INTERPOLATION OF SUBCOUPLES AND QUOTIENT COUPLES

Svante Janson

Abstract. We extend recent results by Pisier on K-subcouples, i.e. subcouples of an interpolation couple that preserve the K-functional (up to constants) and corresponding notions for quotient couples. Examples include interpolation (in the pointwise sense) and a reinterpretation of the Adamyan-Arov-Krein theorem for Hankel operators.

Introduction

It is well-known that even if we know the interpolation spaces of a certain couple of spaces, by the real or complex method, say, there is no general formula that enables us to directly obtain the interpolation spaces of a couple of subspaces or quotient spaces of a given couple. Indeed, there are examples that show that interpolation of subspaces (or quotient spaces) in general may be ill-behaved, see Triebel [26] and Wallstén [27].

Nevertheless, in many natural examples, there are very simple relations (often with less simple proofs) between the interpolation spaces of a couple and a subcouple (or quotient couple). We will here consider only the real method of interpolation, and the crucial property then is that the K-functional for a subcouple equals (within constants) the Kfunctional for the supercouple. While this property was recognized a long time ago by Peetre [17], and has been proven in many concrete cases, it has not been studied in detail until Pisier [21] exploited several abstract properties, including relations with quotient couples, duality and approximation. (See also [22], [23], [3].) The purpose of the present paper is to emphasize this part of Pisier’s work and to develop it in greater detail.

The basic definition and some simple consequences of it are given in Section 2. Quotient couples are studied in Sections 3 and 4. Section 5 treats simultaneous approximation in several norms. Duality is studied in Section 6. Various examples and counter examples are collected in Section 7, and one example, a reinterpretation of the Adamyan-Arov-Krein theorem for Hankel operators, is studied further in Section 8. Finally, a couple of remarks suggesting possible future extensions are given in Section 9.

1. Preliminaries

For the reader’s convenience, we give some definitions and results that will be used later. For details see for example Bergh and Löfström [2] or Brudnyĭ and Krugljak [4].

Although we are interested in vector spaces (real or complex), it is sometimes convenient to forget the scalar multiplication and regard the spaces as Abelian groups (under addition). We have to distinguish, however, between quasi-normed (Abelian) groups, defined as in [2], and quasi-normed vector spaces, which satisfy the further requirement that the quasi-norm be homogeneous (of degree 1), i.e. ∥tx∥ = |t| ∥x∥.

Typeset by AMS-TEX

This work was done at the Mittag-Leffler Institute. I am particularly grateful to Richard Rochberg for helpful discussions.

For example, the vector space S_{0} of all finite rank operators on a given Hilbert space, with ∥T ∥_{S_{0}} = rank(T), is a quasi-normed group but not a quasi-normed vector space.

It turns out that most of our results are naturally stated for quasi-normed groups, so we will work in that setting; the reader may well assume that all spaces are quasinormed vector spaces, or even Banach spaces. We will also abuse language by saying space, subspace, etc. instead of group, subgroup, etc.

A quasi-normed group is a metrizable topological group and a quasi-normed vector space is a metrizable topological vector space; the space S_{0} defined above is not a topological vector space.

We will sometimes assume that the spaces are complete (it may be simpler to assume so always). A quasi-Banach space is a complete quasi-normed vector space. Note that the closed graph theorem holds for such spaces.

Two non-negative functions f and g on some set are equivalent if

cf ≤ g ≤ Cf

for some constants c, C > 0. In particular, this defines equivalence between two quasinorms on the same space. We will usually not distinguish between equivalent quasi-norms and let X = Y, where X and Y are quasi-normed spaces, signify that X equals Y as sets (and algebraically) and that their quasi-norms are equivalent.

If X is a quasi-normed space and p > 0, we let (X)^{p} denote X with the new quasi-norm ∥ ∥_{X}^{p}. Note that (X)^{p} and X have the same topology, but (in general) non-equivalent quasi-norms. X is p-normed if (X)^{p} is normed. Every quasi-norm is equivalent to a p-norm for some p > 0.

We define ∥x∥_{X} = ∞ for x ∉ X.

A quasi-normed couple is a couple X̄ = (X_{0}, X_{1}) of two quasi-normed spaces that are continuously included as subspaces of some Hausdorff topological group. We may then define Σ(X̄) = X_{0} + X_{1} and Δ(X̄) = X_{0} ∩ X_{1} as quasi-normed spaces. We will write ∥ ∥_{j} for ∥ ∥_{X_{j}} when no confusion may occur.

The main functionals of real interpolation and approximation are defined by

K(t, x; X̄) = inf {∥x_{0}∥_{0} + t∥x_{1}∥_{1} : x = x_{0} + x_{1}}, x ∈ Σ(X̄), t > 0,
J(t, x; X̄) = max {∥x∥_{0}, t∥x∥_{1}}, x ∈ Δ(X̄), t > 0,
E(t, x; X̄) = inf {∥x_{1}∥_{1} : x = x_{0} + x_{1}, ∥x_{0}∥_{0} ≤ t}, x ∈ Σ(X̄), t > 0.

(The E-functional may be infinite.)

The K-method is defined as follows. Let Φ be a quasi-Banach lattice on (0, ∞) that contains the function 1 ∧ t. (Such lattices are called parameters of the K-method.) Then

K_{Φ}(X̄) = {x ∈ Σ(X̄) : ∥x∥_{K_{Φ}(X̄)} = ∥K(·, x; X̄)∥_{Φ} < ∞}.

K_{Φ}(X̄) is a quasi-normed space. It is complete if X_{0}, X_{1} are, and a Banach space if X_{0}, X_{1}, Φ are. In particular, the choice Φ = L^{q}(t^{−θ}, dt/t) = {f : t^{−θ}f ∈ L^{q}(dt/t)} yields the usual interpolation spaces (X̄)_{θq}, 0 < θ < 1, 0 < q ≤ ∞.

For the general definition of the J-method we refer to [4] (which also discusses the E-method). This method is fully developed for Banach spaces only; the interpolation space J_{Φ}(X̄) is defined for a Banach couple X̄ and a Banach lattice Φ ⊂ L^{1}(1 ∧ t^{−1}, dt/t) with Φ ≠ {0}. For certain parameters Φ, the method can be defined for all quasi-normed couples. In particular, this is the case for Φ = L^{q}(t^{−θ}, dt/t), (we may here take 0 < θ < 1 and 0 < q ≤ ∞), and then J_{Φ}(X̄) = K_{Φ}(X̄) = (X̄)_{θq}, see [2].

If X is an intermediate space, i.e. Δ(X̄) ⊂ X ⊂ Σ(X̄), we define X^{0} to be the closure of Δ(X̄) in X. In particular,

x ∈ X_{0}^{0} ⇔ x ∈ X_{0} and lim_{t→0}K(t, x; X̄) = 0,  (1.1)

and similarly for X_{1}^{0}. The couple X̄ is regular if X_{0}^{0} = X_{0} and X_{1}^{0} = X_{1}, i.e. if Δ(X̄) is dense in X_{j}, j = 0, 1.

If X is an intermediate space we also define its Gagliardo completion X^{c} to be the set of all limits in X_{0} +X_{1} of sequences that are bounded in X. A space X is Gagliardo complete if X^{c} = X; the couple X̄ is Gagliardo complete if X_{0} and X_{1} are. (We do not require the quasi-norms to be equal, but that may be assumed by renorming.) Note that X_{0}^{c} and X_{1}^{c} are special cases of the K-method, obtained by choosing Φ = L^{∞} and L^{∞}(t^{−1}), respectively.

If X̄ is a Banach couple we define the dual couple X̄′ = (X′_{0}, X′_{1}) by

X′_{j} = {x^{∗} ∈ (X_{0} ∩ X_{1})^{∗} : ∥x^{∗}∥_{X′_{j}} = sup {|⟨x^{∗}, x⟩| : x ∈ X_{0} ∩ X_{1} with ∥x∥_{j} ≤ 1} < ∞}.

Then X′_{0} + X′_{1} = (X_{0} ∩ X_{1})^{∗} (isometrically). If X̄ is regular, we furthermore have natural isometries X′_{j} ≅ X_{j}^{∗} and X′_{0} ∩ X′_{1} ≅ (X_{0} + X_{1})^{∗}.

It is easily seen that a dual couple is Gagliardo complete.

2. Subcouples

We say that Ȳ = (Y_{0}, Y_{1}) is a subcouple of a quasi-normed couple X̄ = (X_{0}, X_{1}) if Y_{j} is a subspace of X_{j} with the induced quasi-norm, j = 0, 1. Obviously, Ȳ then is a quasi-normed couple and

K(t, y, Ȳ) ≥ K(t, y, X̄), t > 0, y ∈ Y_{0} + Y_{1}.  (2.1)

In general, there is no converse inequality; as remarked in the introduction, we will study the case when there is.

Definition. A K-subcouple of a quasi-normed couple X̄ is a subcouple Ȳ such that for some C < ∞,

K(t, y; Ȳ) ≤ CK(t, y; X̄), t > 0, y ∈ Y_{0} + Y_{1}.  (2.2)

Because of (2.1), the condition can equivalently be given as

K(t, y; Ȳ) ≍ K(t, y; X̄), t > 0, y ∈ Y_{0} + Y_{1}.  (2.3)

Remark 2.1. This notion was introduced by Peetre [17], although he imposed the stronger requirement

K(t, y, Ȳ) = K(t, y, X̄),  (2.4)

i.e. C = 1 in (2.2). (Our definition is the same as Pisier’s [21], although he uses the term K-closed subcouple.) There are several reasons for allowing a constant in the definition; for example, it means that the property is preserved if the norms are replaced by any equivalent ones, and it is (to our knowledge) needed in many of the results and examples below.

We call subcouples satifying (2.4) exact K-subcouples.

Problem 2.1. If Ȳ is a K-subcouple of X̄, is it possible to renorm X̄ (with equivalent quasi-norms) such that Ȳ becomes an exact K-subcouple?

Remark 2.2. If X̄ and Ȳ are quasi-Banach spaces, condition (2.1) for a fixed t is equivalent to Y_{0} + Y_{1} being a closed subspace of X_{0} + X_{1}. The main point of the definition is, thus, that the constant C does not depend on t.

Remark 2.3. We may similarly define (exact) K_{p}-subcouples (0 < p ≤ ∞) by replacing K by K_{p}. Obviously, Ȳ is a K_{p}-subcouples of X̄ if and only if it is a K-subcouple. For Banach couples and p ≥ 1, it follows by Holmstedt and Peetre [11] that the best constant in (2.2) is independent of p; in particular Ȳ is an exact K_{p}-subcouple if and only if it is an exact K-subcouple. This may fail for quasi-normed couples, see Example 7.9.

There is a similar definition using the E-functional.

Definition. An E-subcouple of a quasi-normed couple X̄ is a subcouple Ȳ such that for some C_{1}, C_{2} < ∞,

E(C_{1}t, y; Ȳ) ≤ C_{2}E(t, y; X̄), t > 0, y ∈ Y_{0} + Y_{1}.  (2.5)

In fact, the definitions are equivalent.

Proposition 2.1. Ȳ is a K-subcouple of X̄ if and only if it is an E-subcouple.

Proof. Suppose that Ȳ is a K-subcouple of X̄ and let y ∈ Y_{0} + Y_{1} and t > 0. Let s > E(t, y; X̄). By definition there exists x_{0} ∈ X_{0} with ∥x_{0}∥_{X_{0}} ≤ t and ∥y − x_{0}∥_{X_{1}} < s. Hence

K(t/s, y; X̄) ≤ ∥x_{0}∥_{X_{0}} + \frac{t}{s}∥y − x_{0}∥_{X_{1}} < 2t

and thus K(t/s, y; Ȳ) < 2Ct. Consequently y = y_{0} + y_{1} with ∥y_{0}∥_{Y_{0}} + \frac{t}{s}∥y_{1}∥_{Y_{1}} < 2Ct, which implies E(2Ct, y; Ȳ) ≤ ∥y_{1}∥_{Y_{1}} < 2Cs. Thus E(2Ct, y; Ȳ) ≤ 2CE(t, y; X̄).

