# A new interpolation approach to spaces of Triebel-Lizorkin type

## Transcript Of A new interpolation approach to spaces of Triebel-Lizorkin type

A new interpolation approach to spaces of Triebel-Lizorkin type

Peer Christian Kunstmann Karlsruhe Institute of Technologie (KIT)

Institute for Analysis Kaiserstr. 89, D – 76128 Karlsruhe, Germany

e-mail: [email protected]

October 7, 2014

Abstract

We introduce in this paper new interpolation methods for closed subspaces of Banach function spaces. For q ∈ [1, ∞] the lq-interpolation method allows to interpolate linear operators that have bounded lq-valued extensions. For q = 2 and if the Banach function spaces are r-concave for some r < ∞, the method coincides with the Rademacher interpolation method that has been used to characterize boundedness of the H∞-functional calculus. As a special case, we obtain Triebel-Lizorkin spaces Fp2,θq(Rd) by lq-interpolation between Lp(Rd) and Wp2(Rd) where p ∈ (1, ∞). A similar result holds for the recently introduced generalized Triebel-Lizorkin spaces associated with Rq-sectorial operators in Banach function spaces. So, roughly speaking, for the scale of Triebel-Lizorkin spaces our method thus plays the role the real interpolation method plays in the theory of Besov spaces.

AMS subject classiﬁcation (MSC2010): 46 B 70, 47 A 60, 42 B 25 keywords: interpolation, sectorial operators, Triebel-Lizorkin spaces, real interpolation method, H∞-functional calculus, Rademacher interpolation method, γinterpolation method

1 Introduction

Many of the classical function spaces on Rd can be subsumed in the scales of Besov and Triebel-Lizorkin spaces (see, e.g., [21]). These two types of spaces are usually deﬁned via Littlewood-Paley decomposition and this common feature leads to many parallels in their theory. One important diﬀerence, however, is that Besov spaces Bps,q(Rd) arise as real interpolation spaces between Lebesgue spaces Lp(Rd) and Sobolev spaces Wpm(Rd) where m ∈ N. On the one hand this means that one can use the powerful machinery of real

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interpolation for their study. On the other hand one can easily deﬁne “abstract” Besov type spaces by real interpolation between a Banach space X and the domain of a sectorial operator A in X. These spaces allow for natural descriptions of Littlewood-Paley type where the decomposition operators are not deﬁned via Fourier transform but, e.g., via a suitable functional calculus for the operator A.

To be more precise, we recall that a linear operator A in a Banach space X is called sectorial of type ω ∈ [0, π) if its spectrum σ(A) is contained in {z ∈ C \ {0} : |arg z| ≤ ω} ∪ {0} and, for all σ ∈ (ω, π), the sets of operators {λ(λ + A)−1 : λ ∈ C \ {0}, |arg λ| < π − σ} are bounded in L(X). The inﬁmum of all such angles (which actually is a minimum) is denoted by ω(A). If we denote X1(A) the domain D(A) equiped with the graph norm then, for θ ∈ (0, 1) and q ∈ [1, ∞],

∞

R

1−θ

−1 q dt 1/q

(X, X1(A))θ,q = {x ∈ X : x θ,q := (

0

t A(1 + tA) x X t )

< ∞}

or, in case ω(A) < π/2,

T

∞ 1−θ

−tA q dt 1/q

(X, X1(A))θ,q = {x ∈ X : x θ,q := (

0

t Ae

x X t)

< ∞},

and

· X+

·

R θ,q

or

· X+

·

T θ,q

are

equivalent

to

the

abstract

“real

interpolation

norms” obtained, e.g., via the K-method or the J-method. The special case A = −∆ in

X = Lp(Rd), p ∈ (1, ∞), gives back classical Besov spaces Bp2,θq(Rd). The functional calculus

point of view on abstract Besov spaces has been extensively developed in [6] where these

spaces are called McIntosh-Yagi spaces. Remarkable is G. Dore’s result that a sectorial

operator always has a bounded H∞-functional calculus in its associated abstract Besov

spaces [2].

For Triebel-Lizorkin spaces Fp2,θq on Rd where p ∈ (1, ∞), q ∈ [1, ∞] and θ ∈ (0, 1), one

has as an equivalent norm, cf. [20],

f Lp + ( ∞ |t1−θ(−∆)et∆f |q dt )1/q Lp. (1)

0

t

Recently, a generalization of Triebel-Lizorkin spaces has been introduced, cf. [12], replacing in (1) the space Lp by a Banach function space X and the operator −∆ in Lp(Rd) by a

sectorial operator A in X. Provided the operator A is Rq-sectorial in X (cf. Deﬁnition 2.7 below), it is shown in [12] that the scale Xqθ,A of generalized Triebel-Lizorkin spaces thus obtained has a nice theory analogous to the one for the scale of abstract Besov spaces

associated with A. In particular, it was shown in [12] that an Rq-sectorial operator A in a Banach space X always has a bounded H∞-calculus in its associated generalized TriebelLizorkin spaces Xqθ,A, i.e. the analog of Dore’s result holds for the scale of generalized Triebel-Lizorkin spaces. A key issue in [12] had been an adapted version ([12, Proposition

3.9]) of the norm equivalence result for square functions due to C. Le Merdy [17]. For X = Lp, these square functions had been introduced in [1] to give a characterization for the boundedness of the H∞-calculus of a sectorial operator. Later, these characterizations had

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been extended to general classes of Banach spaces via Rademacher and Gaussian random sums [9, 10, 11, 8], and corresponding interpolation methods have been constructed, cf. [8, 19].

It is, however, clear that there cannot be a general interpolation method, meaning an interpolation functor from the category of interpolation couples of Banach spaces into the category of Banach spaces the sense of [23, 1.2.2], underlying generalized Triebel-Lizorkin spaces in a way real interpolation is “underlying” abstract Besov spaces. This has two reasons. The ﬁrst one is obvious: one cannot make sense of expressions like ( j |x|q)1/q X in arbitrary Banach spaces X (although q = 2 is an exception due to random sums, cf. Section 3 below). For this purpose one would need, e.g., a Banach function space. The second reason is less obvious: if X and Y are Banach function spaces then a bounded linear operator X → Y need not have a bounded extension X(lq) → Y (lq) (again, q = 2 is an exception, cf. below). This phenomenon is responsible for the additional technical diﬃculties that arise in the study of F -spaces compared to B-spaces. Looking at the theory developed in [12] it seems unlikely that an arbitrary linear operator T that is bounded X → X and X1(A) → X1(A) acts boundedly Xqθ,A → Xqθ,A. Even being certainly wrong as it stands, [12, Proposition 4.20] gives the hint that this interpolation property holds under boundedness assumptions on lq-extensions of T .

