and let D_{n}(𝕋), n ≥ 0, be the subset of the 2^{n} dyadic intervals of length 2^{−n}.

We define the dyadic distance δ(x, y) between two points in 𝕋 by

δ(x, y) = inf{|I| : x, y ∈ I ∈ D(𝕋)}.

The space Q_{p}^{d}, 0 < p < 1, may now be defined as the space of all (measurable) functions f on 𝕋 such that

sup_{I∈D} |I|^{−p} ∫_{I} ∫_{I} \frac{|f(x) − f(y)|^{2}}{δ(x, y)^{2−p}} dx dy < ∞.  (3)

Again we define a seminorm ∥f∥_{Q_{p}^{d}} as the square root of the supremum, and a norm as this seminorm + | ∫_{𝕋} f|. Some equivalent definitions, and corresponding equivalent (semi)norms, are given in Section 4.

Since δ(x, y) ≥ |x − y|, and the supremum in (3) is over a subset of the set of intervals in (2), it follows immediately that Q_{p} ⊆ Q_{p}^{d}. The inclusion is strict; for example, it is easily seen that if f(x) = log x, 0 < x < 1, then f ∈ Q_{p}^{d} (cf. the similar analytic example log(1 − z) in [4]), but f ∉ BMO because of the infinite jump at 0, and thus f ∉ Q_{p}.

It is also immediate that Q_{p}^{d} ⊂ BMOd, which is obtained by taking p = 2 in (3). (In fact, the John–Nirenberg theorem again shows that any p with 1 < p ≤ 2 in (3) defines BMOd.)

It is clear that one reason for the discrepancy between Q_{p} and Q_{p}^{d} is that Q_{p} is translation (i.e. rotation) invariant whereas Q_{p}^{d} is not. Indeed, the next theorems show that a function belongs to Q_{p} if and only if all its translates belong to Q_{p}^{d}.

Remark. Although the theorems are phrased in terms of translating the function, it is obviously equivalent, and sometimes more natural, to instead consider the function as fixed and translate the dyadic partition used to define Q_{p}^{d}.

Let τ_{t} denote the translation operator τ_{t}f(x) = f(x − t).

Theorem 1. Let 0 < p < 1. Then f ∈ Q_{p} if and only if τ_{t}f ∈ Q_{p}^{d} for all t ∈ 𝕋 and sup_{t} ∥τ_{t}f∥_{Q_{p}^{d}} < ∞. Moreover, ∥f∥_{Q_{p}} ≍ sup_{t} ∥τ_{t}f∥_{Q_{p}^{d}}.

(Here ≍ means that the two sides are equivalent within constant factors that may depend on p.)

The condition that τ_{t}f ∈ Q_{p}^{d} for all t may be relaxed considerably.

Theorem 2. Let 0 < p < 1. Then the following are equivalent.

(i) f ∈ Q_{p} (ii) τ_{t}f ∈ Q_{p}^{d} for all t ∈ 𝕋 (iii) τ_{t}f ∈ Q_{p}^{d} for almost all t ∈ 𝕋 (iv) τ_{t}f ∈ Q_{p}^{d} for t ∈ E ⊆ 𝕋, with |E| > 0.

It follows easily from the proof below that the median (or any other fixed quantile) of t ↦ ∥τ_{t}f∥_{Q_{p}^{d}} is an equivalent seminorm on Q_{p}.

We state two immediate corollaries of Theorem 2.

Corollary 3. Let 0 < p < 1 and let f be a function on 𝕋. If f ∈ Q_{p} then τ_{t}f ∈ Q_{p}^{d} for all t ∈ 𝕋, while if f ∉ Q_{p} then τ_{t}f ∉ Q_{p}^{d} for a.e. t ∈ 𝕋.

(The corresponding result for BMO is easy, see Lemma 7 below; the dual result for H^{1} (where ‘all’ and ‘a.e.’ are interchanged) is given in [2].)

For 0 < q ≤ ∞, define L^{q}(Q_{p}^{d}) as the space of all measurable functions F on 𝕋 × 𝕋, such that F (t, ·) ∈ Q_{p}^{d} for a.e. t and ∥F (t, ·)∥_{Q_{p}^{d}} ∈ L^{q}(𝕋).