Conversely, suppose that Ȳ is an E-subcouple, y ∈ Y_{0} +Y_{1} and t > 0. If s > K(t, y; X̄), then y = x_{0} + x_{1} with ∥x_{0}∥_{X_{0}} + t∥x_{1}∥_{X_{1}} < s and thus E(s, y; X̄) ≤ ∥x_{1}∥_{X_{1}} < s/t. Hence E(C_{1}s, y; Ȳ) ≤ C_{2}s/t, and y = y_{0} + y_{1} with ∥y_{0}∥_{Y_{0}} ≤ C_{1}s and ∥y_{1}∥_{Y_{1}} < C_{2}s/t, which yields K(t, y; Ȳ) ≤ C_{1}s + C_{2}s. Consequently K(t, y; Ȳ) ≤ (C_{1} + C_{2})K(t, y; X̄). □

Remark 2.4. If we use K_{∞} in (2.2), the best constant there equals the infinum of max(C_{1}, C_{2}) for (2.5). In particular, Ȳ is an exact K_{∞}-subcouple if and only if C_{1} may be chosen arbitrarily close to 1, and C_{2} equal to 1. We do not know whether this is equivalent to having C_{1} = C_{2} = 1 in (2.5), i.e. E(t, y; Ȳ) = E(t, y; X̄) (such couples are called exact E-subcouples).

Examples of K-subcouples (and thus E-subcouples) are given in Section 7, as well as some counter examples.

We begin with some easy consequences of the definition.

Proposition 2.2.

(i) (Y_{0}, Y_{1}) is a K-subcouple of (X_{0}, X_{1}) ⇔ (Y_{1}, Y_{0}) is a K-subcouple of (X_{1}, X_{0})

(ii) If Ȳ is a K-subcouple of X̄ and Z̄ is a K-subcouple of Ȳ, then Z̄ is a K-subcouple

of X̄. (iii) If Z̄ is a subcouple of Ȳ, Ȳ is a subcouple of X̄ and Z̄ is a K-subcouple of X̄, then

Z̄ is a K-subcouple of Ȳ. (iv) If (X_{0}, X_{1}) is a quasi-normed couple, then (X_{0}^{0}, X_{1}) (X_{0}, X_{1}^{0}) and (X_{0}^{0}, X_{1}^{0}) are

K-subcouples of it.

(v) If Ȳ is a K-subcouple of X̄, then (Y_{0}^{0}, Y_{1}) is a K-subcouple of X̄ and of (X_{0}^{0}, X_{1}), and similarly for (Y_{0}, Y_{1}^{0}) and (Y_{0}^{0}, Y_{1}^{0}).

(vi) If Ȳ is a K-subcouple of X̄, then (\overline{Y}_{0}, \overline{Y}_{1}) is too. Here \overline{Y}_{j} is the closure of Y_{j} in

X_{j}, j = 0, 1.) (vii) If Ȳ is a K-subcouple of X̄, then Y_{j}^{c} is a subspace of X_{j}^{c} with an equivalent norm,

j = 0, 1, and (Y_{0}^{c}, Y_{1}^{c}) is a K-subcouple of (X_{0}^{c}, X_{1}^{c}). (viii) If (Y_{0}, Y_{1}) is a K-subcouple of (X_{0}, X_{1}) and p_{0}, p_{1} > 0, then ((Y_{0})^{p_{0}}, (Y_{1})^{p_{1}}) is a

K-subcouple of ((X_{0})^{p_{0}}, (X_{1})^{p_{1}}).

Proof. (i), (ii) and (iii) follow directly from the definition.

(iv) follows because, for example, if x ∈ X_{0}^{0} + X_{1}, then any splitting x = x_{0} + x_{1} with x_{j} ∈ X_{j} has x_{0} ∈ X_{0}^{0}.

(v) follows from (ii), (iii) and (iv).

(vi) is easy.

For (vii), the definition yields ∥y∥_{Y_{0}^{c}} = sup_{t} K(t, y; Ȳ) ≍ sup_{t} K(t, y; X̄) = ∥y∥_{X_{0}^{c}} for y ∈ Y_{0}^{c}, and similarly for Y_{1}^{c}. This proves the first statement. Hence, if Y_{j}^{c′} denotes Y_{j}^{c} with the quasi-norm of X_{j}^{c},

K(t, y; Y_{0}^{c′}, Y_{1}^{c′}) ≍ K(t, y; Y_{0}^{c}, Y_{1}^{c}) = K(t, y; Y_{0}, Y_{1}) ≍ K(t, y; X_{0}, X_{1})
= K(t, y; X_{0}^{c}, X_{1}^{c})

for y ∈ Y_{0}^{c} + Y_{1}^{c} = Y_{0} + Y_{1} and t > 0 (assuming, as we may, that the quasi-norms are p-norms).

(viii) follows from Proposition 2.1 and the definition of E-subcouple. □

One important, immediate consequence of the definition is an interpolation formula.

Theorem 2.1. If Ȳ is a K-subcouple of a quasi-normed couple X̄, then

K_{Φ}(Ȳ) = K_{Φ}(X̄) ∩ (Y_{0} + Y_{1})

for every Φ. The quasi-norm in K_{Φ}(Ȳ) is equivalent to the quasi-norm inherited from K_{Φ}(X̄). In particular,

(Ȳ)_{θq} = (X̄)_{θq} ∩ (Y_{0} + Y_{1}), 0 < θ < 1, 0 < q ≤ ∞. □

We can also use reiteration. Recall that a couple Ȳ is said to be complete when both Y_{0} and Y_{1} are.

Theorem 2.2. Suppose that Ȳ is a complete K-subcouple of a quasi-normed couple X̄.

(i) (K_{Φ_{0}}(Ȳ), K_{Φ_{1}}(Ȳ)) is a K-subcouple of (K_{Φ_{0}}(X̄), K_{Φ_{1}}(X̄)) for any two parameters Φ_{0} and Φ_{1};

(ii) (Y_{0}, K_{Φ}(Ȳ)) is a K-subcouple of (X_{0}, K_{Φ}(X̄)) for any parameter Φ that contains

an unbounded positive concave function. (iii) In particular, if 0 < θ_{0} < θ_{1} < 1 and 0 < q_{0}, q_{1} ≤ ∞, then ((Ȳ)_{θ_{0}q_{0}}, (Ȳ)_{θ_{1}q_{1}})

is a K-subcouple of ((X̄)_{θ_{0}q_{0}}, (X̄)_{θ_{1}q_{1}}), and similarly for (Y_{0}, (Ȳ)_{θq}), ((Ȳ)_{θq}, Y_{1}), 0 < θ < 1, 0 < q ≤ ∞.

Proof. We observe first that, by Theorem 2.1, the couples constructed from Ȳ are subcouples of the couples constructed from X̄ (up to equivalent norms).

(i). The generalized Holmstedt formula

K(t, y; K_{Φ_{0}}(Ȳ), K_{Φ_{1}}(Ȳ)) ≍ K(t, K(·, y; Ȳ); Φ̂_{0}, Φ̂_{1}),  (2.6)

where Φ̂_{j} = K_{Φ_{j}}(L_{∞}, L_{∞}(1/s)), holds for every complete quasi-normed couple Ȳ by Kdivisibility, cf. [4, proof of 3.3.24 and (3.9.9)]. Part (i) now follows by applying this formula to both X̄ and Ȳ (provided also X̄ is complete; in general we obtain at least an inequality which suffices).

(ii). Choosing Φ_{0} = L^{∞}, it follows in particular that (Y_{0}^{c}, K_{Φ}(Ȳ)) is a K-subcouple of (X_{0}^{c}, K_{Φ}(X̄)), which gives the estimate, for y ∈ Y_{0}^{c} + K_{Φ}(Y) and t > 0,

K(t, y; Y_{0}^{c}, K_{Φ}(Ȳ)) ≤ C_{1}K(t, y; X_{0}^{c}, K_{Φ}(X̄)) ≤ C_{1}K(t, y; X_{0}, K_{Φ}(X̄)).  (2.7)

We claim that

K(t, y; Y_{0}, K_{Φ}(Ȳ)) ≤ C_{2}∥y∥_{Y_{0}^{c}}, t > 0, y ∈ Y_{0}^{c}  (2.8)

for some C_{2} < ∞, which implies that if y = y_{0} + y_{1} then

K(t, y; Y_{0}, K_{Φ}(Ȳ)) ≤ C_{3}K(t, y_{0}; Y_{0}, K_{Φ}(Ȳ)) + C_{3}K(t, y_{1}; Y_{0}, K_{Φ}(Ȳ))
≤ C_{4}(∥y_{0}∥_{Y_{0}^{c}} + t∥y_{1}∥_{K_{Φ}(Ȳ)})

and thus

K(t, y; Y_{0}, K_{Φ}(Ȳ)) ≤ C_{4}K(t, y; Y_{0}^{c}, K_{Φ}(Ȳ)), t > 0, y ∈ Y_{0}^{c} + K_{Φ}(Y),

which together with (2.7) gives the sought estimate.

In order to prove (2.8), we assume for simplicity that Ȳ is normed and that ∥y∥_{Y_{0}^{c}} = 1; the general case is similar. Then there exists a sequence y_{n} ∈ Y_{0} with ∥y_{n}∥_{Y_{0}} ≤ 2 and ∥y − y_{n}∥_{Y_{1}} ≤ 1/n. Let ω be a fixed unbounded positive concave function in Φ. Then, if n, m ≥ N,

K(t, y_{n} − y_{m}; Ȳ) ≤ ∥y_{n} − y_{m}∥_{0} ∧ t∥y_{n} − y_{m}∥_{1} ≤ 4(1 ∧ \frac{t}{N}) ≤ \frac{4}{ω(N)} ω(t)

and thus

∥y_{n} − y_{m}∥_{K_{Φ}(Ȳ)} ≤ \frac{4}{ω(N)}∥ω∥_{Φ} → 0 as N → ∞.

Hence (y_{n}) is a Cauchy sequence in K_{Φ}(Ȳ). Since K_{Φ}(Ȳ) is complete and y_{n} converges to y in Σ(Ȳ), y_{n} → y in K_{Φ}(Ȳ) too, which yields

K(t, y; Y_{0}, K_{Φ}(Ȳ)) ≤ sup_{n} ∥y_{n}∥_{0} ≤ 2,

thus proving (2.8).

Part (iii) follows from (i) and (ii). □

Remark 2.5. The condition in (ii) that Φ contains an unbounded positive concave function fails if and only if K_{Φ}(X̄) ⊂ X_{0}^{c} for every couple X̄. The result is not true without this condition, see Example 7.8.

Remark 2.6. Theorem 2.2(iii) follows from the traditional Holmstedt formula, [10] or e.g. [2, Section 3.6]. One can easily check that the formula holds for all quasi-normed couples (and q_{0}, q_{1} > 0); hence the last part of the theorem holds also without assuming completeness of Ȳ. (We do not know whether completeness really is required for the first, more general, parts.)

Problem 2.2. Is there a converse to Theorem 2.2, similar to Wolff’s theorem? More precisely, if Ȳ is a subcouple of X̄ such that (Y_{0}, (Ȳ)_{θ_{1}q_{1}}) and ((Ȳ)_{θ_{0}q_{0}}, Y_{1}) are K-subcouples of (X_{0}, (X̄)_{θ_{1}q_{1}}) and ((X̄)_{θ_{0}q_{0}}, X_{1}), where 0 < θ_{0} < θ_{1} < 1 and 0 < q_{0}, q_{1} ≤ ∞, is Ȳ a K-subcouple of X̄?

We end this section with some more technical results for K-subcouples.

Proposition 2.3. If Ȳ is a K-subcouple of X̄, and X_{0} ∩ X_{1} is dense in X_{0}, then Y_{0} ∩ Y_{1} is dense in Y_{0}.

Proof. By the assumptions and (1.1),

y ∈ Y_{0} ⇒ y ∈ Y_{0} and y ∈ X_{0} = X_{0}^{0} ⇒ y ∈ Y_{0} and lim_{t→0} K(t, y; X̄) = 0

⇒ y ∈ Y_{0} and lim_{t→0} K(t, y; Ȳ) = 0 ⇒ y ∈ Y_{0}^{0}. □

Proposition 2.4. Suppose that Ȳ is a K-subcouple of X̄, with X_{j} complete and Y_{j} a closed subspace of X_{j}, j = 0, 1. Then Y_{0} + Y_{1} is a closed subspace of X_{0} + X_{1} (with an equivalent quasi-norm).

Proof. The standard quasi-norms on X_{0} + X_{1} and Y_{0} + Y_{1} are K(1, ·; X̄) and K(1, ·; Ȳ), respectively, which are assumed to be equivalent on Y_{0} + Y_{1}. Since Y_{0} + Y_{1} is complete in its own quasi-norm, and thus also in the one inherited from X_{0} + X_{1}, it is closed in X_{0} + X_{1}. □

Proposition 2.5. If Ȳ is a Gagliardo complete K-subcouple of X̄, then Y_{j} = X_{j} ∩ (Y_{0} + Y_{1}), j = 0, 1.