The purpose of the present paper is to show that these two obstructions are the only ones. In other words: Taking these two aspects into account we develop an interpolation method that plays for (generalized and classical) Triebel-Lizorkin spaces the same role real interpolation does for Besov spaces. It is clear that, in order to do so, we also have to make sense of expressions like ( j |x|q)1/q X1(A) where X1(A) is, in general, not a Banach function space, even if X is, think of Wp2 and Lp where p ∈ (1, ∞). The natural way out is to consider Wp2 as a closed subspace of another Lp-space. Thus we are led to the class of closed subspaces of Banach function spaces. However already this simple example shows that there are several natural embeddings, e.g., induced by the norms f Lp + |α|≤2 ∂αf Lp, f Lp + ∆f Lp, (1 − ∆)f Lp, moreover, in the last two expressions the operator ∆ can be replaced by a countless variety of other second order elliptic operators.

We thus prefer to make these embeddings explicit. We also take the point of view that the primary object is the Banach space X, and that this Banach space is given an additional structure by considering an embedding J : X → E into a Banach function space E. We require J to be isometric, merely for simplicity of notation, understanding that we might have to change to an equivalent norm on X (we are not interested in the isometric theory of Banach spaces here). We then call the triple (X, J, E) a structured Banach space and (J, E) a function space structure on X. It is important, that function space structures may be “non-equivalent” (in a certain sense) even if the induced norms on X are equivalent (cf. Section 2 below). The issue as such has been noted in the context of square functions in [16] where it is less virulent (cp. the remarks in Section 3 on q = 2). Another point that has been essential in [16] and that we encounter here, too, is that within the class of closed subspaces X of Banach function spaces we cannot do duality arguments: since the dual X is not a subspace of a Banach function space, in general, but a quotient space we cannot

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give expressions like ( j |xj|q )1/q X a meaning. The paper is organized as follows: In Section 2 we introduce the lq-interpolation method

by a suitable modiﬁcation of the K-method (discrete version) for real interpolation. In the case q = 2 there is a relation to the Rademacher interpolation method from [8] and to the γ-interpolation method from [10, 19], both working for arbitrary interpolation couples of Banach spaces. We study the relation to these methods in Section 3. This is done via a reformulation of the lq-interpolation method in the spirit of the J-method (discrete version) for real interpoation. In Section 4, we introduce a subclass of interpolation couples for which lq-interpolation spaces can be given a function space structure. For this structure, an interpolated linear operator is not only bounded but also has a bounded lq-extension. Finally we relate in Section 5 the interpolation theory presented here to the generalized Triebel-Lizorkin spaces from [12]. We restrict ourselves here to homogeneous generalized Triebel-Lizorkin spaces X˙ qθ,A and to θ ∈ (0, 1), but only for simplicity of presentation. In particular, we show that all function space structures that are induced by the equivalent norms from [12] on these spaces are lq-equivalent. We close this introduction with a few remarks on what we do not do in this paper.

Remark 1.1. (a) We do not recover Triebel-Lizorkin spaces F∞s ,q(Rd) or F˙∞s ,q(Rd) for q ∈ [1, ∞) or – the case q = 2 – BM O(Rd). In fact, these spaces are not deﬁned via vertical expressions in L∞(Lq) but one has to study expressions in tent spaces. It is also possible to do this for more general sectorial operators A in L2 but this needs more assumptions, e.g., a bounded H∞-calculus for A, a metric structure on the measure space and some oﬀ-diagonal estimates (at least of Davies-Gaﬀney type) for the semigroup operators e−tA, see, e.g., [7, 4].

(b) We do not contribute to the study of Hardy spaces associated with operators, which can be deﬁned via conical square functions or atomic decompositions. Again this is related to tent spaces and uses suitable decay assumptions for resolvents or semigroup operators, see, e.g., [7, 3].

(c) We do not go into details about suﬃcient conditions for the existence of lq-bounded extensions of bounded operators T : Lp → Lp here. This is a classical topic in harmonic analysis (cf., e.g., [5]). We only want to mention that domination by a positive operator or by the Hardy-Littlewood maximal operator is suﬃcient, and that classical Caldero´nZygmund operators have bounded lq-extensions in any Lp for p, q ∈ (1, ∞).

2 lq-interpolation for structured Banach spaces

Let X be a Banach space. We have to make sense of expressions like ( j |xj|q)1/q . To this end we recall the notion of a Banach function space over a σ-ﬁnite measure space (Ω, µ). We ﬁx an increasing sequence (Ωn)n∈N of µ-measurable subsets of Ω of ﬁnite measure whose union is Ω, and call this a localizing sequence. A µ-measurable M ⊂ Ω is called bounded if M ⊂ Ωn for some n. The usual choice on Ω = Rd (with Lebesgue measure) will be bounded Ωn, the usual choice on Ω = Z (with counting measure) will be ﬁnite Ωn. We will consider

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complex-valued function spaces here. However, this is only important in our applications to sectorial operators.

Deﬁnition 2.1. Let (Ω, µ) be a σ-ﬁnite measure space with localizing sequence (Ωn). Let M (µ) be the space of (equivalence classes of) measurable functions and M +(µ) := {f ∈ M (µ) : f ≥ 0}. A Banach space (E, · E) is called a Banach function space over (Ω, µ) if there is a functional ρ : M +(µ) → [0, ∞] having the following properties for f, g ∈ M +(µ), α > 0, sequences (fn) in M +(µ) and µ-measurable M ⊂ Ω:

(i) ρ(f ) = 0 if and only if f = 0 µ-a.e. , ρ(αf ) = αρ(f ) and ρ(f + g) ≤ ρ(f ) + ρ(g) (norm properties),

(ii) 0 ≤ g ≤ f µ-a.e. implies ρ(g) ≤ ρ(f ) (monotonicity),

(iii) 0 ≤ fn f µ-a.e. implies ρ(fn) ρ(f ) (Fatou property), (iv) if M is bounded then ρ(1M ) < ∞, (v) if M is bounded then M f dµ ≤ CM ρ(f ) for a constant CM > 0 independent of f ,

such that E = {f ∈ M (µ) : ρ(|f |) < ∞} and f E = ρ(|f |).

Remark 2.2. (a) If, for ν = 0, 1, Eν is a Banach function space over (Ων, µν) then E0 × E1 is a Banach function space over (Ω0∪˙ Ω1, µ0+˙ µ1) where Ω0∪˙ Ω1 denotes the disjoint union of Ω0 and Ω1 (which may be realized by Ω0×{0}∪Ω1×{1} if necessary) and µ0+˙ µ1(B0∪˙ B1) = µ0(B0) + µ1(B1) for µν-measurable subsets Bν ⊂ Ων, ν = 0, 1.

(b) If E is a Banach function spaces over (Ω, µ) and q ∈ [1, ∞] then E(lq) is the space of all sequences (fj)j∈Z in E such that (fj)j E(lq) := ( j∈Z |fj|q)1/q E < ∞. The space (E(lq), · E(lq)) is a Banach function space over (Ω × Z, µ ⊗ δ) where δ denotes the couting measure on Z. If (Ωn) is the localizing sequence in Ω then we consider Ωn × {|j| ≤ n} as localizing sequence in Ω × Z.