Remark. This is not the usual Lebesgue space of Banach space valued functions, defined as the closure of simple functions in the obvious norm. The problem is that Q_{p}^{d} is not separable, and it is easily seen that if e.g. F (t, x) = 1[0 < x < t] (where 1 denotes the indicator function), then F belongs to L^{q}(Q_{p}^{d}) for any q ≤ ∞ with our definition, but ∥F (t, ·) − F (s, ·)∥_{Q_{p}^{d}} ≥ 1/4 for a.e. s and t, and thus there is no separable subspace of Q_{p}^{d} that contains F (t, ·) for a.e. t, as required by the standard definition, see e.g. [3].

Corollary 4. Let 0 < p < 1 and 0 < q ≤ ∞. Then f ∈ Q_{p} if and only if τ_{t}f(x) ∈ L^{q}(Q_{p}^{d}).

Again, an equivalent seminorm on Q_{p} is given by ∥ ∥τ_{t}f∥_{Q_{p}^{d}}∥_{L^{q}}.

Conversely, if q > 1, starting with any function in L^{q}(Q_{p}^{d}), we may construct a function in Q_{p} as a suitable average.

Theorem 5. Let 0 < p < 1 and 1 < q ≤ ∞. Suppose that F (t, x) ∈ L^{q}(Q_{p}^{d}), and define g(x) = ∫_{𝕋} F (t, x + t) dt. Then g ∈ Q_{p}.

Theorem 5 is an extension to Q_{p} of a result by Garnett and Jones [5] for BMO. That result has important applications, since it enables several deep results for BMO to be shown as consequences of similar but much simpler results for BMOd [5]. We do not know if there are similar applications of Theorem 5.

If we introduce the linear operator T f(t, x) = f(x − t) (mapping functions on 𝕋 to functions on 𝕋^{2}), Corollary 4 shows that T defines an embedding of Q_{p} into L^{q}(Q_{p}^{d}). Conversely, Theorem 5 shows if q > 1, then the adjoint averaging operator T^{∗} : F ↦ ∫_{𝕋} F (t, x + t) dt maps L^{q}(Q_{p}^{d}) into Q_{p}. Moreover, T^{∗}T is the identity and T T^{∗} is a projection, which yields the following corollaries.

Corollary 6. If 0 < p < 1 and 1 < q ≤ ∞, then Q_{p} is isomorphic to the complemented subspace T (Q_{p}) of L^{q}(Q_{p}^{d}).

Corollary 7. If 0 < p < 1 and 1 < q ≤ ∞, then T^{∗} maps L^{q}(Q_{p}^{d}) onto Q_{p}.

The final theorem in this section gives a different relation between Q_{p} and Q_{p}^{d}. The example log x above of a function in Q_{p}^{d} \ Q_{p} failed to be in Q_{p} because it did not belong to BMO; in fact, this is the only way to get such an example.

Theorem 8. Let 0 < p < 1. Then Q_{p} = Q_{p}^{d} ∩ BMO.

3. Preliminaries

Throughout the paper, p is a fixed number with 0 < p < 1. We let c and C denote unspecified positive constants that may depend on p (but not on anything else).

We define, for any interval I and integrable function f,

f(I) = \frac{1}{|I|} ∫_{I} f,

the mean of f on I, and

φ_{f}(I) = \frac{1}{|I|} ∫_{I} |f − f(I)|^{2},

the square mean oscillation of f on I. Obviously, φ_{f}(I) < ∞ ⇔ f ∈ L^{2}(I); we extend the definition to all measurable functions f on I by letting φ_{f}(I) = ∞ when f ∉ L^{1}(I). Recall that f ∈ BMO if and only if sup_{I} φ_{f}(I) < ∞. Note the well-known identities

\frac{1}{|I|} ∫_{I} |f − a|^{2} = φ_{f}(I) + |f(I) − a|^{2}  (4)

for any complex number a, and

\frac{1}{|I|^{2}} ∫_{I} ∫_{I} |f(x) − f(y)|^{2} dx dy = 2φ_{f}(I).  (5)