Proof. If y ∈ X_{0} ∩ (Y_{0} + Y_{1}), then

K(t, y; Ȳ) ≤ CK(t, y; X̄) ≤ C∥y∥_{X_{0}}

and thus y ∈ Y_{0}^{c} = Y_{0}. Hence Y_{0} = X_{0} ∩ (Y_{0} + Y_{1}), and similarly Y_{1} = X_{1} ∩ (Y_{0} + Y_{1}). □

Remark 2.7. Conversely, if Y_{j} = X_{j} ∩ (Y_{0} + Y_{1}), j = 0, 1, and X̄ is Gagliardo complete, then Ȳ is Gagliardo complete.

Remark 2.8. We will see in Example 7.8 that Proposition 2.5 is not true without the hypothesis that Ȳ is Gagliardo complete, even for Banach couples.

3. Definition of quotient couples

Given a closed subcouple (Y_{0}, Y_{1}) of a quasi-normed couple (X_{0}, X_{1}) (i.e. Y_{j} is a closed subspace of X_{j}), we can form the two quotient spaces X_{0}/Y_{0} and X_{1}/Y_{1}. These are quasinormed spaces, but in order to regard them as a quasi-normed couple, we also have to regard them as subspaces of a common containing space. (A simple example of the problem that may arise is given by X_{0} = X_{1} = R^{2}, Y_{0} = {(x, 0)}, Y_{1} = {(0, x)}, where natural definitions would give X_{0}/Y_{0}+X_{1}/Y_{1} = {0}.) We, of course, require the quotient mappings X_{j} → X_{j}/Y_{j} to define a mapping (X_{0}, X_{1}) → (X_{0}/Y_{0}, X_{1}/Y_{1}) of the couples, which means that they are restrictions of a continuous map π : X_{0} + X_{1} → (X_{0}/Y_{0}) + (X_{1}/Y_{1}). This map is onto and thus it defines a continuous bijection (X_{0} +X_{1})/Z → (X_{0}/Y_{0})+(X_{1}/Y_{1}), where Z = ker(π) is a closed subspace of X_{0} + X_{1}. Furthermore, the diagram

X_{0} ⊕ X_{1} ⟶ (X_{0}/Y_{0}) ⊕ (X_{1}/Y_{1})
↓  ↓
X_{0} + X_{1} ⟶^{π} (X_{0}/Y_{0}) + (X_{1}/Y_{1})

commutes, all mappings in it are surjective, and all except π, by definition, induce the quasi-norms on their respective ranges as the corresponding quotient norm. Hence, if u ∈ (X_{0}/Y_{0}) + (X_{1}/Y_{1}),

∥u∥ = inf {∥u_{0}∥_{X_{0}/Y_{0}} + ∥u_{1}∥_{X_{1}/Y_{1}} : u = u_{0} + u_{1}}
= inf {∥x_{0}∥_{X_{0}} + ∥x_{1}∥_{X_{1}} : u = π(x_{0} + x_{1})} = inf {∥x∥_{X_{0}+X_{1}} : u = π(x)};

that is, the mapping (X_{0} + X_{1})/Z → (X_{0}/Y_{0}) + (X_{1}/Y_{1}) is an isometry.

This shows that we may take the containing space for the quotient couple as (X_{0}+X_{1})/Z for some closed subspace Z of X_{0} + X_{1}. Conversely, if Z ⊂ X_{0} + X_{1} is closed and π : X_{0} + X_{1} → (X_{0} + X_{1})/Z is the quotient mapping, π induces continuous embeddings X_{j}/(Z ∩X_{j}) → (X_{0} +X_{1})/Z, (j = 0, 1); if Y_{j} = Z ∩X_{j}, we thus have defined a compatible couple (X_{0}/Y_{0}, X_{1}/Y_{1}).

Given X_{0}, X_{1}, Y_{0}, Y_{1}, the conditions Z ∩ X_{j} = Y_{j} may be satisfied by several subspaces Z of X_{0} +X_{1}. This happens already when Y_{0} = Y_{1} = {0}, where we may take any Z which intersects X_{0} and X_{1} in {0} only, and regard X_{0} and X_{1} as subspaces of (X_{0} + X_{1})/Z; this introduces new identifications of elements in X_{0} and X_{1}. This should be avoided, and returning to the general case, two elements in X_{0}/Y_{0} and X_{1}/Y_{1} should be identified only when we can choose a common representative in X_{0} ∩ X_{1} for them; in other words π should map X_{0} ∩ X_{1} onto (X_{0}/Y_{0}) ∩ (X_{1}/Y_{1}). The following (purely algebraic) lemma shows that this requires Z to equal Y_{0} + Y_{1}.

Lemma 3.1. Suppose that (Y_{0}, Y_{1}) is a subcouple of (X_{0}, X_{1}) and that Z is a subspace of X_{0} + X_{1}. Then X_{0}/Y_{0} and X_{1}/Y_{1} may be regarded as subspaces of (X_{0} + X_{1})/Z if and only if Y_{0} = Z ∩ X_{0} and Y_{1} = Z ∩ X_{1}. Furthermore, in this case π (defined above) maps X_{0} ∩ X_{1} onto (X_{0}/Y_{0}) ∩ (X_{1}/Y_{1}) if and only if Z = Y_{0} + Y_{1}.

Proof. The first part is obvious. For the second part, assume first that π maps X_{0}∩X_{1} onto (X_{0}/Y_{0})∩(X_{1}/Y_{1}). If z ∈ Z, then z = x_{0} −x_{1} with x_{j} ∈ X_{j}. Since π(x_{0} −x_{1}) = π(z) = 0, π(x_{0}) = π(x_{1}) ∈ (X_{0}/Y_{0})∩(X_{1}/Y_{1}) and thus π(x_{0}) = π(x_{1}) = π(x) for some x ∈ X_{0} ∩X_{1}. This means that x_{j} − x ∈ Z ∩ X_{j} = Y_{j} and z = x_{0} − x_{1} = (x_{0} − x) − (x_{1} − x) ∈ Y_{0} + Y_{1}. Conversely, if Z = Y_{0} + Y_{1} and π(x_{0}) = π(x_{1}) with x_{j} ∈ X_{j}, then x_{0} − x_{1} ∈ Z and thus x_{0} − x_{1} = y_{0} + y_{1} for some y_{j} ∈ Y_{j} ⊂ X_{j}. Thus x_{0} − y_{0} = x_{1} + y_{1} ∈ X_{0} ∩ X_{1} and π(x_{0}) = π(x_{0} − y_{0}) ∈ π(X_{0} ∩ X_{1}). □

The subcouples that allow us to form nice quotient couples are thus those described in the following definition.

Definition. A normal subcouple of a quasi-normed couple (X_{0}, X_{1}) is a subcouple (Y_{0}, Y_{1}) such that

(i) Y_{0} = X_{0} ∩ (Y_{0} + Y_{1}) and Y_{1} = X_{1} ∩ (Y_{0} + Y_{1}); (ii) Y_{0} + Y_{1} is closed in X_{0} + X_{1}.

(It follows that Y_{0} is closed in X_{0} and Y_{1} in X_{1}.)

Proposition 3.1. Suppose that (Y_{0}, Y_{1}) is a normal subcouple of (X_{0}, X_{1}). Then the quotient spaces Q_{0} = X_{0}/Y_{0} and Q_{1} = X_{1}/Y_{1} may be regarded as an interpolation couple Q̄ = (Q_{0}, Q_{1}) (also denoted by X̄/Ȳ), with Q_{0} + Q_{1} = (X_{0} + X_{1})/(Y_{0} + Y_{1}) isometrically and Q_{0} ∩ Q_{1} = (X_{0} ∩ X_{1})/(Y_{0} ∩ Y_{1}) algebraically. □

Note that (ii) in the definition above only says that Y_{0} + Y_{1} is a closed subset of X_{0} + X_{1}; we do not require its intrinsic quasi-norm to be equivalent to the one induced by X_{0} + X_{1}, although that is implied by the closed graph theorem whenever X_{0} and X_{1} are quasi-Banach spaces.

Similarly, if X_{0} and X_{1} are quasi-Banach spaces in Proposition 3.1, it follows by the closed graph theorem that Q_{0} ∩ Q_{1} and (X_{0} ∩ X_{1})/(Y_{0} ∩ Y_{1}) have equivalent quasi-norms, but the quasi-norms are in general not equal. This is related to the notion of J-quotient couples discussed in the next section.

The first condition in the definition of normal subcouples has several equivalent formulations.

Proposition 3.2. Let (Y_{0}, Y_{1}) be a subcouple of (X_{0}, X_{1}). Then the following are equivalent.

(i) Y_{j} = X_{j} ∩ (Y_{0} + Y_{1}), j = 0, 1, (ii) Y_{j} = X_{j} ∩ Z, j = 0, 1, for some Z ⊂ X_{0} + X_{1} (iii) Y_{0} ∩ X_{1} = Y_{0} ∩ Y_{1} = X_{0} ∩ Y_{1} (iv) Y_{0} ∩ X_{1} ⊂ Y_{1} and Y_{1} ∩ X_{0} ⊂ Y_{0}.

Proof. (i)⇒(ii)⇒(iii)⇒(iv) are obvious.

Assume (iv) and suppose that x ∈ X_{0} ∩ (Y_{0} + Y_{1}). Then x = y_{0} + y_{1} with y_{j} ∈ Y_{j}. Since also y_{1} = x − y_{0} ∈ X_{0}, y_{1} ∈ Y_{1} ∩ X_{0} ⊂ Y_{0} and thus x = y_{0} + y_{1} ∈ Y_{0}. Hence X_{0} ∩ (Y_{0} + Y_{1}) ⊂ Y_{0} and since the converse inclusion is obvious, and X_{1} ∩ (Y_{0} + Y_{1}) = Y_{1} follows by symmetry, (i) holds. □

Corollary 3.1. Let (X_{0}, X_{1}) be a quasi-normed couple with X_{0} ⊂ X_{1} (continuous inclusion). Then a subcouple (Y_{0}, Y_{1}) is a normal subcouple if and only if

(i) Y_{0} = X_{0} ∩ Y_{1}

(ii) Y_{1} is a closed subspace of X_{1}. □

4. Interpolation of quotient couples

An important part of Pisier’s work [21], is the equivalence between the K-subcouple condition for a subcouple and a ‘dual’ condition for quotient couples. We will in this section describe this equivalence; several of the results below are contained in [21], at least implicity, and are included here for completeness.

Lemma 4.1 (Pisier). Let Ȳ be a subcouple of a X̄. Then the following are equivalent. (i) Ȳ is a K-subcouple of X̄. (ii) There exists C < ∞ such that for every x ∈ X_{0} ∩ X_{1}, y_{0} ∈ Y_{0}, y_{1} ∈ Y_{1} and t > 0,

there exists y ∈ Y_{0} ∩ Y_{1} such that

∥x − y∥_{X_{0}} ∨ t∥x − y∥_{X_{1}} ≤ C(∥x − y_{0}∥_{X_{0}} ∨ t∥x − y_{1}∥_{X_{1}}).  (4.1)

(iii) Let \overline{\overline{X}} = X_{0} × X_{1}, \overline{\overline{Y}} = Y_{0} × Y_{1} ⊂ \overline{\overline{X}} and \overline{\overline{Δ}} = \{(x, x) : x ∈ X_{0} ∩ X_{1}\} ⊂ \overline{\overline{X}}. Then there exist C < ∞ such that if t > 0 and \overline{\overline{X}} is given the norm ∥(x_{0}, x_{1})∥_{t} = ∥x_{0}∥_{X_{0}} ∨ t∥x_{1}∥_{X_{1}}, every \overline{\overline{x}} ∈ \overline{\overline{Y}} + \overline{\overline{Δ}} has a splitting \overline{\overline{x}} = \overline{\overline{y}} + \overline{\overline{z}} with \overline{\overline{y}} ∈ \overline{\overline{Y}}, \overline{\overline{z}} ∈ \overline{\overline{Δ}} and

∥\overline{\overline{y}}∥_{t} + ∥\overline{\overline{z}}∥_{t} ≤ C∥\overline{\overline{x}}∥_{t}.  (4.2)

Proof. (i)⇔(iii). By the quasi-triangular inequality, (4.2) may be replaced by ∥\overline{\overline{y}}∥_{t} ≤ C∥\overline{\overline{x}}∥_{t} (for a different C). Since (x_{0}, x_{1}) = (y_{0}, y_{1}) + \overline{\overline{z}} with \overline{\overline{z}} ∈ \overline{\overline{Δ}} if and only if x_{0} − x_{1} = y_{0} − y_{1}, (iii) can be restated as:

If x_{0} − x_{1} ∈ Y_{0} + Y_{1}, then x_{0} − x_{1} = y_{0} − y_{1} for some y_{j} ∈ Y_{j} with

∥y_{0}∥_{0} ∨ t∥y_{1}∥_{1} ≤ C(∥x_{0}∥_{0} ∨ t∥x_{1}∥_{1}).  (4.3)

This is equivalent to (2.2) (or, more directly, the definition of K_{∞}-subcouple, cf. Remark 2.3).