For a Banach function space E, one can make sense of expressions like ( j |fj|q)1/q E where fj ∈ M (µ), but for later applications we have to allow for greater ﬂexibility, so we take closed subspaces of Banach function spaces. We ﬁnd it, however, helpful to keep the embedding in notation. Therefore, we deﬁne the basic objects for our interpolation method as follows.

Deﬁnition 2.3. A structured Banach space is a triple (X, J, E) where X is a Banach space, E is a Banach function space and J : X → E is a linear map such that x X = Jx E for all x ∈ X, i.e. J : X → E is isometric, thus injective, but not necessarily surjective. For a given Banach space X we call a pair (J, E) a function space structure on X if (X, J, E) is a structured Banach space.

It will not be essential that J is isometric, it would be suﬃcient that Jx E is equivalent to the norm in X, but things are easier written down this way. For a structured Banach

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space (X, J, E), we can thus make sense of expressions like (

n j=1

|J

xj

|q

)1/q

E, and – via

the Fatou property – we can take limits

∞

n

( |J xj|q)1/q E = lim ( |J xj|q)1/q E.

n→∞

j=1

j=1

In this paper we always understand that ( j |fj|q)1/q means supj |fj| in case q = ∞. We extend the notion of Rq-bounded (sets of) operators to our setting.

Deﬁnition 2.4. Let X = (X, J, E) and Y = (Y, K, F ) be structured Banach spaces and q ∈ [1, ∞]. A set T of linear operators X → Y is called lq-bounded or Rq-bounded (w.r.t. the function space structures (J, E) on X and (K, F ) on Y ) if there exists a constant C

such that, for all n ∈ N, x1, . . . , xn ∈ X and T1, . . . , Tn ∈ T ,

n

n

( |KTjxj|q)1/q F ≤ C ( |J xj|q)1/q E.

j=1

j=1

The least constant C is denoted Rq(T ) and called the Rq-bound of T . A single linear operator T : X → Y is called lq-bounded if the set {T } is lq-bounded. The least constant

is denoted Rq(T ) in this case. Occasionally we shall say that T or T is Rq-bounded X → Y which is a more precise notation.

Denoting the set of Rq-bounded operators T : X → Y by RqL(X, Y ) it can be shown that RqL(X, Y ) is a Banach space for the norm Rq(·) (cf. [12, Proposition 2.6]).

Remark 2.5. The notion has been called Rq-boundedness in [24] in the context of Rboundedness of sets of operators in general Banach spaces. For the purpose of this paper it seems more natural to call it lq-boundedness here. However, we shall use the (somehow established) notion of Rq-sectorial operators below. Hence we use both terms lq-boundedness and Rq-boundedness in this paper, the choice depending on the context.

A fact we have to accept is that a single operator T : X → Y need not be lq-bounded in general if q = 2. Related is the phenomenon that the notion of lq-boundedness depends on the function space structures on X and Y .

Example 2.6. The Rademacher sequence (rk) is an orthonormal system in L2[0, 1]. Let X denote their closed span in E = L2[0, 1]. Then X = (X, J, E) is a structured Banach space

where J denotes the inclusion map X → E. Now let (hk) be the normalized characteristic functions on intervals (2−k, 21−k] which also form an orthonormal sequence in E = L2[0, 1] and deﬁne J˜ : X → E by k akrk → k akhk. Then also X˜ = (X, J˜, E) is a structured Banach space. We have (cf., e.g., [12, Example 2.16]):

n

( |rj|q)1/q L2 = n1/q,

j=1

n

( |hj|q)1/q L2 = n1/2.

j=1

Hence the identity is not lq-bounded X → X˜ for 1 ≤ q < 2 and not lq-bounded X˜ → X for q > 2. The example can be made to work on L2[0, 1] as well.

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Therefore, on a Banach space X, we call function space structures (J, E) and (J˜, E˜) that give rise to equivalent norms lq-equivalent if there exists a constant C such that, for

all n ∈ N and x1, . . . , xn ∈ X,

n

C−1 ( |J xj|q)1/q E ≤

j=1

n

n

( |J˜xj|q)1/q E˜ ≤ C ( |J xj|q)1/q E,

j=1

j=1

i.e. if the identity operator (X, J, E) → (X, J˜, E˜) is lq-bounded in both directions. It is clear that lq-boundedness is preserved (with equivalent lq-bounds) if we change to lq-equivalent function space structures.

As an example that shall become important lateron, we extend the notion of Rq-sectorial operators (cf. [24], [12]) from Banach function spaces to structured Banach spaces.

Deﬁnition 2.7. Let X = (X, J, E) be a structured Banach space and q ∈ [1, ∞]. A sectorial operator A in X is called Rq-sectorial of type ω ∈ [0, π) (w.r.t. to the function space structure (J, E) on X) if it is sectorial of type ω and for all σ ∈ (ω, π) the set

{λ(λ + A)−1 : λ ∈ C \ {0}, |arg λ| < π − σ}

is Rq-bounded in X. The inﬁmum of all such angles ω is denoted ωq(A).

Deﬁnition 2.8. We call a pair (X0, X1) = ((X0, J0, E0), (X1, J1, E1)) an interpolation couple if X0 → Z and X1 → Z with continuous injections for some Hausdorﬀ topological vector space Z, i.e. if (X0, X1) is an interpolation couple in the usual sense (cf. [23]).

We now introduce the lq-interpolation method for interpolation couples of structured Banach spaces.

Deﬁnition 2.9. For q ∈ [1, ∞], an interpolation couple (X0, X1) = ((X0, J0, E0), (X1, J1, E1)), and θ ∈ (0, 1) we let

x θ,lq := inf{ ( |2−jθJ0x0(j)|q)1/q E0 + ( |2j(1−θ)J1x1(j)|q)1/q E1 :

j∈Z

j∈Z

∀j ∈ Z : x = x0(j) + x1(j), x0(j) ∈ X0, x1(j) ∈ X1}

for x ∈ X0 + X1 and deﬁne

(X0, X1)θ,lq := Xθ,lq := {x ∈ X0 + X1 : x θ,lq < ∞}

with norm · θ,lq .

Proposition 2.10. For θ ∈ (0, 1) and q ∈ [1, ∞] the normed space (Xθ,lq , · θ,lq ) is a Banach space.

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Proof. One has to show essentially that , for ν = 0, 1,

Uν := {(xν(j))j∈Z ⊂ Xν : (2j(ν−θ)Jνxν(j))j ∈ Eν(lq(Z))}

is complete for the associated norm and that

D := {((x0(j), x1(j)) ∈ U0 × U1 : ∀j ∈ Z : x0(j) = −x1(j)}

is closed in U0 × U1. This is easy. The following is the basic interpolation property.