Moreover, if I is divided into two subintervals I′ and I′′ of the same length \frac{1}{2}|I|, then |f(I) − f(I′)| = |f(I) − f(I′′)| = \frac{1}{2}|f(I′) − f(I′′)| and

\begin{aligned}φ_{f}(I) & = \frac{1}{2|I′|} ∫_{I′} |f − f(I)|^{2} + \frac{1}{2|I′′|} ∫_{I′′} |f − f(I)|^{2} \\ & = \frac{1}{2}(φ_{f}(I′) + |f(I) − f(I′)|^{2}) + \frac{1}{2}(φ_{f}(I′′) + |f(I) − f(I′′)|^{2}) \\ & = \frac{1}{2}φ_{f}(I′) + \frac{1}{2}φ_{f}(I′′) + \frac{1}{4}|f(I′) − f(I′′)|^{2}. \end{aligned}  (6)

In particular,

|f(I) − f(I′)| = \frac{1}{2}|f(I′) − f(I′′)| ≤ φ_{f}(I)^{1/2}.  (7)

Furthermore, if I ⊂ J, then by (4),

φ_{f}(I) ≤ \frac{1}{|I|} ∫_{I} |f − f(J)|^{2} ≤ \frac{|J|}{|I|}φ_{f}(J).  (8)

We defined above dyadic intervals in 𝕋. Similarly, if I is any interval, dyadic or not, we let D_{n}(I), n ≥ 0, denote the set of the 2^{n} subintervals of length 2^{−n}|I| obtained by n successive bipartitions of I.

We define, for any interval I and a measurable function f on I,

ψ_{f}(I) = ∑_{k=0}^{∞} ∑_{J∈D_{k}(I)} 2^{−pk}φ_{f}(J).  (9)

By (9) and (5),

\begin{aligned}ψ_{f}(I) & = ∑_{k=0}^{∞} ∑_{J∈D_{k}(I)} 2^{−pk} \frac{1}{2}(2^{−k}|I|)^{−2} ∫_{J} ∫_{J} |f(x) − f(y)|^{2} dx dy \\ & = ∫_{𝕋} ∫_{𝕋} α_{I}(x, y)|f(x) − f(y)|^{2} dx dy, \end{aligned}  (10)

where

α_{I}(x, y) = \frac{1}{2} ∑_{k=0}^{∞} ∑_{J∈D_{k}(I)} 2^{(2−p)k}|I|^{−2}1[x, y ∈ J].  (11)

Since x, y ∈ J ∈ D_{k}(I) implies |x − y| ≤ |J| = 2^{−k}|I|, and thus 2^{k} ≤ |I|/|x − y|,

α_{I}(x, y) ≤ ∑_{2^{k}≤|I|/|x−y|} 2^{(2−p)k}|I|^{−2} ≤ C(\frac{|I|}{|x − y|})^{2−p}|I|^{−2} = C|I|^{−p}|x − y|^{p−2};

furthermore α_{I}(x, y) = 0 unless x, y ∈ I. Consequently,

ψ_{f}(I) ≤ C|I|^{−p} ∫_{I} ∫_{I} \frac{|f(x) − f(y)|^{2}}{|x − y|^{2−p}} dx dy.  (12)

The converse is less obvious, but will be proved in Lemma 3 below.

If I is a dyadic interval, and x, y ∈ I, then x, y ∈ J for some J ∈ D_{k}(I) if and only if δ(x, y) ≤ 2^{−k}|I|, and thus, by (11),

α_{I}(x, y) = \frac{1}{2} ∑_{2^{k}≤|I|/δ(x,y)} 2^{(2−p)k}|I|^{−2} ≍ (\frac{|I|}{δ(x, y)})^{2−p}|I|^{−2} = |I|^{−p}δ(x, y)^{p−2}.

Consequently, if I is a dyadic interval,

ψ_{f}(I) ≍ |I|^{−p} ∫_{I} ∫_{I} \frac{|f(x) − f(y)|^{2}}{δ(x, y)^{2−p}} dx dy.  (13)

4. The dyadic space

The dyadic space Q_{p}^{d} was defined above by (3). We give here several equivalent definitions.

Theorem 9. Let 0 < p < 1. Then f ∈ Q_{p}^{d} if and only if sup_{I∈D} ψ_{f}(I) < ∞. Moreover, sup_{I∈D} ψ_{f}(I)^{1/2} is a seminorm on Q_{p}^{d}, equivalent to ∥f∥_{Q_{p}^{d}} as defined above.

Proof. Immediate by the definition and (13).

We let F_{n} denote the σ-field generated by the partition D_{n}(𝕋); thus, if f ∈ L^{1}(𝕋), then E(f | F_{n}) is the function that is constant f(I) on each dyadic interval I ∈ D_{n}(𝕋).