(ii)⇔(iii). Similarly, (4.2) can be replaced by ∥\overline{\overline{z}}∥_{t} ≤ C∥\overline{\overline{x}}∥_{t}. Since (x_{0}, x_{1}) = \overline{\overline{y}} + (x, x) with \overline{\overline{y}} ∈ \overline{\overline{Y}} if and only if x_{0} − x ∈ Y_{0} and x_{1} − x ∈ Y_{1}, (iii) can be restated as:

If x ∈ X_{0} ∩ X_{1}, y_{0} ∈ Y_{0} and y_{1} ∈ Y_{1}, there exists z with

(x − y_{0}, x − y_{1}) − (z, z) ∈ \overline{\overline{Y}}  (4.4)

and ∥(z, z)∥_{t} ≤ C∥(x − y_{0}, x − y_{1})∥_{t}.

Since (4.4) is equivalent to x − z ∈ Y_{0} and x − z ∈ Y_{1}, this gives (ii) by letting y = x − z (and conversely). □

Theorem 4.1 (Pisier). Suppose that Ȳ is a normal subcouple of X̄ and let Q̄ = (X_{0}/Y_{0}, X_{1}/Y_{1}). The following are equivalent.

(i) Ȳ is a K-subcouple of X̄.

(ii) There exists C < ∞ such that for every x ∈ X_{0} ∩ X_{1} there exists y ∈ Y_{0} ∩ Y_{1} with

∥x − y∥_{X_{j}} ≤ C inf {∥x − u∥_{X_{j}} : u ∈ Y_{j}}, j = 0, 1.  (4.5)

(iii) There exists C < ∞ such that for every x ∈ X_{0} + X_{1}, there exists y ∈ Y_{0} + Y_{1} with

∥x − y∥_{X_{j}} ≤ C inf {∥x − u∥_{X_{j}} : u ∈ Y_{0} + Y_{1}}, j = 0, 1.  (4.6)

(iv) There exists C < ∞ such that for every z ∈ Q_{0} ∩ Q_{1} and t > 0, there exists x ∈ X_{0} ∩ X_{1} with π(x) = z and

J(t, x; X̄) ≤ CJ(t, z; Q̄).  (4.7)

(v) There exists C < ∞ such that if z ∈ Q_{0} ∩ Q_{1} then there exists x ∈ X_{0} ∩ X_{1} with

π(x) = z and

∥x∥_{X_{j}} ≤ C∥z∥_{Q_{j}}, j = 0, 1.  (4.8)

(vi) There exists C < ∞ such that if z ∈ Q_{0} + Q_{1} then there exists x ∈ X_{0} + X_{1} with

π(x) = z and

∥x∥_{X_{j}} ≤ C∥z∥_{Q_{j}}, j = 0, 1.  (4.9)

(The norms in (iii) and (vi) may be infinite.)

Proof. (i)⇒(ii). If x ∈ Y_{0} or x ∈ Y_{1}, then x ∈ Y_{0} ∩ Y_{1} because Ȳ is a normal subcouple, and we may take y = x. Otherwise, the infima in the right hand side of (4.5) are positive, because Y_{j} is closed in X_{j}, and we may choose y_{j} ∈ Y_{j} with ∥x−y_{j}∥_{X_{j}} ≤ 2 inf{∥x−u∥_{X_{j}} : u ∈ Y_{j}}. We apply Lemma 4.1 (ii) with t = ∥x − y_{0}∥_{Y_{0}}/∥x − y_{1}∥_{Y_{1}} and obtain y ∈ Y_{0} ∩ Y_{1} satisfying (4.1), which easily yields (4.5).

(ii)⇒(i). By Lemma 4.1, since (4.5)⇒(4.1).

(ii)⇔(v). Writing x′ = x − y, (ii) may be restated as:

For every x ∈ X_{0} ∩ X_{1}, there exists x′ ∈ X_{0} ∩ X_{1} with π(x′) = π(x) and ∥x′∥_{X_{j}} ≤ C∥π(x)∥_{Q_{j}}, j = 0, 1. Replacing π(x) by z ∈ Q_{0} ∩Q_{1} (recalling that π(X_{0} ∩X_{1}) = Q_{0} ∩Q_{1} by Proposition 3.1), this is (v) (with x′ instead of x).

(iii)⇔(vi). Similar.

(v)⇒(vi). Clear if z ∈ Q_{0} ∩ Q_{1}, and otherwise trivial.

(vi)⇒(v). Trivial.

(v)⇒(iv). Trivial.

(iv)⇒(v). If z ≠ 0, let t = ∥z∥_{Q_{0}}/∥z∥_{Q_{1}}; (4.7) then implies (4.8). □

We say that Q̄ = X̄/Ȳ is a J-quotient couple of X̄ if (iv) holds. (Pisier [21] uses the term J-closed.) Hence, if Ȳ is a normal subcouple of X̄, Ȳ is a K-subcouple if and only if X̄/Ȳ is a J-quotient couple of X̄.

Remark 4.1. Although x in Theorem 4.1 (iv) a priori may depend on t, it can, in fact, be chosen independently of t by (v).

Remark 4.2. The condition (iv) for a fixed t says that (X_{0}/Y_{0})∩(X_{1}/Y_{1}) = (X_{0}∩X_{1})/(Y_{0}∩ Y_{1}) with equivalent quasi-norms. Again, the main point of the condition is that C does not depend on t, cf. Remark 2.2.

We obtain easily an interpolation theorem for quotient couples using the J-method.

Theorem 4.2. Suppose that Ȳ is a normal K-subcouple of a complete quasi-normed couple X̄. Then

(X̄/Ȳ)_{θq} = (X̄)_{θq}/(Ȳ)_{θq}, 0 < θ < 1, 0 < q ≤ ∞.  (4.10)

More precisely, the quotient mapping π : X̄ → X̄/Ȳ maps (X̄)_{θq} onto (X̄/Ȳ)_{θq}, and induces an isomorphism (X̄)_{θq}/(Ȳ)_{θq} ≅ (X̄/Ȳ)_{θq} with equivalence of quasi-norms.

More generally, if X̄ is a Banach couple,

J_{Φ}(X̄/Ȳ) = J_{Φ}(X̄)/(J_{Φ}(X̄) ∩ (Y_{0} + Y_{1}))  (4.11)

for every parameter Φ.

Proof. Use the discrete definition of the J-method [2], [4] and Theorem 4.1 (iv); for (4.10) we also use Theorem 2.1. We leave the details to the reader. □

We can also consider several interpolation methods simultaneously.

Theorem 4.3. Let Ȳ be a normal K-subcouple of a Banach couple X̄. Then there exists C < ∞ such that if {Φ_{α}}_{α∈A} is a finite set of parameters for the J-method and z ∈ Σ(X̄/Ȳ), there exists x ∈ Σ(X̄) with π(x) = z and

∥x∥_{J_{Φα(X̄)}} ≤ C∥z∥_{J_{Φα(X̄/Ȳ)}}, α ∈ A.  (4.12)

Proof. We only have to consider α with 0 < ∥z∥_{J_{Φα(X̄/Ȳ)}} < ∞, and changing the norm in Φ_{α} by a constant factor we may assume that ∥z∥_{J_{Φα(X̄/Ȳ)}} = 1. We may also assume that Φ_{α} is an interpolation space for (L^{1}(dt/t), L^{1}(dt/t^{2})) [4, Corollary 3.4.6]. We now apply Theorem 4.2 to the parameter Φ = Δ(Φ_{α}) = ∩_{α}Φ_{α}, using J_{Δ(Φ_{α})} = Δ(J_{Φ_{α}}) [4, Theorem 3.4.9], observing that the equivalence constants in these theorems do not depend on the set {Φ_{α}}, which gives ∥z∥_{J_{Φ}(X̄/Ȳ)} ≤ C_{1} and the existence of an x ∈ π^{−1}(z) with ∥x∥_{J_{Φ(X̄)}} ≤ C_{2}. □

We next consider the K-method.

Definition. Let Ȳ be a normal subcouple of X̄ Then X̄/Ȳ is a K-quotient couple of X̄ if there exists C < ∞ such that for every z ∈ Σ(X̄/Ȳ) there exists x ∈ Σ(X̄) with π(x) = z and

K(t, x; X̄) ≤ CK(t, z; X̄/Ȳ), t > 0.  (4.13)

Remark 4.3. If π(x) = z then K(t, x; X̄) ≥ K(t, z; X̄/Ȳ). Hence (4.13) says that the K-functionals are equivalent (uniformly in z).

Remark 4.4. We may similarly define E-quotient couples, cf. (2.5), and prove as in Proposition 2.1 that E-quotient couples are the same as K-quotient couples.

Theorem 4.4. Let Ȳ be a normal subcouple of a complete and Gagliardo complete couple X̄. Then the following are equivalent,

(i) X̄/Ȳ is a K-quotient couple of X̄.

(ii) X̄/Ȳ is a Gagliardo complete J-quotient couple of X̄.

(iii) Ȳ is a K-subcouple of X̄ and X̄/Ȳ is Gagliardo complete.

The implications (ii) ⟺ (iii) ⟹ (i) hold also if X̄ is not Gagliardo complete.

Proof. (ii)⇔(iii) holds by Theorem 4.1.

(i)⇒(ii). Let Q_{j} = X_{j}/Y_{j} and assume that z ∈ Q_{0} ∩ Q_{1}. By assumption there exists x ∈ π^{−1}(z) such that

K(t, x; X̄) ≤ CK(t, z; Q̄) ≤ C(∥z∥_{Q_{0}} ∧ t∥z∥_{Q_{1}}).

Hence x ∈ X_{0}^{c} ∩ X_{1}^{c} and ∥x∥_{X_{j}^{c}} ≤ C∥z∥_{Q_{j}}, j = 0, 1. Since X_{j}^{c} = X_{j}, this shows that (4.8) holds, and thus Q̄ is a J-quotient couple. Similarly, if z ∈ Q_{j}^{c}, z = π(x) with ∥x∥_{X_{j}^{c}} ≤ C∥z∥_{Q_{j}^{c}}, and since X_{j} is Gagliardo complete, Q_{j} is too.

(ii)⇒(i). By first replacing the quasi-norms on X_{0}, X_{1} by equivalent p-norms, for some small p > 0, and then raising all quasi-norms to power p, we may assume that the spaces are normed groups. We use K-divisibility in the form of Cwikel’s lemma [5, Theorem V.3.4] (which holds for Gagliardo complete normed couples by the same proof) and obtain that z = ∑_{−∞}^{∞} z_{k} (in Q_{0} + Q_{1}) for some sequence {z_{k}} such that

∑_{k} ∥z_{k}∥_{Q_{0}} ∧ t∥z_{k}∥_{Q_{1}} ≤ C_{1}K(t, z; Q̄).  (4.14)

By Theorem 4.1 (vi) there exist x_{k} ∈ X_{0} + X_{1} with π(x_{k}) = z_{k} and ∥x_{k}∥_{X_{j}} ≤ C_{2}∥z_{k}∥_{Q_{j}}, and thus

∑_{k} K(t, x_{k}; X̄) ≤ ∑_{k} ∥x_{k}∥_{X_{0}} ∧ t∥x_{k}∥_{X_{1}}
≤ C_{2} ∑_{k} ∥z_{k}∥_{Q_{0}} ∧ t∥z_{k}∥_{Q_{1}} ≤ C_{3}K(t, z; Q̄).  (4.15)

In particular, ∑_{k} ∥x_{k}∥_{X_{0}+X_{1}} = ∑_{k} K(1, x_{k}; X̄) < ∞, and thus (since X_{0} + X_{1} is complete) x = ∑ x_{k} exists in X_{0} + X_{1}. Clearly, π(x) = ∑ π(x_{k}) = z and, by (4.15),

K(t, x; X̄) ≤ ∑_{k} K(t, x_{k}; X̄) ≤ C_{3}K(t, z; Q̄). □

Interpolation results for K-quotient couples follow directly from the definition. In particular, Theorem 4.4 yields, with Theorem 2.1, the following theorems.