Theorem 2.11. Let (X0, X1) = ((X0, J0, E0), (X1, J1, E1)) and (Y0, Y1) = ((Y0, K0, F0), (Y1, K1, F1)) be interpolation couples of structured Banach spaces. Let q ∈ [1, ∞] and let T : X0 + X1 → Y0 + Y1 be a linear operator such that T : X0 → Y0 and T : X1 → Y1 are Rq-bounded with Rq-bounds M0 and M1, respectively. Then for any θ ∈ (0, 1) the operator T acts as a bounded linear operator Xθ,lq → Yθ,lq with norm ≤ cθM01−θM1θ where cθ = 2θ.

As will become apparent in the proof, the constant cθ is the price to pay for the use of discrete lq-norms in the deﬁnition of our method.

Proof. Let x ∈ Xθ,lq and ε > 0. For each j ∈ Z we choose a decomposition x = x0(j)+x1(j) with xν(j) ∈ Xν, ν = 0, 1, such that

( |2−jθJ0x0(j)|q)1/q E0 + ( |2j(1−θ)J1x1(j)|q)1/q E1 ≤ x θ,lq + ε.

j∈Z

j∈Z

We choose an integer m such that 2m−1 < M1/M0 ≤ 2m. Letting x˜ν(j) := xν(j + m) for j ∈ Z and ν = 0, 1, we have T x = T x˜0(j) + T x˜1(j) with T x˜ν(j) ∈ Yν for ν = 0, 1. Hence

T x θ,lq ≤ ( |2−jθK0T x˜0(j)|q)1/q F0 + ( |2j(1−θ)K1T x˜1(j)|q)1/q F1

j∈Z

j∈Z

which is

≤ ( |2−jθM0J0x˜0(j)|q)1/q E0 + ( |2j(1−θ)M1J1x˜1(j)|q)1/q E1

j∈Z

j∈Z

by assumption. Now we introduce m:

= 2mθM0 ( |2−(j+m)θJ0x0(j + m)|q)1/q E0

j∈Z

+2m(θ−1)M1 ( |2(j+m)(1−θ)J1x1(j + m)|q)1/q E1 .

j∈Z

We shift the summation index by m and obtain

≤ max{2mθM0, 2−m(1−θ)M1}( x θ,lq + ε).

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We let ε → 0 and observe

2mθM0 ≤ 2θM01−θM1θ, 2−m(1−θ)M1 ≤ M01−θM1θ. We thus have shown the claim with cθ = 2θ.

A short comment on the functor property of our interpolation method seems to be in order.

Remark 2.12. Let q ∈ [1, ∞]. We consider the category C of all Banach spaces with bounded linear operators as morphisms (cf. [23, 1.2.2]). Now let the category Cq be given

by taking as objects interpolation couples (X0, X1) of structured Banach spaces and as morphisms between two couples (X0, X1) and (Y0, Y1) linear operators T : X0+X1 → Y0+Y1 such that T : X0 → Y0 and T : X1 → Y1 are lq-bounded. Then lq-interpolation is a covariant functor from Cq to C which is of type θ.

We note several simple properties.

Proposition 2.13. Let (X0, X1) be an interpolation couple of structured Banach spaces. For q ∈ [1, ∞] and θ ∈ (0, 1) we have

(a) (X0, X1)θ,lq = (X1, X0)1−θ,lq , (b) (X0, X0)θ,lq = X0, (c) if q < q˜ ≤ ∞ then (X0, X1)θ,lq → (X0, X1)θ,lq˜, (d) (X0, X1)θ,1 → (X0, X1)θ,lq → (X0, X1)θ,∞ where (X0, X1)θ,r denote real interpolation

spaces.

Proof. (a) is obvious, and (c) follows from lq → lq˜. For the proof of (b) we decompose x

by taking x0(j) = x for j ≥ 0, x0(j) = 0 for j < 0 and x1(j) = 0 for j ≥ 0, x1(j) = x for j < 0. This gives X0 → (X0, X0)θ,lq , but the reverse embedding is clear since always (X0, X1)θ,lq → X0 + X1. For the proof of (d) we notice

sup 2j(ν−θ)xν (j) Xν ≤

j

≤

≤

≤

sup |2j(ν−θ)Jν xν (j)| Eν

j

( |2j(ν−θ)Jν xν (j)|q)1/q Eν

j

|2j(ν−θ)Jν xν (j)| Eν

j

2j(ν−θ)xν (j) Xν .

j

Remark 2.14. Proposition 2.13(d) implies for 0 < θ0 < θ1 < 1, q0, q1, q ∈ [1, ∞] and λ ∈ (0, 1) by reiteration

((X0, X1)θ0,lq0 , (X0, X1)θ1,lq1 )λ,q = (X0, X1)(1−λ)θ0+λθ1,q.

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3 The case q = 2 and Rademacher interpolation

The case q = 2 in lq-interpolation is a special one: If in X = (X, J, E) the Banach function space E is qE-concave for some qE < ∞ then (cf. [18, Thm 1.d.6(i)]) we have equivalence of expressions

n

( |J xj|2)1/2 E and

j=1

1n

n

rj(u)J xj E du = E

rj xj X

0 j=1

j=1

uniformly in n where the rj denote Rademacher functions on [0, 1]. Moreover, such a space E is of ﬁnite cotype and we have equivalence of expressions

n

E

rj xj X

j=1

n

E

γj xj X

j=1

uniformly in n where the γj are independent Gaussian variables. This has two consequences. The ﬁrst one is well known: If, in addition, in Y = (Y, K, F )

the space F is qF -concave for some qF < ∞, then a set of operators T from X → Y is R2-bounded X → Y if and only if it is R-bounded, i.e. if and only if there exists a constant C such that, for all n ∈ N, x1, . . . , xn ∈ X, and T1, . . . , Tn ∈ T , we have

n

n

E

rjTjxj Y ≤ CE

rjxj X .

j=1

j=1

Since singletons {T } are always R-bounded, we obtain under these assumptions that each bounded operator T : X → Y is R2-bounded X → Y. The same holds for the related notion of γ-boundedness.

The second consequence is that l2-interpolation spaces for an interpolation couple (X0, X1), for which Eν is qν-concave for some qν < ∞ and ν = 0, 1, coincide with the spaces obtained for the couple (X0, X1) by Rademacher interpolation or by γ-interpolation (which are equivalent in this case). In order to see this, we present the following reformulation of our method for general q. For q = 2, the relation to the Rademacher interpolation spaces from [8, Deﬁnition 7.1] is then obvious.

Theorem 3.1. Let q ∈ [1, ∞] and let (X0, X1) be an interpolation couple. For θ ∈ (0, 1) and x ∈ X0 + X1 we let

x

J θ,lq

:=

inf{ (

|2−jθJ0xj |q)1/q E0 + (

|2j(1−θ)J1xj |q)1/q E1 :

j∈Z

j∈Z

∀j ∈ Z : xj ∈ X0 ∩ X1 and x = xj in X0 + X1}.

j∈Z

Then (X0, X1)θ,lq = {x ∈ X0 + X1 :

x

J θ,lq

<

∞}

and

x

J θ,lq

is

an

equivalent

norm

on

(X0, X1)θ,lq .