Theorem 10. Let 0 < p < 1. If f ∈ L^{1}(𝕋) and f_{n} = E(f | F_{n}), then the following are equivalent.

(i) f ∈ Q_{p}^{d}. (ii) For some M < ∞ and every n ≥ 0,

∑_{k=0}^{∞} 2^{(1−p)k} E(|f − f_{n+k}|^{2} | F_{n}) ≤ M a.s.

(iii) For some M < ∞ and every n ≥ 0,

∑_{k=0}^{∞} 2^{(1−p)k} E(|f_{n+k+1} − f_{n+k}|^{2} | F_{n}) ≤ M a.s.

Proof. If I ∈ D_{n}(𝕋) and J ∈ D_{k}(I) ⊆ D_{n+k}, then φ_{f}(J) = |J|^{−1} ∫_{J} |f − f_{n+k}|^{2}, and |J| = 2^{−k}|I|. Hence, by the definition (9),

ψ_{f}(I) = ∑_{k=0}^{∞} 2^{−pk} \frac{2^{k}}{|I|} ∫_{I} |f − f_{n+k}|^{2} = ∑_{k=0}^{∞} 2^{(1−p)k} E(|f − f_{n+k}|^{2} | F_{n})(x)

for x ∈ I, which together with Theorem 9 shows (i) ⇔ (ii).

Furthermore,

E(|f − f_{n+k}|^{2} | F_{n}) = ∑_{j=k}^{∞} E(|f_{n+j+1} − f_{n+j}|^{2} | F_{n}),

and thus, interchanging the order of summation,

\begin{aligned}∑_{k=0}^{∞} 2^{(1−p)k} E(|f − f_{n+k}|^{2} | F_{n}) & = ∑_{j=0}^{∞} ∑_{k=0}^{j} 2^{(1−p)k} E(|f_{n+j+1} − f_{n+j}|^{2} | F_{n}) \\ & ≍ ∑_{j=0}^{∞} 2^{(1−p)j} E(|f_{n+j+1} − f_{n+j}|^{2} | F_{n}), \end{aligned}

which yields the equivalence of (ii) and (iii).

Again, equivalent seminorms on Q_{p}^{d} may defined from (ii) and (iii), by taking the square roots of the smallest possible M.

5. Proofs

The proofs are based on the following fundamental lemma. Note that this lemma and the two following ones are valid for (finite) intervals in ℝ as well as in 𝕋; we will pass between the two cases without further comment. Lemma 1. Let I be an interval and let I′ and I′′ be two intervals such that I′ and I′′ are adjacent, |I′| = |I′′| = |I| and I ⊂ I′ ∪ I′′. Then, for any f ∈ L^{1}(I′ ∪ I′′),

φ_{f}(I) ≤ φ_{f}(I′) + φ_{f}(I′′) + |f(I′) − f(I′′)|^{2},  (14)
ψ_{f}(I) ≤ C(ψ_{f}(I′) + ψ_{f}(I′′) + |f(I′) − f(I′′)|^{2}).  (15)

Proof. It follows from (8) and (6) that

φ_{f}(I) ≤ \frac{|I′ ∪ I′′|}{|I|}φ_{f}(I′ ∪ I′′) = φ_{f}(I′) + φ_{f}(I′′) + \frac{1}{2}|f(I′) − f(I′′)|^{2},

proving (14).

For (15), we assume for simplicity that I′ = [0, 1) and I′′ = [1, 2); this is no loss of generality by homogeneity. For each j ≥ 0, let {I_{j,i}}_{i=1}^{2^{j+1}} be the 2^{j+1} dyadic intervals of length 2^{−j} contained in I′ ∪ I′′ (i.e., D_{j}(I′) ∪ D_{j}(I′′)), arranged in the natural order. If J ∈ D_{j}(I), then J ⊂ I_{j,i} ∪ I_{j,i+1} for some i, and thus by (14) applied to J,

φ_{f}(J) ≤ φ_{f}(I_{j,i}) + φ_{f}(I_{j,i+1}) + |f(I_{j,i}) − f(I_{j,i+1})|^{2}.