Theorem 4.5. Let Ȳ be a normal K-subcouple of a complete quasi-normed couple X̄ such that the couple X̄/Ȳ is Gagliardo complete. Then, for any parameter Φ,

K_{Φ}(X̄/Ȳ) = K_{Φ}(X̄)/(K_{Φ}(X̄) ∩ (Y_{0} + Y_{1})) = K_{Φ}(X̄)/K_{Φ}(Ȳ). □  (4.16)

Theorem 4.6. Let Ȳ and X̄ be as in Theorem 4.5. Then there exists C < ∞ such that if z ∈ ∑(X̄/Ȳ), there exists x ∈ X_{0} + X_{1} such that π(x) = z and

∥x∥_{K_{Φ}(X̄)} ≤ C∥z∥_{K_{Φ}(X̄/Ȳ)}  (4.17)

for any parameter Φ of the K-method. □

The condition that X̄/Ȳ be Gagliardo complete in the last theorems is annoying for several reasons: it is often difficult to check in applications; we do not know whether it really is required for the results; we do not know whether it can fail to hold in Theorem 4.4.

Problem 4.1. If Ȳ is a normal K-subcouple of a complete, Gagliardo complete couple X̄, is X̄/Ȳ necessarily Gagliardo complete?

We give some results that may help to show Gagliardo completeness in applications.

Lemma 4.2. Let X̄ be a quasi-normed couple and let 0 < θ < 1, 0 < q ≤ ∞. Then X_{0} is Gagliardo complete in (X_{0}, X_{1}) iff X_{0} is Gagliardo complete in (X_{0}, (X̄)_{θq}).

Proof. By Holmstedt’s formula [10], for x ∈ X_{0} + (X̄)_{θq},

K(t, x; X_{0}, (X̄)_{θq}) ≍ t(∫_{t^{1/θ}}^{∞} (s^{−θ}K(s, x; X̄))^{q} \frac{ds}{s})^{1/q}

(with the obvious modification if q = ∞), and it follows directly that K(·, x; X_{0}, (X̄)_{θq}) is bounded if and only if K(·, x; X_{0}, X_{1}) is. Since X_{0}^{c} = X_{0} + (X_{0} ∩ X_{1})^{c} ⊂ X_{0} + (X̄)_{θq}, this completes the proof. □

Proposition 4.1. Suppose that Ȳ is a normal K-subcouple of a complete quasi-normed couple X̄.

(i) If 0 < θ_{0}, θ_{1} < 1 and 0 < q_{0}, q_{1} < ∞, then ((X̄)_{θ_{0}q_{0}}/(Ȳ)_{θ_{0}q_{0}}, (X̄)_{θ_{1}q_{1}}/(Ȳ)_{θ_{1}q_{1}}) is a Gagliardo complete K-quotient couple of ((X̄)_{θ_{0}q_{0}}, (X̄)_{θ_{1}q_{1}}).

(ii) If (X_{0}/Y_{0}, (X̄)_{θq}/(Ȳ)_{θq}) is Gagliardo complete for some θ, q with 0 < θ < 1, 0 < q ≤ ∞, then (X_{0}/Y_{0}, (X̄)_{θq}/(Ȳ)_{θq}) is Gagliardo complete for all such θ, q, and it is a K-quotient couple of (X_{0}, (X̄)_{θq}).

Proof. (i). For any couple Z̄, (Z̄)_{θ_{j}q_{j}} is Gagliardo complete in (Z_{0}, Z_{1}), and thus also in ((Z̄)_{θ_{0}q_{0}}, (Z̄)_{θ_{1}q_{1}}). Taking Z̄ = X̄/Ȳ, the result follows by Theorems 4.2 and 4.4.

(ii). By Lemma 4.2 and Theorem 4.2, X_{0}/Y_{0} is Gagliardo complete in (X_{0}/Y_{0}, X_{1}/Y_{1}) and thus in (X_{0}/Y_{0}, (X̄)_{θq}/(Ȳ)_{θq}) for any θ, q. The result now follows as for (i). □

We can obtain partial results even without Gagliardo completeness of X̄/Ȳ.

Theorem 4.7. Let Ȳ be a normal K-subcouple of a complete quasi-normed couple X̄.

(i) There exists C < ∞ such that if z ∈ ∑(X̄/Ȳ) and ω(t) is any positive function on (0, ∞) with

ω(t) → ∞ as t → ∞  and  \frac{ω(t)}{t} → ∞ as t → 0,  (4.18)

then there exist x ∈ X_{0} + X_{1} with π(x) = z and

K(t, x; X̄) ≤ CK(t, z; X̄/Ȳ) + ω(t), t > 0.  (4.19)

(ii) For any parameter Φ that contains a function ω as in (4.18),

K_{Φ}(X̄/Ȳ) = K_{Φ}(X̄)/K_{Φ}(Ȳ).  (4.20)

(iii) If (Φ_{α})_{α∈A} is a set of parameters of the K-method such that ∩_{α}Φ_{α} contains a function ω as in (4.18), there exist C_{α} < ∞ such that if z ∈ ∑(X̄/Ȳ), there exists x ∈ X_{0} + X_{1} such that π(x) = z and

∥x∥_{K_{Φα} (X̄)} ≤ C_{α}∥z∥_{K_{Φα} (X̄/Ȳ)}, α ∈ A.  (4.21)

Proof. (i). The Gagliardo completeness of X̄/Ȳ is used in the proof of Theorem 4.4 only through Cwikel’s lemma, where it is needed when K(t, z; Q̄)/(1 ∧ t) → 0 as t → ∞ or as t → 0; however, a weaker version with the right hand side of (4.14) replaced by C_{1}K(t, z; Q̄) + ω(t) holds in any case. (Check the proof in [5], first replacing ω(t) by a smaller positive increasing concave function.) The same argument as for Theorem 4.4 now yields (4.19) (with ω replaced by Cω).

(ii) and (iii) follow from (i) by replacing ω by e.g. K(1, z; X̄/Ȳ)ω(t) in (4.19). □

Remark 4.5. The parameters Φ that are excluded in Theorem 4.7 (ii) are the degenerate ones for which K_{Φ}(X̄) ⊂ X_{0}^{c} or X_{1}^{c} for every couple X̄. In particular, part (ii) applies to the usual methods ( )_{θq}, which also was proved in Theorem 4.2 using the J-method. Part (iii) applies, for example, to any finite or countable set of parameters that satisfy the condition in (ii), and to the whole scale ( )_{θq}, 0 < θ < 1, 0 < q ≤ ∞.

Remark 4.6. If, say, X_{0}/Y_{0} is Gagliardo complete in X̄/Ȳ, (4.18) may be weakened to ω(t)/t → ∞ as t → 0. In particular, we may then take Φ = L^{∞} in Theorem 4.7 (ii) and (iii).

5. Simultaneous approximation

In this section we consider the problem of approximating an element x in a quasinormed space by an element y in a subspace Y such that the error ∥x − y∥ is small. We do not look for best approximations, where the error attains its infinum, but we require the error to be at most a constant times this.

Definition. Let Y be a subspace of some vector space (or abelian group) V and let ∥ ∥ be a quasi-norm defined on another subspace of V. An element y ∈ Y is a good approximation in Y with respect to ∥ ∥, of an element x ∈ V, if

∥x − y∥ ≤ C inf_{u∈Y} ∥x − u∥  (5.1)

for some constant C < ∞. (We include the case when the right hand side of (5.1) is infinite; then every y ∈ Y is a good approximation of x.) When ∥ ∥ = ∥ ∥_{X} for a quasinormed space X, we also say good approximation with respect to X.

The definition is stated somewhat informally. It is meaningless as it stands for a single pair y and x (unless x ∈ Ȳ), but we will use it for a set of approximations, meaning that the same error constant C can be chosen for the whole set.

As long as we consider only a single quasi-normed space X and a closed subspace Y, every element in X has trivially a good approximation in Y (for any C > 1 in (5.1)). The situation becomes more complicated if we consider two or more different quasi-norms; in general, the sets of good approximation in the different quasi-norms may be very different, and there is no guarantee that there exists a simultaneous good approximation with respect to the different quasi-norms. The following result for two quasi-norms is a reformulation of Theorem 4.1 (i)–(iii).

Theorem 5.1 (Pisier). Suppose that Ȳ is a normal subcouple of a quasi-normed couple X̄. The following are equivalent.

(i) Ȳ is a K-subcouple of X̄.

(ii) Every x ∈ X_{0} ∩ X_{1} has a simultaneous good approximation y ∈ Y_{0} ∩ Y_{1}, which is

a good approximation in Y_{j} with respect to ∥ ∥_{X_{j}}, j = 0, 1. (iii) Every x ∈ X_{0} +X_{1} has a simultaneous good approximation y ∈ Y_{0} +Y_{1} with respect

to both ∥ ∥_{X_{0}} and ∥ ∥_{X_{1}}. □

We next consider simultaneous approximations with respect to more than two quasinorms, assuming that these quasi-norms come from an interpolation family. Note that if y is a good approximation of x with respect to both X_{0} and X_{1}, we have, for example, the estimate

∥x − y∥_{(X̄)_{θq}} ≤ C_{1}∥x − y∥_{X_{0}}^{1−θ}∥x − y∥_{X_{1}}^{θ} ≤ C_{2}d_{X_{0}}(x, Y_{0} + Y_{1})^{1−θ}d_{X_{1}}(x, Y_{0} + Y_{1})^{θ}(5.2)

for the error in (X̄)_{θq}. Nevertheless, y does not have to be a good approximation with respect to this quasi-norm, because much better approximations may exist, see Example 7.10.

Theorem 5.2. Let Ȳ be a normal K-subcouple of a complete quasi-normed couple X̄ such that the couple X̄/Ȳ is Gagliardo complete. Then every x ∈ X_{0} + X_{1} has a simultaneous good approximation y ∈ Y_{0} + Y_{1} with respect to the quasi-norms ∥ ∥_{K_{Φ}(X̄)}, where Φ ranges over all parameters for the K-method. (The error constant in (5.1) may be chosen the same for all Φ.) Furthermore, y ∈ K_{Φ}(Ȳ) for all parameters Φ such that x ∈ K_{Φ}(X̄).

Remark 5.1. If X̄ is Gagliardo complete, the original quasi-norms ∥ ∥_{0} and ∥ ∥_{1} are included among ∥ ∥_{K_{Φ}(X̄)}.

Proof. Let z = π(x), where π : X̄ → X̄/Ȳ is the quotient mapping. By Theorem 4.6 there exists x′ ∈ X_{0} + X_{1} with π(x′) = z = π(x) and

∥x′∥_{K_{Φ}(X̄)} ≤ C∥z∥_{K_{Φ(X̄/Ȳ)}} ≤ C∥x − u∥_{K_{Φ(X̄)}},  (5.3)

for every u ∈ Y_{0} + Y_{1}. Then y = x − x′ ∈ π^{−1}{0} = Y_{0} + Y_{1}, and (5.3) shows that y is a good approximation with respect to every K_{Φ}(X̄).

If x ∈ K_{Φ}(X̄), we may choose u = 0 in (5.3) and obtain a finite right hand side. Hence x′ ∈ K_{Φ}(X̄) and y ∈ K_{Φ}(X̄) ∩ (Y_{0} ∩ Y_{1}) = K_{Φ}(Ȳ) by Theorem 2.1. □

As remarked in the preceding section, we do not know whether the condition that X̄/Ȳ be Gagliardo complete really is needed here. We can give some partial results without this condition, proved in the same way using Theorems 4.7 and 4.3.

Theorem 5.3. Let Ȳ be a normal K-subcouple of a complete quasi-normed couple X̄.

(i) If (Φ_{α})_{α∈A} is a set of parameters of the K-method such that there exists a positive function ω as in (4.18) with ω ∈ ∩_{α}Φ_{α}, then every x ∈ X_{0}+X_{1} has a simultaneous good approximation y ∈ Y_{0} + Y_{1} with respect to the quasi-norms ∥ ∥_{K_{Φα} (X̄)}, α ∈ A. (The error constant in (5.1) may depend on α.) In particular, this applies to the quasi-norms ∥ ∥_{(X̄)_{θq}}, 0 < θ < 1, 0 < 1 ≤ ∞.

(ii) If X̄ is a Banach couple, the same conclusion holds for the J-method and any finite set of parameters. □

Remark 5.2. If X_{0}/Y_{0} is Gagliardo complete in X̄/Ȳ, we may weaken (4.18) as in Remark 4.6. In particular, if furthermore X_{0} is Gagliardo complete in X̄, we may also require that y be a good approximation with respect to X_{0}. (By symmetry, the same thing is true for X_{1} if X_{1} and X_{1}/Y_{1} are Gagliardo complete.)