10

Peer Christian Kunstmann Karlsruhe Institute of Technologie (KIT)

Institute for Analysis Kaiserstr. 89, D – 76128 Karlsruhe, Germany

e-mail: [email protected]

October 7, 2014

Abstract

We introduce in this paper new interpolation methods for closed subspaces of Banach function spaces. For q ∈ [1, ∞] the lq-interpolation method allows to interpolate linear operators that have bounded lq-valued extensions. For q = 2 and if the Banach function spaces are r-concave for some r < ∞, the method coincides with the Rademacher interpolation method that has been used to characterize boundedness of the H∞-functional calculus. As a special case, we obtain Triebel-Lizorkin spaces Fp2,θq(Rd) by lq-interpolation between Lp(Rd) and Wp2(Rd) where p ∈ (1, ∞). A similar result holds for the recently introduced generalized Triebel-Lizorkin spaces associated with Rq-sectorial operators in Banach function spaces. So, roughly speaking, for the scale of Triebel-Lizorkin spaces our method thus plays the role the real interpolation method plays in the theory of Besov spaces.

AMS subject classiﬁcation (MSC2010): 46 B 70, 47 A 60, 42 B 25 keywords: interpolation, sectorial operators, Triebel-Lizorkin spaces, real interpolation method, H∞-functional calculus, Rademacher interpolation method, γinterpolation method

1 Introduction

Many of the classical function spaces on Rd can be subsumed in the scales of Besov and Triebel-Lizorkin spaces (see, e.g., [21]). These two types of spaces are usually deﬁned via Littlewood-Paley decomposition and this common feature leads to many parallels in their theory. One important diﬀerence, however, is that Besov spaces Bps,q(Rd) arise as real interpolation spaces between Lebesgue spaces Lp(Rd) and Sobolev spaces Wpm(Rd) where m ∈ N. On the one hand this means that one can use the powerful machinery of real

1

interpolation for their study. On the other hand one can easily deﬁne “abstract” Besov type spaces by real interpolation between a Banach space X and the domain of a sectorial operator A in X. These spaces allow for natural descriptions of Littlewood-Paley type where the decomposition operators are not deﬁned via Fourier transform but, e.g., via a suitable functional calculus for the operator A.

To be more precise, we recall that a linear operator A in a Banach space X is called sectorial of type ω ∈ [0, π) if its spectrum σ(A) is contained in {z ∈ C \ {0} : |arg z| ≤ ω} ∪ {0} and, for all σ ∈ (ω, π), the sets of operators {λ(λ + A)−1 : λ ∈ C \ {0}, |arg λ| < π − σ} are bounded in L(X). The inﬁmum of all such angles (which actually is a minimum) is denoted by ω(A). If we denote X1(A) the domain D(A) equiped with the graph norm then, for θ ∈ (0, 1) and q ∈ [1, ∞],

∞

R

1−θ

−1 q dt 1/q

(X, X1(A))θ,q = {x ∈ X : x θ,q := (

0

t A(1 + tA) x X t )

< ∞}

or, in case ω(A) < π/2,

T

∞ 1−θ

−tA q dt 1/q

(X, X1(A))θ,q = {x ∈ X : x θ,q := (

0

t Ae

x X t)

< ∞},

and

· X+

·

R θ,q

or

· X+

·

T θ,q

are

equivalent

to

the

abstract

“real

interpolation

norms” obtained, e.g., via the K-method or the J-method. The special case A = −∆ in

X = Lp(Rd), p ∈ (1, ∞), gives back classical Besov spaces Bp2,θq(Rd). The functional calculus

point of view on abstract Besov spaces has been extensively developed in [6] where these

spaces are called McIntosh-Yagi spaces. Remarkable is G. Dore’s result that a sectorial

operator always has a bounded H∞-functional calculus in its associated abstract Besov

spaces [2].

For Triebel-Lizorkin spaces Fp2,θq on Rd where p ∈ (1, ∞), q ∈ [1, ∞] and θ ∈ (0, 1), one

has as an equivalent norm, cf. [20],

f Lp + ( ∞ |t1−θ(−∆)et∆f |q dt )1/q Lp. (1)

0

t

Recently, a generalization of Triebel-Lizorkin spaces has been introduced, cf. [12], replacing in (1) the space Lp by a Banach function space X and the operator −∆ in Lp(Rd) by a

sectorial operator A in X. Provided the operator A is Rq-sectorial in X (cf. Deﬁnition 2.7 below), it is shown in [12] that the scale Xqθ,A of generalized Triebel-Lizorkin spaces thus obtained has a nice theory analogous to the one for the scale of abstract Besov spaces

associated with A. In particular, it was shown in [12] that an Rq-sectorial operator A in a Banach space X always has a bounded H∞-calculus in its associated generalized TriebelLizorkin spaces Xqθ,A, i.e. the analog of Dore’s result holds for the scale of generalized Triebel-Lizorkin spaces. A key issue in [12] had been an adapted version ([12, Proposition

3.9]) of the norm equivalence result for square functions due to C. Le Merdy [17]. For X = Lp, these square functions had been introduced in [1] to give a characterization for the boundedness of the H∞-calculus of a sectorial operator. Later, these characterizations had

2

been extended to general classes of Banach spaces via Rademacher and Gaussian random sums [9, 10, 11, 8], and corresponding interpolation methods have been constructed, cf. [8, 19].

It is, however, clear that there cannot be a general interpolation method, meaning an interpolation functor from the category of interpolation couples of Banach spaces into the category of Banach spaces the sense of [23, 1.2.2], underlying generalized Triebel-Lizorkin spaces in a way real interpolation is “underlying” abstract Besov spaces. This has two reasons. The ﬁrst one is obvious: one cannot make sense of expressions like ( j |x|q)1/q X in arbitrary Banach spaces X (although q = 2 is an exception due to random sums, cf. Section 3 below). For this purpose one would need, e.g., a Banach function space. The second reason is less obvious: if X and Y are Banach function spaces then a bounded linear operator X → Y need not have a bounded extension X(lq) → Y (lq) (again, q = 2 is an exception, cf. below). This phenomenon is responsible for the additional technical diﬃculties that arise in the study of F -spaces compared to B-spaces. Looking at the theory developed in [12] it seems unlikely that an arbitrary linear operator T that is bounded X → X and X1(A) → X1(A) acts boundedly Xqθ,A → Xqθ,A. Even being certainly wrong as it stands, [12, Proposition 4.20] gives the hint that this interpolation property holds under boundedness assumptions on lq-extensions of T .