The 2^{j} different J ∈ D_{j}(I) yield different values of i, and summing over all j and J we thus obtain

\begin{aligned}ψ_{f}(I) & = ∑_{j=0}^{∞} ∑_{J∈D_{j}(I)} 2^{−pj}φ_{f}(J) \\ & ≤ 2 ∑_{j=0}^{∞} ∑_{i=1}^{2^{j+1}} 2^{−pj}φ_{f}(I_{j,i}) + ∑_{j=0}^{∞} ∑_{i=1}^{2^{j+1}−1} 2^{−pj}|f(I_{j,i}) − f(I_{j,i+1})|^{2}. \end{aligned}  (16)

The first double sum on the right hand side of (16) is just ψ_{f}(I′)+ψ_{f}(I′′). In order to estimate the final sum, consider a pair (j, i) with j ≥ 0 and 1 ≤ i < 2^{j+1} − 1. Let I^{∗} be the smallest dyadic interval that contains I_{j,i} ∪ I_{j,i+1}, and let the length of I^{∗} be 2^{−j+m}, where m ≥ 1. (Recall that |I_{j,i}| = 2^{−j}.) Moreover, for 0 ≤ l ≤ m, let J_{l} and K_{l} be the dyadic intervals of length 2^{−j+l} that contain I_{j,i} and I_{j,i+1}, respectively; thus I_{j,i} = J_{0} ⊂ J_{1} ⊂ ⋯ ⊂ J_{m} = I^{∗} and I_{j,i+1} = K_{0} ⊂ ⋯ ⊂ K_{m} = I^{∗}. Using the Cauchy–Schwarz inequality and (7), we obtain

\begin{matrix}|f(I_{j,i})− & f(I_{j,i+1})|^{2} ≤ (∑_{l=1}^{m} |f(J_{l−1}) − f(J_{l})| + ∑_{l=1}^{m} |f(K_{l}) − f(K_{l−1})|)^{2} \\ ≤ (2 ∑_{l=1}^{∞} l^{−2})(∑_{l=1}^{m} l^{2}|f(J_{l}) − f(J_{l−1})|^{2} + ∑_{l=1}^{m} l^{2}|f(K_{l}) − f(K_{l−1})|^{2}) \\ ≤ C ∑_{l=1}^{m} l^{2}(φ_{f}(J_{l}) + φ_{f}(K_{l})). \end{matrix}(17)

If i ≠ 2^{j}, then I_{j,i} ∪ I_{j,i+1} ⊆ I′ or I′′, and thus |I^{∗}| ≤ 1 and m ≤ j. If i = 2^{j}, however, then I^{∗} = [0, 2) and m = j + 1; in this case we modify (17) by observing that J_{j} = I′ and K_{j} = I′′ and thus

|f(I_{j,i}) − f(I_{j,i+1})| ≤ ∑_{l=1}^{j} |f(J_{l−1}) − f(J_{l})| + ∑_{l=1}^{j} |f(K_{l}) − f(K_{l−1})| + |f(I′) − f(I′′)|

which by the same argument yields

|f(I_{j,i}) − f(I_{j,i+1})|^{2} ≤ C ∑_{l=1}^{j} l^{2}(φ_{f}(J_{l}) + φ_{f}(K_{l})) + C|f(I′) − f(I′′)|^{2}. (18)

We now keep j ≥ 0 fixed and sum (17) or (18) (when i = 2^{j}) for 1 ≤ i ≤ 2^{j+1}−1. We observe that the intervals J_{l} and K_{l} that appear belong to D_{j−l}(I′) ∪ D_{j−l}(I′′), with 1 ≤ l ≤ j. Moreover, each dyadic interval J in D_{j−l}(I′) ∪ D_{j−l}(I′′) appears at most four times as a J_{l} or a K_{l} (viz. when, in D_{l}(J), I_{j,i} is the rightmost interval, I_{j,i+1} is the leftmost interval or I_{j,i} and I_{j,i+1} are the two middle intervals). Consequently, since (18) is used only once,

∑_{i=1}^{2^{j+1}−1} |f(I_{j,i}) − f(I_{j,i+1})|^{2} ≤ C ∑_{l=1}^{j} ∑_{J∈D_{j−l}(I′)∪D_{j−l}(I′′)} l^{2}φ_{f}(J) + C|f(I′) − f(I′′)|^{2}.