6. Duality

In this section we exclusively consider Banach couples. The dual of a Banach couple X̄ was defined in Section 1. If furthermore X̄ is regular and Ȳ is a regular closed subcouple of X̄, we define

(Y_{0} ∩ Y_{1})^{⊥} = {x^{∗} ∈ (X_{0} ∩ X_{1})^{∗} : ⟨x^{∗}, y⟩ = 0 for y ∈ Y_{0} ∩ Y_{1}}  (6.1)

and

Y_{j}^{⊥} = X′_{j} ∩ (Y_{0} ∩ Y_{1})^{⊥} = {x^{∗} ∈ X′_{j} : ⟨x^{∗}, y⟩ = 0 for y ∈ Y_{0} ∩ Y_{1}}, j = 0, 1.  (6.2)

Then (Y_{0} ∩ Y_{1})^{⊥} is a closed subspace of (X_{0} ∩ X_{1})^{∗} = X′_{0} + X′_{1} and Y_{j}^{⊥} is a closed subspace of X′_{j}; thus Ȳ^{⊥} = (Y_{0}^{⊥}, Y_{1}^{⊥}) is a closed subcouple of X̄′. Moreover, there are natural isometries

(Y_{0} ∩ Y_{1})^{⊥} ≅ (X_{0} ∩ X_{1}/Y_{0} ∩ Y_{1})^{∗}

and

Y_{j}^{⊥} ≅ {x^{∗} ∈ X_{j}^{∗} : ⟨x^{∗}, y⟩ = 0 for y ∈ Y_{j}} ≅ (X_{j}/Y_{j})^{∗}, j = 0, 1.

Note that, although not shown in our notation, the spaces Y_{0}^{⊥}, Y_{1}^{⊥} and (Y_{0} ∩ Y_{1})^{⊥} depend on X̄ as well as on Ȳ.

Proposition 6.1. Suppose that Ȳ is a normal regular subcouple of a regular Banach couple X̄. Then the isomorphism

(Δ(X̄/Ȳ))^{∗} = (X_{0} ∩ X_{1}/Y_{0} ∩ Y_{1})^{∗} ≅ (Y_{0} ∩ Y_{1})^{⊥}

yields an isometry between the couples (X̄/Ȳ)′ and Ȳ^{⊥}.

Proof. By definition, identifying (Δ(X̄/Ȳ))^{∗} with (Y_{0} ∩ Y_{1})^{⊥} ⊂ (X_{0} ∩ X_{1})^{∗} by the isomorphism above, for x^{∗} ∈ (Y_{0} ∩ Y_{1})^{⊥}

∥x^{∗}∥_{(X_{j}/Y_{j})′} = sup{|⟨x^{∗}, x⟩| : x ∈ X_{0} ∩ X_{1}, ∥π(x)∥_{X_{j}/Y_{j}} ≤ 1}

and

∥x^{∗}∥_{Y_{j}^{⊥}} = ∥x^{∗}∥_{X′_{j}} = sup{|⟨x^{∗}, x⟩| : x ∈ X_{0} ∩ X_{1}, ∥x∥_{X_{j}} ≤ 1}.

Hence ∥x^{∗}∥_{Y_{j}^{⊥}} ≤ ∥x^{∗}∥_{(X_{j}/Y_{j})′}. On the other hand, let x^{∗} ∈ Y_{j}^{⊥} and x ∈ X_{0} ∩ X_{1} with ∥π(x)∥_{X_{j}/Y_{j}} < 1; then x = x_{1} + y_{1} with ∥x_{1}∥_{X_{j}} < 1 and y_{1} ∈ Y_{j}. Since Ȳ is regular, there exists y_{2} ∈ Y_{0} ∩ Y_{1} with ∥y_{1} − y_{2}∥_{Y_{j}} < 1 − ∥x_{1}∥_{X_{j}}, and thus ∥x − y_{2}∥_{X_{j}} ≤ ∥x_{1}∥_{X_{j}} + ∥y_{1} − y_{2}∥_{X_{j}} < 1. Since x − y_{2} ∈ X_{0} ∩ X_{1}, this yields

|⟨x^{∗}, x⟩| = |⟨x^{∗}, x − y_{2}⟩| ≤ ∥x^{∗}∥_{Y_{j}^{⊥}},

which implies ∥x^{∗}∥_{(X_{j}/Y_{j})′} ≤ ∥x^{∗}∥_{Y_{j}^{⊥}}. □

Suppose that Ȳ and X̄ are as in Proposition 6.1. It is then easy to see that Ȳ^{⊥} ≅ (X̄/Ȳ)′ is a K-subcouple of X̄′ if and only if X̄/Ȳ is a J-quotient couple of X̄, which, by Theorem 4.1, holds if and only if Ȳ is a K-subcouple of X̄.

We give a different proof that works also when Ȳ is not a normal subcouple.

We begin with a preliminary result.

Proposition 6.2. Suppose that Ȳ is a regular closed subcouple of a regular Banach couple X̄. Then the following are equivalent.

(i) Y_{0} + Y_{1} is a closed subspace of X_{0} + X_{1}.

(ii) Y_{0}^{⊥} + Y_{1}^{⊥} is a closed subspace of (X_{0} ∩ X_{1})^{∗} = X′_{0} + X′_{1}.

(iii) Y_{0}^{⊥} + Y_{1}^{⊥} is a closed subspace of (Y_{0} ∩ Y_{1})^{⊥}.

(iv) Y_{0}^{⊥} + Y_{1}^{⊥} = (Y_{0} ∩ Y_{1})^{⊥}.

(v) Ȳ^{⊥} is a normal subcouple of X̄′.

Proof. (i)⇔(iv). We apply Lemma 3.1 to the subcouple Ȳ^{⊥} of X̄′ with Z = (Y_{0} ∩ Y_{1})^{⊥}. We identify (X′_{0} +X′_{1})/Z = (X_{0} ∩X_{1})^{∗}/(Y_{0} ∩Y_{1})^{⊥} with (Y_{0} ∩Y_{1})^{∗}. Then the quotient map π : X′_{0} + X′_{1} → (X′_{0} + X′_{1})/Z is the natural (restriction) map X′_{0} + X′_{1} = (X_{0} ∩ X_{1})^{∗} → Y′_{0} + Y′_{1} = (Y_{0} ∩ Y_{1})^{∗}, and it induces the natural identifications X′_{j}/Y_{j}^{⊥} ≅ Y′_{j} ≅ Y_{j}^{∗}. By Lemma 3.1, (iv) holds if and only if π maps X′_{0}∩X′_{1} ≅ (X_{0}+X_{1})^{∗} onto Y′_{0}∩Y′_{1} ≅ (Y_{0}+Y_{1})^{∗}. Since π : (X_{0} + X_{1})^{∗} → (Y_{0} + Y_{1})^{∗} is the adjoint of the embedding Y_{0} + Y_{1} → X_{0} + X_{1}, this happens if and only if Y_{0} + Y_{1} is a closed subspace of X_{0} + X_{1}.

(iv)⇒(iii) is trivial.

(iii)⇒(iv). Let T : (X_{0} ∩ X_{1})/(Y_{0} ∩ Y_{1}) → (X_{0}/Y_{0}) ⊕ (X_{1}/Y_{1}) be the natural mapping. Then T is injective; furthermore

T^{∗} : (X_{0}/Y_{0})^{∗} ⊕ (X_{1}/Y_{1})^{∗} ≅ Y_{0}^{⊥} ⊕ Y_{1}^{⊥} → ((X_{0} ∩ X_{1})/(Y_{0} ∩ Y_{1}))^{∗} ≅ (Y_{0} ∩ Y_{1})^{⊥}

may be identified with the mapping (x′_{0}, x′_{1}) → x′_{0} +x′_{1} of Y_{0}^{⊥} ⊕Y_{1}^{⊥} into (Y_{0} ∩Y_{1})^{⊥}. Hence (iii) ⇒ im(T^{∗}) is closed ⇒ im(T) is closed. ⇒ T is an isomorphism into ⇒ T^{∗} is onto ⇒ (iv).

(iii)⇔(ii) is trivial.

(ii)⇔(v) follows by the definitions. □

Theorem 6.1 (Pisier). Suppose that Ȳ is a closed subcouple of a regular Banach couple X̄. Then Ȳ is a K-subcouple of X̄ if and only if Ȳ is regular and (Y_{0}^{⊥}, Y_{1}^{⊥}) is a Ksubcouple of (X′_{0}, X′_{1}).

Proof. We may assume that Ȳ is regular, because Proposition 2.3 implies that Ȳ is regular if it is a K-subcouple of X̄. Similarly, by Proposition 6.2 (i)⇒(v) (Ȳ is a K-subcouple) or (ii)⇒(v) (Ȳ^{⊥} is a K-subcouple), we may assume that Ȳ^{⊥} is a normal subcouple of X̄′. It is easily seen (cf. the proof of Proposition 6.2) that the quotient couple X̄′/Ȳ^{⊥} is isometric to Ȳ′, and that the quotient map

π : X′_{0} ∩ X′_{1} ≅ (X_{0} + X_{1})^{∗} → (X′_{0}/Y_{0}^{⊥}) ∩ (X′_{1}/Y_{1}^{⊥}) ≅ Y′_{0} ∩ Y′_{1} ≅ (Y_{0} + Y_{1})^{∗}

is the adjoint of the embedding Y_{0}+Y_{1} → X_{0}+X_{1}. Since the dual of the norm K(t, ·; X̄) on X_{0} +X_{1} is J(t^{−1}, ·; X̄′), and similarly for Ȳ, it follows that, for each t and C, K(t, y; Ȳ) ≤ CK(t, y; X̄) for all y ∈ Y_{0} + Y_{1}, if and only if each y′ ∈ Y′_{0} ∩ Y′_{1} can be lifted to an x′ ∈ X′_{0} ∩ X′_{1} with J(t^{−1}, x′; X̄′) ≤ CJ(t^{−1}, y′; Ȳ′). Consequently, Ȳ is a K-subcouple of X̄ if and only if Ȳ′ ≅ X̄′/Ȳ^{⊥} is a J-quotient couple of X̄′, and the result follows by Theorem 4.1. □

We collect some further consequences of the argument above.

Theorem 6.2. Suppose that Ȳ is a closed K-subcouple of a regular Banach couple X̄. Then Ȳ^{⊥} is a normal K-subcouple of X̄′ and there is a natural isometry between the couples Ȳ′ and X̄′/Ȳ^{⊥} which allows us to regard Ȳ′ as a J-quotient couple and K-quotient couple of X̄′.

Proof. All assertions follow by Theorem 6.1 and its proof except that Ȳ′ is a K-quotient couple. For this we use Theorem 4.4 and the fact that dual couples are Gagliardo complete. □

Remark 6.1. It does not follow from the assumptions in Theorem 6.2 that Ȳ is a normal subcouple of X̄, see Example 7.8.

7. Examples

Example 7.1. A complemented subcouple, i.e. a subcouple Ȳ of X̄ such that there is a bounded projection X̄ → Ȳ, is a K-subcouple. Ȳ is normal and I − P induces a linear lifting X̄/Ȳ → X̄ that yields liftings as in the definitions of J-quotient couple and Kquotient couple. P gives a linear good approximation. Conversely, if, for example, X̄ and Ȳ are regular and there is a linear lifting satisfying the J-quotient condition, or a linear good approximation, then Ȳ is a complemented subcouple of X̄.

Example 7.2. Regard the Hardy space H^{p} on the unit disc as a closed subspace of L^{p} on the circle. Then (H^{p}, H^{q}) is a K-subcouple of (L^{p}, L^{q}), 0 < p, q ≤ ∞, see Pisier [21] and (for p = 1) Bennett and Sharpley [5]. (See also [22], [23], [3] for generalizations and applications.) The subcouple is normal, e.g. by Corollary 3.1.

By Theorem 4.1, in particular, (L^{p}/H^{p}, L^{∞}/H^{∞}) is a J-quotient couple of (L^{p}, L^{∞}). If p > 1, this is a dual couple and hence Gagliardo complete. By Proposition 4.1 applied to (L^{∞}/H^{∞}, L^{p_{0}}/H^{p_{0}}) for some p_{0} < p, it follows that (L^{p}/H^{p}, L^{∞}/H^{∞}) is Gagliardo complete for each p > 0, and thus a K-quotient couple of (L^{p}, L^{∞}).

Theorem 5.2 yields, for example, that for each p_{0} < 0, every f ∈ L^{p_{0}} has a simultaneous good analytic approximation with respect to all L^{p}, p_{0} ≤ p ≤ ∞. The error constants may depend on p here, since the equalities (L^{p_{0}}, L^{∞})_{θp} = L^{p} are not isometric; it seems likely that they can be taken to depend on p_{0} only, but we have not made any detailed analysis. We do not know whether the result corresponding to the limit p_{0} → 0 also holds, i.e., whether every f ∈ ∪_{p>0}L^{p} has a simultaneous good analytic approximation with respect to

all L^{p}, 0 < p ≤ ∞.