The purpose of the present paper is to show that these two obstructions are the only ones. In other words: Taking these two aspects into account we develop an interpolation method that plays for (generalized and classical) Triebel-Lizorkin spaces the same role real interpolation does for Besov spaces. It is clear that, in order to do so, we also have to make sense of expressions like ( j |x|q)1/q X1(A) where X1(A) is, in general, not a Banach function space, even if X is, think of Wp2 and Lp where p ∈ (1, ∞). The natural way out is to consider Wp2 as a closed subspace of another Lp-space. Thus we are led to the class of closed subspaces of Banach function spaces. However already this simple example shows that there are several natural embeddings, e.g., induced by the norms f Lp + |α|≤2 ∂αf Lp, f Lp + ∆f Lp, (1 − ∆)f Lp, moreover, in the last two expressions the operator ∆ can be replaced by a countless variety of other second order elliptic operators.

We thus prefer to make these embeddings explicit. We also take the point of view that the primary object is the Banach space X, and that this Banach space is given an additional structure by considering an embedding J : X → E into a Banach function space E. We require J to be isometric, merely for simplicity of notation, understanding that we might have to change to an equivalent norm on X (we are not interested in the isometric theory of Banach spaces here). We then call the triple (X, J, E) a structured Banach space and (J, E) a function space structure on X. It is important, that function space structures may be “non-equivalent” (in a certain sense) even if the induced norms on X are equivalent (cf. Section 2 below). The issue as such has been noted in the context of square functions in [16] where it is less virulent (cp. the remarks in Section 3 on q = 2). Another point that has been essential in [16] and that we encounter here, too, is that within the class of closed subspaces X of Banach function spaces we cannot do duality arguments: since the dual X is not a subspace of a Banach function space, in general, but a quotient space we cannot

3

give expressions like ( j |xj|q )1/q X a meaning. The paper is organized as follows: In Section 2 we introduce the lq-interpolation method

by a suitable modiﬁcation of the K-method (discrete version) for real interpolation. In the case q = 2 there is a relation to the Rademacher interpolation method from [8] and to the γ-interpolation method from [10, 19], both working for arbitrary interpolation couples of Banach spaces. We study the relation to these methods in Section 3. This is done via a reformulation of the lq-interpolation method in the spirit of the J-method (discrete version) for real interpoation. In Section 4, we introduce a subclass of interpolation couples for which lq-interpolation spaces can be given a function space structure. For this structure, an interpolated linear operator is not only bounded but also has a bounded lq-extension. Finally we relate in Section 5 the interpolation theory presented here to the generalized Triebel-Lizorkin spaces from [12]. We restrict ourselves here to homogeneous generalized Triebel-Lizorkin spaces X˙ qθ,A and to θ ∈ (0, 1), but only for simplicity of presentation. In particular, we show that all function space structures that are induced by the equivalent norms from [12] on these spaces are lq-equivalent. We close this introduction with a few remarks on what we do not do in this paper.

Remark 1.1. (a) We do not recover Triebel-Lizorkin spaces F∞s ,q(Rd) or F˙∞s ,q(Rd) for q ∈ [1, ∞) or – the case q = 2 – BM O(Rd). In fact, these spaces are not deﬁned via vertical expressions in L∞(Lq) but one has to study expressions in tent spaces. It is also possible to do this for more general sectorial operators A in L2 but this needs more assumptions, e.g., a bounded H∞-calculus for A, a metric structure on the measure space and some oﬀ-diagonal estimates (at least of Davies-Gaﬀney type) for the semigroup operators e−tA, see, e.g., [7, 4].

(b) We do not contribute to the study of Hardy spaces associated with operators, which can be deﬁned via conical square functions or atomic decompositions. Again this is related to tent spaces and uses suitable decay assumptions for resolvents or semigroup operators, see, e.g., [7, 3].

(c) We do not go into details about suﬃcient conditions for the existence of lq-bounded extensions of bounded operators T : Lp → Lp here. This is a classical topic in harmonic analysis (cf., e.g., [5]). We only want to mention that domination by a positive operator or by the Hardy-Littlewood maximal operator is suﬃcient, and that classical Caldero´nZygmund operators have bounded lq-extensions in any Lp for p, q ∈ (1, ∞).

2 lq-interpolation for structured Banach spaces

Let X be a Banach space. We have to make sense of expressions like ( j |xj|q)1/q . To this end we recall the notion of a Banach function space over a σ-ﬁnite measure space (Ω, µ). We ﬁx an increasing sequence (Ωn)n∈N of µ-measurable subsets of Ω of ﬁnite measure whose union is Ω, and call this a localizing sequence. A µ-measurable M ⊂ Ω is called bounded if M ⊂ Ωn for some n. The usual choice on Ω = Rd (with Lebesgue measure) will be bounded Ωn, the usual choice on Ω = Z (with counting measure) will be ﬁnite Ωn. We will consider

4

complex-valued function spaces here. However, this is only important in our applications to sectorial operators.

Deﬁnition 2.1. Let (Ω, µ) be a σ-ﬁnite measure space with localizing sequence (Ωn). Let M (µ) be the space of (equivalence classes of) measurable functions and M +(µ) := {f ∈ M (µ) : f ≥ 0}. A Banach space (E, · E) is called a Banach function space over (Ω, µ) if there is a functional ρ : M +(µ) → [0, ∞] having the following properties for f, g ∈ M +(µ), α > 0, sequences (fn) in M +(µ) and µ-measurable M ⊂ Ω:

(i) ρ(f ) = 0 if and only if f = 0 µ-a.e. , ρ(αf ) = αρ(f ) and ρ(f + g) ≤ ρ(f ) + ρ(g) (norm properties),

(ii) 0 ≤ g ≤ f µ-a.e. implies ρ(g) ≤ ρ(f ) (monotonicity),

(iii) 0 ≤ fn f µ-a.e. implies ρ(fn) ρ(f ) (Fatou property), (iv) if M is bounded then ρ(1M ) < ∞, (v) if M is bounded then M f dµ ≤ CM ρ(f ) for a constant CM > 0 independent of f ,

such that E = {f ∈ M (µ) : ρ(|f |) < ∞} and f E = ρ(|f |).

Remark 2.2. (a) If, for ν = 0, 1, Eν is a Banach function space over (Ων, µν) then E0 × E1 is a Banach function space over (Ω0∪˙ Ω1, µ0+˙ µ1) where Ω0∪˙ Ω1 denotes the disjoint union of Ω0 and Ω1 (which may be realized by Ω0×{0}∪Ω1×{1} if necessary) and µ0+˙ µ1(B0∪˙ B1) = µ0(B0) + µ1(B1) for µν-measurable subsets Bν ⊂ Ων, ν = 0, 1.

(b) If E is a Banach function spaces over (Ω, µ) and q ∈ [1, ∞] then E(lq) is the space of all sequences (fj)j∈Z in E such that (fj)j E(lq) := ( j∈Z |fj|q)1/q E < ∞. The space (E(lq), · E(lq)) is a Banach function space over (Ω × Z, µ ⊗ δ) where δ denotes the couting measure on Z. If (Ωn) is the localizing sequence in Ω then we consider Ωn × {|j| ≤ n} as localizing sequence in Ω × Z.