Summing over j we finally obtain, substituting j = k + l,

∑_{j=0}^{∞} ∑_{i=1}^{2^{j+1}−1}2^{−pj}|f(I_{j,i}) − f(I_{j,i+1})|^{2}

≤ C ∑_{l=1}^{∞} ∑_{k=0}^{∞} ∑_{J∈D_{k}(I′)∪D_{k}(I′′)} 2^{−pk−pl}l^{2}φ_{f}(J) + C ∑_{j=0}^{∞} 2^{−pj}|f(I′) − f(I′′)|^{2}

= Cψ_{f}(I′) + Cψ_{f}(I′′) + C|f(I′) − f(I′′)|^{2},

which by (16) completes the proof of (15).

We proceed with further results on ψ_{f}(I).

Lemma 2. For any interval I and f ∈ L^{1}(I),

|I|^{−p} ∫_{I} ∫_{I} \frac{|f(x) − f(y)|^{2}}{|x − y|^{2−p}} dx dy ≤ C \frac{1}{2|I|} ∫_{−|I|}^{|I|} ψ_{f}(I + t) dt + Cψ_{f}(I).

Proof. By (10) and Fubini,

\frac{1}{2|I|} ∫_{−|I|}^{|I|} ψ_{f}(I + t) dt = ∫_{𝕋} ∫_{𝕋} \frac{1}{2|I|} ∫_{−|I|}^{|I|} α_{I+t}(x, y) dt |f(x) − f(y)|^{2} dx dy.

This and (10) show that it suffices to verify

\frac{1}{2|I|} ∫_{−|I|}^{|I|} α_{I+t}(x, y) dt + α_{I}(x, y) ≥ c|I|^{−p}|x − y|^{p−2},  x, y ∈ I. (19)

First, suppose that x, y ∈ I with |x − y| ≤ \frac{1}{2}|I| and let l ≥ 0 be such that 2^{−l−2}|I| < |x − y| ≤ 2^{−l−1}|I|. Then, by (11),

\frac{1}{2|I|} ∫_{−|I|}^{|I|} α_{I+t}(x, y) dt ≥ \frac{1}{2|I|} ∫_{−|I|}^{|I|} \frac{1}{2} ∑_{J∈D_{l}(I+t)} 2^{(2−p)l}|I|^{−2}1[x, y ∈ J] dt

= \frac{2^{(2−p)l}}{4|I|^{3}} ∑_{J∈D_{l}(I)} ∫_{−|I|}^{|I|} 1[x, y ∈ J + t] dt

≥ c|x − y|^{p−2}|I|^{2−p−3} ∑_{J∈D_{l}(I)} ∫_{−|I|}^{|I|} 1[x, y ∈ J + t] dt.

It is easily seen that the final integral, for each J, equals |J| − |x − y| ≥ \frac{1}{2}|J|, and thus the sum over J is at least \frac{1}{2}|I|. Hence (19) holds for |x − y| ≤ \frac{1}{2}|I|.

Finally, if x, y ∈ I with |x − y| > \frac{1}{2}|I|, then, taking k = 0 in (11),

α_{I}(x, y) ≥ \frac{1}{2}|I|^{−2} ≥ c|I|^{−p}|x − y|^{p−2}

and (19) holds in this case too.

Lemma 3. For any interval I and f ∈ L^{1}(I),

cψ_{f}(I) ≤ |I|^{−p} ∫_{I} ∫_{I} \frac{|f(x) − f(y)|^{2}}{|x − y|^{2−p}} dx dy ≤ Cψ_{f}(I).

Proof. The left inequality is just (12).

For the right inequality, we may assume that f is defined on ℝ with f constant = f(I) outside I. Let I_{−} and I_{+} be the two intervals of the same length as I that are adjacent to I on the left and right, respectively. Note that then ψ_{f}(I_{−}) = ψ_{f}(I_{+}) = 0 and that f(I_{−}) = f(I_{+}) = f(I).

For every t with |t| < |I|, either I + t ⊂ I_{−} ∪ I or I + t ⊂ I ∪ I_{+}, and in both cases Lemma 1 yields ψ_{f}(I + t) ≤ Cψ_{f}(I). The result follows by Lemma 2.

An immediate consequence of Lemma 3 is another characterization of Q_{p}.

Lemma 4. Q_{p} equals the space of all functions f on 𝕋 such that sup_{I⊆𝕋} ψ_{f}(I) is finite.

We will use the fact that it here suffices to consider intervals I with dyadic lengths. (Note that we do not require I to be dyadic, only its length. Restriction to dyadic intervals would give Q_{p}^{d} by Theorem 9.)