Example 7.3. Let Θ be an inner function in the unit disc. Since multiplication by Θ is an isometry in each L^{p}, it follows from the preceding example that, if 0 < p < q ≤ ∞, (ΘH^{p}, ΘH^{q}) is a K-subcouple of (ΘL^{p}, ΘL^{q}) = (L^{p}, L^{q}); hence (ΘH^{p}, ΘH^{q}) is a Ksubcouple of (H^{p}, H^{q}) by Proposition 2.2 (iii), and (H^{p}/ΘH^{p}, H^{q}/ΘH^{q}) is a J-quotient couple of (H^{p}, H^{q}) by Theorem 4.1.

Suppose now that Θ is a Blaschke product with simple zeros (z_{n})_{1}^{∞}. Then H^{p}/ΘH^{p} may be identified with the sequence space Z_{p} = {(f(z_{n}))_{1}^{∞} : f ∈ H^{p}}. It follows, for example, that (Z_{p_{0}}, Z_{p_{1}})_{θp} = Z_{p} when 1/p = (1 − θ)/p_{0} + θ/p_{1}, 0 < θ < 1, 0 < p_{0}, p_{1} ≤ ∞ (Theorem 4.2), and that if (a_{n})_{1}^{∞} ∈ Z_{p_{0}} ∩ Z_{p_{1}}, then a_{n} = f(z_{n}) for some analytic f with ∥f∥_{H^{p_{j}}} ≤ C∥(a_{n})∥_{Z_{p_{j}}} (Theorem 4.1(v)).

If furthermore Θ is an interpolating Blaschke product, i.e. (z_{n}) is an interpolating sequence, then Z_{p} = ℓ^{p}(dμ), 0 < p ≤ ∞, where μ(n) = 1 − |z_{n}|^{2}, see e.g. [8], [24]. Thus (Z_{p}, Z_{q}) is Gagliardo complete, and is thus a K-quotient couple of (H^{p}, H^{q}), 0 < p ≤ q ≤ ∞. In fact, in this case there is a linear lifting that maps Z_{p} = ℓ^{p}(dμ) boundedly into H^{p} for all p, 0 < p ≤ ∞, for example

(a_{n})_{1}^{∞} → ∑_{1}^{∞} a_{n}f_{n}(z) exp (1 − (\frac{1 − z̄_{k}z}{1 − |z_{k}|^{2}} )^{1/2}),  (7.1)

where f_{n} ∈ H^{∞} are as in the P. Beurling interpolation theorem [8, Theorem VII.2.1] (this is easily verified for p = ∞ or p ≤ 1; for 1 < p < ∞ we use interpolation). Hence (ΘH^{p}, ΘH^{q}) is a complemented subcouple of (H^{p}, H^{q}). We do not know whether this extends to Θ that are not interpolating Blaschke products. (Note that this example combines two different meanings of ‘interpolation’.)

Example 7.4. Consider functions in R^{n} and let k ≥ 1. Define T f = (D^{α}f)_{|α|≤k}, where α ranges over the N = \binom{n+k}{k} multi-indices with |α| ≤ k. Then T yields an isomorphism of the Sobolev space W_{k}^{p} into (L^{p})^{N}; in particular, we may regard (W_{k}^{1}, W_{k}^{∞}) as a closed subcouple of ((L^{1})^{N}, (L^{∞})^{N}). By a result by DeVore and Scherer [7], [5], it is a Ksubcouple.

Example 7.5. Consider again functions on R^{n} and let T (g_{0}, …, g_{n}) = g_{0} + ∑_{i=1}^{n} R_{i}g_{i}, where R_{i} are the Riesz transforms. Then T : (L^{p})^{n+1} → L^{p}, 1 < p < ∞, and T : (L^{∞})^{n+1} → BMO. Miyachi [15] proved (somewhat more generally) that if 1 < p < ∞ and f ∈ L^{p} ∩ BMO, then f = T g for some g ∈ (L^{p} ∩ L^{∞})^{n+1} with ∥g∥_{(L^{p})^{n+1}} ≤ C∥f∥_{L^{p}} and ∥g∥_{(L^{∞})^{n+1}} ≤ C∥f∥_{BMO} (C depends on p but not on f). In other words, (L^{p}, BMO) may be regarded as a J-quotient couple of ((L^{p})^{n+1}, (L^{∞})^{n+1}), 1 < p < ∞. (L^{p}, BMO) is Gagliardo complete, so by Theorem 4.4 it is a K-quotient couple as well. By Theorem 4.1, Ȳ = (ker T ∩ (L^{p})^{n+1}, ker T ∩ (L^{∞})^{n+1}) is a K-subcouple of ((L^{p})^{n+1}, (L^{∞})^{n+1}), 1 < p < ∞.

Note also that T^{∗} : f → (f, −R_{1}f, …, −R_{n}f) yields isomorphisms H^{1} → (L^{1})^{n+1} and L^{p} → (L^{p})^{n+1}, 1 < p < ∞. (Here H^{1} is the real variable Hardy space.) Hence we can regard (H^{1}, L^{p}) as a closed subcouple of ((L^{1})^{n+1}, (L^{p})^{n+1}). It is easily seen that (H^{1⊥}, L^{p⊥}) = (ker T ∩ (L^{∞})^{n+1}, ker T ∩ (L^{p′})^{n+1}), which by the above is a Ksubcouple of ((L^{∞})^{n+1}, (L^{p′})^{n+1}); Theorem 6.1 now yields that (H^{1}, L^{p}) is a K-subcouple of ((L^{1})^{n+1}, (L^{p})^{n+1}).

Example 7.6. Consider the space of all bounded linear operators on ℓ^{2}, regarded as matrices (a_{ij})_{i,j=1}^{∞}. We let S_{∞} be the space of all bounded linear operators, S_{0} the space of finite rank operators with ∥T ∥_{S_{0}} = rank(T), and S_{p}, for 0 < p < ∞, the Schatten p-class. S_{p} can be defined by interpolation (the E-method):

S_{p} = {T : ( ∫_{0}^{∞} E(t, T; S_{0}, S_{∞})^{p} dt)^{1/p} < ∞}
= ((S_{0}, S_{∞})_{p/(p+1),p+1})^{1+1/p}, 0 < p < ∞,  (7.2)

see [2, Section 7.3]. (E(t, T; S_{0}, S_{∞}) is known as the singular number s_{[t]}(T).) S_{p} is a quasi-Banach space for 0 < p ≤ ∞ and a Banach space for p ≥ 1.

Let T_{p} denote the subspace of S_{p} consisting of (upper) triangular matrices, i.e. matrices (a_{ij}) with a_{ij} = 0 when i > j. It is well known that the natural projection τ maps S_{p} onto T_{p} if and only if 1 < p < ∞. In particular, (T_{p}, T_{q}) is a complemented subcouple of (S_{p}, S_{q}) for 1 < p_{0}, p_{1} < ∞.

Pisier [21] proved that (T_{p}, T_{q}) is a K-subcouple of (S_{p}, S_{q}) whenever 0 < p ≤ q ≤ ∞. Let Q_{p} = S_{p}/T_{p}. If p′ < p < q, then Q_{p} = (Q_{p′}, Q_{q})_{θq} for some θ by Theorem 4.2. Hence Q_{p}^{c} = Q_{p} in Q_{p′} + Q_{q} and thus in Q_{p} + Q_{q}. Furthermore, Q_{p} + Q_{q} = Q_{q}, so Q_{q}^{c} = Q_{q} (in Q_{p} + Q_{q}) is trivial. Consequently, (Q_{p}, Q_{q}) is Gagliardo complete for 0 < p < q ≤ ∞. By Theorem 4.4 it is a K-quotient couple of (S_{p}, S_{q}) and by Theorem 5.2, for every p_{0} > 0, every bounded operator has a simultaneous good triangular approximation with respect to (for example) all S_{p}, p_{0} < p ≤ ∞. (This extends results by Pisier [21] and Kaftal, Larson and Weiss [13].)

We do not know whether there exists a simultaneous good triangular approximation with respect to S_{p} for all p > 0.

Example 7.7. In the preceding example we only considered p > 0. We will here show that (T_{0}, T_{∞}) is not a K-subcouple of (S_{0}, S_{∞}). (The same argument shows that (T_{0}, T_{p}) is not a K-subcouple of (S_{0}, S_{p}) for any p > 0.)

Let n ≥ 1 and define A = (a_{ij})_{i,j=1}^{∞} with

a_{ij} = \{ \begin{matrix}n^{3(j−i)} & 1 ≤ i, j ≤ n + 1 \\ 0 & otherwise. \end{matrix}  (7.3)

Let B be the triangular projection of A. Since A − B has n(n − 1)/2 non-zero entries, each of them at most n^{−3}, and rank(A) = 1,

E(1, B; S_{0}, S_{∞}) ≤ ∥B − A∥_{S_{∞}} ≤ n^{2}n^{−3} = n^{−1}.

On the other hand, any triangular matrix of rank ≤ n has at most n non-zero entries on the diagonal. Hence, if C ∈ T_{0} with rank(C) ≤ n, B − C has at least one diagonal entry equal to 1, so ∥B − C∥_{S_{∞}} ≥ 1. Consequently,

E(n, B; T_{0}, T_{∞}) ≥ 1 ≥ nE(1, B; S_{0}, S_{∞}).  (7.4)

Since n is arbitrary, (T_{0}, T_{∞}) is not an E-subcouple of (S_{0}, S_{∞}); by Proposition 2.1 it is not a K-subcouple.

Example 7.8. This example shows that the conditions in Theorem 2.2 (ii) and Proposition 2.5 can not be omitted. Let X_{0} = ℓ^{∞}, Y_{0} = c_{0} and X_{1} = Y_{1} = c_{0}(w) = {(a_{n})_{1}^{∞} : w(n)a_{n} → 0 as n → ∞, where w(n) is a sequence of positive numbers such that w(n) → 0 as n → ∞. Then X_{0} = Y_{0}^{c}, and Ȳ is an exact K-subcouple of X̄. Let Φ = L^{∞}. Then K_{Φ}(Z̄) = Z_{0}^{c} for every couple Z̄, and it is easy to check that (Y_{0}, K_{Φ}(Ȳ)) = (Y_{0}, Y_{0}^{c}) = (c_{0}, ℓ_{∞}) is not a K-subcouple of (X_{0}, K_{Φ}(X̄)) = (ℓ_{∞}, ℓ_{∞}).

Note also that Y_{0} ≠ X_{0} ∩ (Y_{0} + Y_{1}), cf. Proposition 2.5, so Ȳ is not a normal subcouple of X̄. Note that X_{0}/Y_{0} = ℓ^{∞}/c_{0} and X_{1}/Y_{1} = {0} do not form a couple in a natural way, while their duals Y_{0}^{⊥} ≅ c_{0}^{⊥} and Y_{1}^{⊥} = {0} form a normal K-subcouple of X̄′ = ((ℓ^{∞})^{∗}, ℓ^{1}(w^{−1})) by Theorem 6.2.

Example 7.9. Let X_{0} = X_{1} = R^{2} with the norms ∥(x_{1}, x_{2})∥_{0} = ∥(x_{1}, −x_{2})∥_{1} = I(x_{1} ≠ 0) + I(x_{1} ≠ x_{2}), where I is the indicator function. Let Y_{0} = Y_{1} = {(x_{1}, 0)}. Thus ∥(x_{1}, 0)∥_{Y_{j}} = 2I(x_{1} ≠ 0). It is easily seen that for any y = (x_{1}, 0) ≠ 0, K(t, y; Ȳ) = K_{∞}(t, y; Ȳ) = K(t, y; X̄) = 2 ∧ 2t, while K_{∞}(t, y; X̄) = min(2, 2t, 1 ∨ t). Hence Ȳ is an exact K-subcouple of X̄ but not an exact K_{∞}-subcouple. Similarly, Ȳ is not an exact E-subcouple.

Example 7.10. This example shows that a simultaneous good approximation with respect to two norms, may be a bad approximation with respect to an intermediate norm.