For a Banach function space E, one can make sense of expressions like ( j |fj|q)1/q E where fj ∈ M (µ), but for later applications we have to allow for greater ﬂexibility, so we take closed subspaces of Banach function spaces. We ﬁnd it, however, helpful to keep the embedding in notation. Therefore, we deﬁne the basic objects for our interpolation method as follows.

Deﬁnition 2.3. A structured Banach space is a triple (X, J, E) where X is a Banach space, E is a Banach function space and J : X → E is a linear map such that x X = Jx E for all x ∈ X, i.e. J : X → E is isometric, thus injective, but not necessarily surjective. For a given Banach space X we call a pair (J, E) a function space structure on X if (X, J, E) is a structured Banach space.

It will not be essential that J is isometric, it would be suﬃcient that Jx E is equivalent to the norm in X, but things are easier written down this way. For a structured Banach

5

space (X, J, E), we can thus make sense of expressions like (

n j=1

|J

xj

|q

)1/q

E, and – via

the Fatou property – we can take limits

∞

n

( |J xj|q)1/q E = lim ( |J xj|q)1/q E.

n→∞

j=1

j=1

In this paper we always understand that ( j |fj|q)1/q means supj |fj| in case q = ∞. We extend the notion of Rq-bounded (sets of) operators to our setting.

Deﬁnition 2.4. Let X = (X, J, E) and Y = (Y, K, F ) be structured Banach spaces and q ∈ [1, ∞]. A set T of linear operators X → Y is called lq-bounded or Rq-bounded (w.r.t. the function space structures (J, E) on X and (K, F ) on Y ) if there exists a constant C

such that, for all n ∈ N, x1, . . . , xn ∈ X and T1, . . . , Tn ∈ T ,

n

n

( |KTjxj|q)1/q F ≤ C ( |J xj|q)1/q E.

j=1

j=1

The least constant C is denoted Rq(T ) and called the Rq-bound of T . A single linear operator T : X → Y is called lq-bounded if the set {T } is lq-bounded. The least constant

is denoted Rq(T ) in this case. Occasionally we shall say that T or T is Rq-bounded X → Y which is a more precise notation.

Denoting the set of Rq-bounded operators T : X → Y by RqL(X, Y ) it can be shown that RqL(X, Y ) is a Banach space for the norm Rq(·) (cf. [12, Proposition 2.6]).

Remark 2.5. The notion has been called Rq-boundedness in [24] in the context of Rboundedness of sets of operators in general Banach spaces. For the purpose of this paper it seems more natural to call it lq-boundedness here. However, we shall use the (somehow established) notion of Rq-sectorial operators below. Hence we use both terms lq-boundedness and Rq-boundedness in this paper, the choice depending on the context.

A fact we have to accept is that a single operator T : X → Y need not be lq-bounded in general if q = 2. Related is the phenomenon that the notion of lq-boundedness depends on the function space structures on X and Y .

Example 2.6. The Rademacher sequence (rk) is an orthonormal system in L2[0, 1]. Let X denote their closed span in E = L2[0, 1]. Then X = (X, J, E) is a structured Banach space

where J denotes the inclusion map X → E. Now let (hk) be the normalized characteristic functions on intervals (2−k, 21−k] which also form an orthonormal sequence in E = L2[0, 1] and deﬁne J˜ : X → E by k akrk → k akhk. Then also X˜ = (X, J˜, E) is a structured Banach space. We have (cf., e.g., [12, Example 2.16]):

n

( |rj|q)1/q L2 = n1/q,

j=1

n

( |hj|q)1/q L2 = n1/2.

j=1

Hence the identity is not lq-bounded X → X˜ for 1 ≤ q < 2 and not lq-bounded X˜ → X for q > 2. The example can be made to work on L2[0, 1] as well.

6

Therefore, on a Banach space X, we call function space structures (J, E) and (J˜, E˜) that give rise to equivalent norms lq-equivalent if there exists a constant C such that, for

all n ∈ N and x1, . . . , xn ∈ X,

n

C−1 ( |J xj|q)1/q E ≤

j=1

n

n

( |J˜xj|q)1/q E˜ ≤ C ( |J xj|q)1/q E,

j=1

j=1

i.e. if the identity operator (X, J, E) → (X, J˜, E˜) is lq-bounded in both directions. It is clear that lq-boundedness is preserved (with equivalent lq-bounds) if we change to lq-equivalent function space structures.

As an example that shall become important lateron, we extend the notion of Rq-sectorial operators (cf. [24], [12]) from Banach function spaces to structured Banach spaces.

Deﬁnition 2.7. Let X = (X, J, E) be a structured Banach space and q ∈ [1, ∞]. A sectorial operator A in X is called Rq-sectorial of type ω ∈ [0, π) (w.r.t. to the function space structure (J, E) on X) if it is sectorial of type ω and for all σ ∈ (ω, π) the set

{λ(λ + A)−1 : λ ∈ C \ {0}, |arg λ| < π − σ}

is Rq-bounded in X. The inﬁmum of all such angles ω is denoted ωq(A).

Deﬁnition 2.8. We call a pair (X0, X1) = ((X0, J0, E0), (X1, J1, E1)) an interpolation couple if X0 → Z and X1 → Z with continuous injections for some Hausdorﬀ topological vector space Z, i.e. if (X0, X1) is an interpolation couple in the usual sense (cf. [23]).

We now introduce the lq-interpolation method for interpolation couples of structured Banach spaces.

Deﬁnition 2.9. For q ∈ [1, ∞], an interpolation couple (X0, X1) = ((X0, J0, E0), (X1, J1, E1)), and θ ∈ (0, 1) we let

x θ,lq := inf{ ( |2−jθJ0x0(j)|q)1/q E0 + ( |2j(1−θ)J1x1(j)|q)1/q E1 :

j∈Z

j∈Z

∀j ∈ Z : x = x0(j) + x1(j), x0(j) ∈ X0, x1(j) ∈ X1}

for x ∈ X0 + X1 and deﬁne

(X0, X1)θ,lq := Xθ,lq := {x ∈ X0 + X1 : x θ,lq < ∞}

with norm · θ,lq .

Proposition 2.10. For θ ∈ (0, 1) and q ∈ [1, ∞] the normed space (Xθ,lq , · θ,lq ) is a Banach space.

7

Proof. One has to show essentially that , for ν = 0, 1,

Uν := {(xν(j))j∈Z ⊂ Xν : (2j(ν−θ)Jνxν(j))j ∈ Eν(lq(Z))}

is complete for the associated norm and that

D := {((x0(j), x1(j)) ∈ U0 × U1 : ∀j ∈ Z : x0(j) = −x1(j)}

is closed in U0 × U1. This is easy. The following is the basic interpolation property.