Lemma 5. Q_{p} equals the space of all functions f on 𝕋 such that sup_{I} ψ_{f}(I) < ∞, where I ranges over the set of all intervals in 𝕋 with dyadic length 2^{−n}, n = 0, 1, …

Proof. Since every interval I is contained in an interval J with dyadic length |J| < 2|I|, it is obvious that it suffices to consider intervals of dyadic lengths in (2). The proof is completed by Lemma 3.

Proof of Theorem 1. Since every interval of dyadic length is the translate of a dyadic interval, Lemma 5 shows that f ∈ Q_{p} if and only if sup_{t∈𝕋} sup_{I∈D} ψ_{f}(I−t) < ∞. Moreover, using Theorem 9,

∥f∥_{Q_{p}}^{2} ≍ sup_{t∈𝕋} sup_{I∈D} ψ_{f}(I − t) = sup_{t∈𝕋} sup_{I∈D} ψ_{τ_{t}f}(I) ≍ sup_{t∈𝕋}∥τ_{t}f∥_{Q_{p}^{d}}^{2},

and the result follows.

Proof of Theorem 8. As remarked above, the inclusion Q_{p} ⊆ Q_{p}^{d} ∩ BMO follows directly from the definitions.

Conversely, suppose that f ∈ Q_{p}^{d} ∩ BMO. Let I be an interval of dyadic length. Then there exist two adjacent dyadic intervals I′ and I′′ of the same length |I| such that I ⊂ I′ ∪ I′′. Lemma 1 yields

ψ_{f}(I) ≤ Cψ_{f}(I′) + Cψ_{f}(I′′) + C|f(I′) − f(I′′)|^{2}.

The first two terms on the right hand side are bounded by C∥f∥_{Q_{p}^{d}}^{2} by Theorem 9, and the last term is bounded by Cφ_{f}(I′ ∪ I′′) ≤ C∥f∥_{BMO}^{2} by (7). Hence ψ_{f}(I) is bounded uniformly for all intervals I of dyadic length, and the result follows by Lemma 5.

Lemma 6. Let I ⊂ 𝕋 be an interval of length 2^{−n}, n ≥ 1, and let m(t), for t ∈ 𝕋, be the smallest integer such that the translated interval I + t is contained in a dyadic interval of length 2^{−n+m(t)}. Then

|{t : m(t) > M}| ≤ 2^{−M},  M = 0, 1, …  (20)

In particular, for every r < ∞,

∫_{𝕋} m(t)^{r} dt ≤ ∑_{m=1}^{∞} m^{r}2^{1−m} < ∞.

Proof. Clearly, m(t) ≤ n, so (20) is trivial for M ≥ n. For 0 ≤ M < n, it is easily seen that {t : m(t) > M} consists of 2^{n−M} intervals of length 2^{−n}, and thus there is equality in (20).

In particular, |{t : m(t) = M}| ≤ 2^{1−M}, and the final estimate follows.

Lemma 7. Suppose that τ_{t}f ∈ BMOd for t ∈ E, where E ⊆ 𝕋 is a set with positive measure. Then f ∈ BMO.

Proof. Let M ≥ 1 be such that 2^{−M} < |E|. Then, since t ↦ ∥τ_{t}f∥_{BMOd} is measurable, there exists a number A < ∞ and a subset E′ ⊂ E with |E′| > 2^{−M} such that ∥τ_{t}f∥_{BMOd} ≤ A for t ∈ E′.

Suppose that I is an interval of dyadic length 2^{−n} with n ≥ M, and let m(t) be as in Lemma 6. By (20) and our assumptions, |{t : m(t) > M}| ≤ 2^{−M} < |E′|, and thus there exists a t ∈ E′ such that m(t) ≤ M. Then ∥τ_{t}f∥_{BMOd} ≤ A and I + t is contained in a dyadic interval J with |J|/|I| = 2^{m(t)} ≤ 2^{M}, and thus, using (8),

φ_{f}(I) = φ_{τ_{t}f}(I + t) ≤ \frac{|J|}{|I|}φ_{τ_{t}f}(J) ≤ 2^{M}∥τ_{t}f∥_{BMOd}^{2} ≤ 2^{M}A^{2}.