Let Ȳ be a normal K-subcouple of a Banach couple X̄ and assume that Y_{0}^{0} ⊄ Y_{1} and Q_{0}^{0} ⊄ Q_{1}, where Q̄ = X̄/Ȳ; for example Ȳ = (H^{p_{0}}, H^{p_{1}}) and X̄ = (L^{p_{0}}, L^{p_{1}}), 1 ≤ p_{0} < p_{1} ≤ ∞, as in Example 7.2. Fix θ ∈ (0, 1) and q ≤ ∞. Then ∥y∥_{(Ȳ)_{θq}}/∥y∥_{Y_{0}} can be arbitrarily large for y ∈ Y_{0} ∩ Y_{1}, because otherwise K(t, y; Ȳ) ≤ C_{1}t^{θ}∥y∥_{(Ȳ)_{θq}} ≤ C_{2}t^{θ}∥y∥_{Y_{0}} ≤ \frac{1}{2}∥y∥_{Y_{0}} if t is small enough, and this would, as is easily seen, imply Y_{0}^{0} ⊂ Y_{1}. Consequently there exist y_{n} ∈ Y_{0} ∩ Y_{1} with ∥y_{n}∥_{Y_{0}} = 1 and ∥y_{n}∥_{(Ȳ)_{θq}} ≥ n. Similarly there exist z_{n} ∈ Q_{0} ∩ Q_{1} with ∥z_{n}∥_{Q_{0}} = 1 and ∥z_{n}∥_{Q_{1}} = λ_{n} → ∞; we may further assume that λ_{n} ≥ ∥y_{n}∥_{Y_{1}}^{1/(1−θ)}.

By Theorem 4.1 there exist u_{n} ∈ X_{0} ∩X_{1} with z_{n} = π(u_{n}) and ∥u_{n}∥_{X_{j}} ≍ ∥z_{n}∥_{Q_{j}} = λ_{n}^{j}, j = 0, 1. Define x_{n} = λ_{n}^{−θ}u_{n} + u_{1}. Then (if n is large enough)

∥x_{n}∥_{X_{0}} ≍ 1 ≍ ∥π(x_{n})∥_{Q_{0}} = d_{X_{0}}(x_{n}, Y_{0})

and

∥x_{n}∥_{X_{1}} ≍ λ_{n}^{1−θ} ≍ ∥π(x_{n})∥_{Q_{1}} = d_{X_{1}}(x_{n}, Y_{1}).

Since also ∥x_{n} − y_{n}∥_{X_{0}} ≤ C and ∥x_{n} − y_{n}∥_{X_{1}} ≤ Cλ_{n}^{1−θ}, y_{n} is a good approximation of x_{n} with respect to both X_{0} and X_{1}. On the other hand,

∥x_{n}∥_{(X̄)_{θq}} ≤ Cλ_{n}^{−θ}∥u_{n}∥_{X_{0}}^{1−θ}∥u_{n}∥_{X_{1}}^{θ} + ∥u_{1}∥_{(X̄)_{θq}} ≤ C

and thus

∥x_{n} − y_{n}∥_{(X̄)_{θq}} ≥ ∥y_{n}∥_{(X̄)_{θq}} − C ≥ c∥y_{n}∥_{(Ȳ)_{θq}} − C ≥ cn − C → ∞,

while

d_{(X̄)_{θq}}(x_{n}, (Ȳ)_{θq}) ≤ ∥x_{n}∥_{(X̄)_{θq}} ≤ C.

Hence y_{n} is not a good approximation of x_{n} with respect to (X̄)_{θq}.

8. The Adamyan-Arov-Krein theorem

Consider again the spaces S_{p} of matrices acting on ℓ^{2} defined in Example 7.6, 0 ≤ p ≤ ∞, and let Γ_{p} be the subspaces of Hankel matrices, i.e. matrices of the form (a_{i+j})_{i,j=1}^{∞}. By a well-known unitary equivalence, we may equivalently let S_{p} be the Schatten classes of operators from H^{2} to H_{−}^{2} = L^{2} ⊖ H^{2}, where H^{2} ⊂ L^{2} = L^{2}(T) is the Hardy space as in Example 7.2, and Γ_{p} = Γ_{∞} ∩ S_{p} consists of the Hankel operators

Γ_{∞} = {T : T f = P_{−}(φf) for some φ ∈ L^{∞}}.  (8.1)

The Adamyan-Arov-Krein theorem (or AAK for short), see e.g. [16], says that if T is a Hankel operator, then

inf{∥T − K∥_{S_{∞}} : rank(K) ≤ n} = inf{∥T − K∥_{S_{∞}} : rank(K) ≤ n, K ∈ Γ_{0}}, (8.2)

i.e., in our notation,

E(t, T; S_{0}, S_{∞}) = E(t, T; Γ_{0}, Γ_{∞}), t > 0  (8.3)

(and furthermore, the infima are attained). In other words, (Γ_{0}, Γ_{∞}) is an exact Esubcouple of (S_{0}, S_{∞}). Proposition 2.1 yields the following.

Theorem 8.1. (Γ_{0}, Γ_{∞}) is a K-subcouple of (S_{0}, S_{∞}). □

Conversely, Theorem 8.1 yields the AAK theorem up to constants and may thus be regarded as a weak AAK theorem.

By reiteration (Theorems 2.1 and 2.2 and Proposition 2.2 (viii)) we easily obtain a generalization. This result is due to Peller [18], [19] (at least for q = ∞), where also applications to interpolation of Besov spaces are given.

Theorem 8.2. (Γ_{p}, Γ_{q}) is a K-subcouple of (S_{p}, S_{q}), 0 ≤ p ≤ q ≤ ∞. □

Theorem 8.2 implies in particular, using the equivalence of K-subcouples and E-sub-couples again, that (8.3) and (8.2) hold within constants (in the norm and the rank) also if S_{∞} is replaced by S_{p} for any p > 0, which can be regarded as a weak AAK theorem for S_{p}. In this case the constants are really needed; the (strong) AAK theorem, with equalities in (8.2) and (8.3), does not hold for S_{p}, p < ∞, as the following example shows.

Example 8.1. Let A be the Hankel matrix (a_{ij}) with a_{11} = 1, a_{12} = a_{21} = ε > 0 (small) and a_{ij} = 0, i + j > 3, and let B = (b_{ij}) with b_{11} = 1, b_{12} = b_{21} = ε, b_{22} = ε^{2} and b_{ij} = 0, i ∨ j > 2. Then B has rank 1 so E(1, A; S_{0}, S_{p}) ≤ ∥A − B∥_{S_{p}} = ε^{2}.

On the other hand, suppose C = (c_{ij}) is any rank 1 Hankel matrix with ∥A−C∥_{S_{p}} ≤ 2ε^{2}. Then c_{11} = a_{11} + O(ε^{2}) = 1 + O(ε^{2}), c_{12} = ε + O(ε^{2}), and c_{12} = ε + O(ε^{2}). Hence, since C has rank 1, c_{22} = c_{12}c_{21}/c_{11} = ε^{2} + O(ε^{3}). Since A and C are Hankel, this gives a_{13} − c_{13} = a_{22} − c_{22} = a_{31} − c_{31} = −ε^{2} + O(ε^{3}), and, for example, ∥A − C∥_{S_{2}} ≥ \sqrt{3}|a_{22} − c_{22}| = \sqrt{3}ε^{2} + O(ε^{3}). Consequently E(1, A; Γ_{0}, Γ_{2}) ≥ \sqrt{3}ε^{2} + O(ε^{3}), which shows that (Γ_{0}, Γ_{2}) is not an exact E-subcouple of (S_{0}, S_{2}). In fact, it is not difficult to show that ∥A − C∥_{S_{p}} ≥ 3^{1/(p∨1)}ε^{2} + O(ε^{3}), and thus (Γ_{0}, Γ_{p}) is not an exact E-subcouple of (S_{0}, S_{p}) for any p < ∞.

We do not know whether the AAK theorem for S_{p} holds in the form

E(n, T; Γ_{0}, Γ_{p}) ≤^{?} CE(n, T; S_{0}, S_{p}),  (8.4)

i.e., if we may take C_{1} = 1 in (2.5) in this case.

It follows as in Example 7.6 that (S_{p}/Γ_{p}, S_{∞}/Γ_{∞}) is Gagliardo complete for every p > 0.

Remark 8.1. Christian Le Merdy (personal communication) has shown that S_{0}/Γ_{0} is Gagliardo complete in S_{∞}/Γ_{∞}, which answers a question that was asked in a draft of this paper. More precisely he showed that if R_{n} = {T ∈ S_{0} : rank(T) ≤ n}, then \overline{Γ_{∞} + R_{n}} ⊂ Γ_{∞} +R_{2n}, which implies that for any T̄ ∈ (S_{0}/Γ_{0})^{c}, ∥T̄∥_{S_{0}/Γ_{0}} ≤ 2∥T̄∥_{(S_{0}/Γ_{0})^{c}}. He also showed that the constant 2 is necessary; in fact, the matrix A = (a_{ij}) with a_{23} = a_{32} = 1 and all other a_{ij} = 0, belongs to \overline{Γ_{∞} + R_{1}} \ (Γ_{∞} + R_{1}), and the corresponding element Ā ∈ S_{∞}/Γ_{∞} satisfies ∥Ā∥_{S_{0}/Γ_{0}} = 2 and ∥Ā∥_{(S_{0}/Γ_{0})^{c}} = 1.

Theorem 5.3 (with Remark 5.2 for p = ∞) yields the following approximation result.

Theorem 8.3. Every bounded operator on ℓ^{2} has a simultaneous good Hankel approximation with respect to all S_{p}, 0 < p ≤ ∞. (The error constants may depend on p.) □

Remark 8.2. There are several similarities between the scales (Γ_{p}) and (T_{p}); for example, the natural projection is bounded S_{p} → Γ_{p} when 1 < p < ∞ [18]. Note however one difference: The AAK theorem does not hold for (T_{p}) (not even weakly), as was shown in Example 7.7.

We have so far dealt with the classical Hankel operators or, equivalently, Hankel matrices. There are several generalizations of Hankel operators where the AAK theorem remains valid, and the considerations above hold for them as well.

One such generalization, due to Ball and Helton [1] and Treil [25], is to vector-valued spaces, where we consider operators from H^{2}(E_{1}) to H_{−}^{2}(E_{2}) for two separable Hilbert spaces E_{1} and E_{2}, taking φ ∈ L^{∞}(B(E_{1}, E_{2})) in (8.1); cf. Peller [20].

Other generalizations are given by Cotlar and Sadosky [6], in particular they prove the AAK theorem for weighted spaces H^{2}(μ) ⊂ L^{2}(μ), where μ is a (finite) measure on the circle.

There are other generalizations of Hankel operators for which it is not known whether the AAK theorem holds (even in its weak version). We do not know of any case where the weak AAK theorem (Theorem 8.1) is proved but not the strong form.

9. Final remarks

Remark 9.1. This paper is in the usual framework of (compatible) couples of spaces. It seems plausible, however, that the notions in this paper could be defined and studied more generally for Doolittle diagrams [14], [4, section 2.7.2], which might simplify the treatment of quotients and duals.

Remark 9.2. We do not know if it is possible to develop analoguous results for the complex method or other interpolation methods. From an abstract point of view, the following attempt seems reasonable.

Definition. Let Ā be a Banach couple and let Ȳ be a closed subcouple of a Banach couple X̄. Then Ȳ is an Ā-subcouple of X̄ if every linear mapping Ȳ → Ā may be extended to a mapping X̄ → Ā, and (if Ȳ is a normal subcouple) X̄/Ȳ is an Ā-quotient couple of X̄ if every linear mapping Ā → X̄/Ȳ may be lifted to a mapping Ā → X̄. (By the closed graph theorem, we automatically have norm estimates.)

The definition is motivated by the following simple result, whose proof we leave to the reader. Here ℓ̄^{p} denotes the couple (ℓ^{p}, ℓ^{p}(2^{n})) of weighted spaces of two-sided sequences.

Proposition 9.1. Let Ȳ be a closed subcouple of a Banach couple X̄. Then Ȳ is a K-subcouple of X̄ if and only if it is an ℓ̄^{∞}-subcouple, and (if Ȳ is normal) X̄/Ȳ is a J-quotient couple of X̄ if and only if it is an ℓ̄^{1}-quotient couple. □

Recalling that the K-method can be defined as a coorbit functor using the couple ℓ̄^{∞}, and that the J-method can be defined as an orbit functor using ℓ̄^{1} [4], it looks promising to consider Ā-subcouples or Ā-quotient couples for other coorbit or orbit functors, for example \overline{F L^{∞}}-subcouples and \overline{F L^{1}}-quotient couples for the complex method, where \overline{F L^{p}} is a weighted couple of Fourier sequence spaces, cf. [12].

Remark 9.3. Interpolation of subspaces and quotient spaces by the complex method for infinite families of spaces have been considered by Hernandez, Rochberg and Weiss [9], who in particular give a duality theorem. We do not know whether the methods of this paper can be extended to families of spaces.

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Department of Mathematics, Uppsala University, P.O. Box 480, 751 06 Uppsala, Sweden.

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