Theorem 2.11. Let (X0, X1) = ((X0, J0, E0), (X1, J1, E1)) and (Y0, Y1) = ((Y0, K0, F0), (Y1, K1, F1)) be interpolation couples of structured Banach spaces. Let q ∈ [1, ∞] and let T : X0 + X1 → Y0 + Y1 be a linear operator such that T : X0 → Y0 and T : X1 → Y1 are Rq-bounded with Rq-bounds M0 and M1, respectively. Then for any θ ∈ (0, 1) the operator T acts as a bounded linear operator Xθ,lq → Yθ,lq with norm ≤ cθM01−θM1θ where cθ = 2θ.

As will become apparent in the proof, the constant cθ is the price to pay for the use of discrete lq-norms in the deﬁnition of our method.

Proof. Let x ∈ Xθ,lq and ε > 0. For each j ∈ Z we choose a decomposition x = x0(j)+x1(j) with xν(j) ∈ Xν, ν = 0, 1, such that

( |2−jθJ0x0(j)|q)1/q E0 + ( |2j(1−θ)J1x1(j)|q)1/q E1 ≤ x θ,lq + ε.

j∈Z

j∈Z

We choose an integer m such that 2m−1 < M1/M0 ≤ 2m. Letting x˜ν(j) := xν(j + m) for j ∈ Z and ν = 0, 1, we have T x = T x˜0(j) + T x˜1(j) with T x˜ν(j) ∈ Yν for ν = 0, 1. Hence

T x θ,lq ≤ ( |2−jθK0T x˜0(j)|q)1/q F0 + ( |2j(1−θ)K1T x˜1(j)|q)1/q F1

j∈Z

j∈Z

which is

≤ ( |2−jθM0J0x˜0(j)|q)1/q E0 + ( |2j(1−θ)M1J1x˜1(j)|q)1/q E1

j∈Z

j∈Z

by assumption. Now we introduce m:

= 2mθM0 ( |2−(j+m)θJ0x0(j + m)|q)1/q E0

j∈Z

+2m(θ−1)M1 ( |2(j+m)(1−θ)J1x1(j + m)|q)1/q E1 .

j∈Z

We shift the summation index by m and obtain

≤ max{2mθM0, 2−m(1−θ)M1}( x θ,lq + ε).

8

We let ε → 0 and observe

2mθM0 ≤ 2θM01−θM1θ, 2−m(1−θ)M1 ≤ M01−θM1θ. We thus have shown the claim with cθ = 2θ.

A short comment on the functor property of our interpolation method seems to be in order.

Remark 2.12. Let q ∈ [1, ∞]. We consider the category C of all Banach spaces with bounded linear operators as morphisms (cf. [23, 1.2.2]). Now let the category Cq be given

by taking as objects interpolation couples (X0, X1) of structured Banach spaces and as morphisms between two couples (X0, X1) and (Y0, Y1) linear operators T : X0+X1 → Y0+Y1 such that T : X0 → Y0 and T : X1 → Y1 are lq-bounded. Then lq-interpolation is a covariant functor from Cq to C which is of type θ.

We note several simple properties.

Proposition 2.13. Let (X0, X1) be an interpolation couple of structured Banach spaces. For q ∈ [1, ∞] and θ ∈ (0, 1) we have

(a) (X0, X1)θ,lq = (X1, X0)1−θ,lq , (b) (X0, X0)θ,lq = X0, (c) if q < q˜ ≤ ∞ then (X0, X1)θ,lq → (X0, X1)θ,lq˜, (d) (X0, X1)θ,1 → (X0, X1)θ,lq → (X0, X1)θ,∞ where (X0, X1)θ,r denote real interpolation

spaces.

Proof. (a) is obvious, and (c) follows from lq → lq˜. For the proof of (b) we decompose x

by taking x0(j) = x for j ≥ 0, x0(j) = 0 for j < 0 and x1(j) = 0 for j ≥ 0, x1(j) = x for j < 0. This gives X0 → (X0, X0)θ,lq , but the reverse embedding is clear since always (X0, X1)θ,lq → X0 + X1. For the proof of (d) we notice

sup 2j(ν−θ)xν (j) Xν ≤

j

≤

≤

≤

sup |2j(ν−θ)Jν xν (j)| Eν

j

( |2j(ν−θ)Jν xν (j)|q)1/q Eν

j

|2j(ν−θ)Jν xν (j)| Eν

j

2j(ν−θ)xν (j) Xν .

j

Remark 2.14. Proposition 2.13(d) implies for 0 < θ0 < θ1 < 1, q0, q1, q ∈ [1, ∞] and λ ∈ (0, 1) by reiteration

((X0, X1)θ0,lq0 , (X0, X1)θ1,lq1 )λ,q = (X0, X1)(1−λ)θ0+λθ1,q.

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3 The case q = 2 and Rademacher interpolation

The case q = 2 in lq-interpolation is a special one: If in X = (X, J, E) the Banach function space E is qE-concave for some qE < ∞ then (cf. [18, Thm 1.d.6(i)]) we have equivalence of expressions

n

( |J xj|2)1/2 E and

j=1

1n

n

rj(u)J xj E du = E

rj xj X

0 j=1

j=1

uniformly in n where the rj denote Rademacher functions on [0, 1]. Moreover, such a space E is of ﬁnite cotype and we have equivalence of expressions

n

E

rj xj X

j=1

n

E

γj xj X

j=1

uniformly in n where the γj are independent Gaussian variables. This has two consequences. The ﬁrst one is well known: If, in addition, in Y = (Y, K, F )

the space F is qF -concave for some qF < ∞, then a set of operators T from X → Y is R2-bounded X → Y if and only if it is R-bounded, i.e. if and only if there exists a constant C such that, for all n ∈ N, x1, . . . , xn ∈ X, and T1, . . . , Tn ∈ T , we have

n

n

E

rjTjxj Y ≤ CE

rjxj X .

j=1

j=1

Since singletons {T } are always R-bounded, we obtain under these assumptions that each bounded operator T : X → Y is R2-bounded X → Y. The same holds for the related notion of γ-boundedness.

The second consequence is that l2-interpolation spaces for an interpolation couple (X0, X1), for which Eν is qν-concave for some qν < ∞ and ν = 0, 1, coincide with the spaces obtained for the couple (X0, X1) by Rademacher interpolation or by γ-interpolation (which are equivalent in this case). In order to see this, we present the following reformulation of our method for general q. For q = 2, the relation to the Rademacher interpolation spaces from [8, Deﬁnition 7.1] is then obvious.

Theorem 3.1. Let q ∈ [1, ∞] and let (X0, X1) be an interpolation couple. For θ ∈ (0, 1) and x ∈ X0 + X1 we let

x

J θ,lq

:=

inf{ (

|2−jθJ0xj |q)1/q E0 + (

|2j(1−θ)J1xj |q)1/q E1 :

j∈Z

j∈Z

∀j ∈ Z : xj ∈ X0 ∩ X1 and x = xj in X0 + X1}.

j∈Z

Then (X0, X1)θ,lq = {x ∈ X0 + X1 :

x

J θ,lq

<

∞}

and

x

J θ,lq

is

an

equivalent

norm

on

(X0, X1)θ,lq .

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