Consequently, φ_{f}(I) is uniformly bounded for all I of dyadic length ≤ 2^{−M}; this easily implies, using (6) and (8), that φ_{f}(I) is uniformly bounded for all intervals I ⊆ 𝕋, i.e. f ∈ BMO.

Proof of Theorem 2. By Theorem 1, it remains only to show that (iv) ⇒ (i). Hence, assume that (iv) holds. Since Q_{p}^{d} ⊂ BMOd, Lemma 7 shows that f ∈ BMO.

Now, choose some t ∈ E. Since BMO is translation invariant, also τ_{t}f ∈ BMO; furthermore, τ_{t}f ∈ Q_{p}^{d} by assumption. Hence τ_{t}f ∈ Q_{p} by Theorem 8, and thus f ∈ Q_{p} since Q_{p} is translation invariant.

Proof of Theorem 5. We write f_{t}(x) = F (t, x) and h_{t}(x) = F (t, x + t).

Suppose that I is an interval of dyadic length 2^{−n}, n ≥ 1. We fix t ∈ T and let (ignoring the case when I + t is dyadic) I′ and I′′ be the two dyadic intervals of length 2^{−n} that intersect I. Then I ⊂ I′ ∪ I′′ and Lemma 1 yields

\begin{aligned}ψ_{h_{t}}(I)^{1/2} & = ψ_{f_{t}}(I + t)^{1/2} ≤ C(ψ_{f_{t}}(I′)^{1/2} + ψ_{f_{t}}(I′′)^{1/2} + |f_{t}(I′) − f_{t}(I′′)|) \\ & ≤ C(∥f_{t}∥_{Q_{p}^{d}} + |f_{t}(I′) − f_{t}(I′′)|). \end{aligned} (21)

Let m(t) be as in Lemma 6, and let, for l = 0, …, m(t), J_{l} and K_{l} be the dyadic intervals of length 2^{−n+l} that contain I′ and I′′, respectively. Then J_{m(t)} = K_{m(t)} and, using (7),

|f_{t}(I′) − f_{t}(I′′)| ≤ ∑_{l=1}^{m(t)}(|f(J_{l−1}) − f(J_{l})| + |f(K_{l}) − f(K_{l−1})|)

≤ ∑_{l=1}^{m(t)}(φ_{f_{t}}(J_{l})^{1/2} + φ_{f_{t}}(K_{l})^{1/2})

≤ m(t)∥f_{t}∥_{Q_{p}^{d}}.

Consequently, (21) yields

ψ_{h_{t}}(I)^{1/2} ≤ Cm(t)∥f_{t}∥_{Q_{p}^{d}}.

Since g = ∫_{𝕋} h_{t} dt and ψ_{f}(I)^{1/2} may be regarded as an L^{2} norm, we may use Minkowski’s inequality and obtain

ψ_{g}(I)^{1/2} ≤ ∫_{𝕋} ψ_{h_{t}}(I)^{1/2} dt ≤ C ∫_{𝕋} m(t)∥f_{t}∥_{Q_{p}^{d}} dt.

Consequently, by Hölder’s inequality, choosing r < ∞ such that 1/r + 1/q = 1,

ψ_{g}(I)^{1/2} ≤ C∥m(t)∥_{L^{r}(𝕋)}∥F ∥_{L^{q}(Q_{p}^{d})}.

By Lemma 6, this shows that ψ_{g}(I) is uniformly bounded when I is an interval of dyadic length ≤ 1/2. The case I = 𝕋 follows easily (we omit the details), and thus g ∈ Q_{p} by Lemma 5.

References

[1] R. Aulaskari, J. Xiao, R.H. Zhao, On subspaces and subsets of BMOA and UBC. Analysis 15 (1995), 101–121.

[2] B. Davis, Hardy spaces and rearrangements. Trans. Amer. Math. Soc. 261 (1980), 211–233.

[3] N. Dunford & J.T. Schwartz, Linear Operators, Part I. Interscience, New York 1958.

[4] M. Essén and J. Xiao, Some results on Q_{p} spaces, 0 < p < 1. J. reine angew. Math. 485 (1997), 173–195.

[5] J.B. Garnett and P.W. Jones, BMO from dyadic BMO. Pacific J. Math. 99 (1982), 351–371.

Department of Mathematics, Uppsala University, PO Box 480, S-751 06 Uppsala, Sweden

  E-mail address: svante.janson@math.uu.se