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Conical square function estimates in UMD Banach spaces and applications to H?-functional calculi

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IN UMD BANACH SPACES AND APPLICATIONS

TOH∞-FUNCTIONAL CALCULI

By

TUOMASHYTONEN¨ , JAN VANNEERVEN ANDPIERREPORTAL Abstract. We study conical square function estimates for Banach-valued functions and introduce a vector-valued analogue of the Coifman–Meyer–Stein tent spaces. Following recent work of Auscher–McIntosh–Russ, the tent spaces

in turn are used to construct a scale of vector-valued Hardy spaces associated with a given bisectorial operatorAwith certain off-diagonal bounds such thatA always has a boundedH∞-functional calculus on these spaces. This provides a new way of proving functional calculus ofAon the Bochner spacesLp(Rn; X)by checking appropriate conical square function estimates and also a conical analogue of Bourgain’s extension of the Littlewood-Paley theory to the UMD-valued context. Even whenX = C, our approach gives refinedp-dependent versions of known results.

1

Introduction

Since the development of the Littlewood-Paley theory, square function estimates of the form   ˆ ∞ 0 t√Δe−t√Δf2dt t 1/2  Lp(Rn) fL p(Rn),

have been widely used in harmonic analysis. By the notationA  B, whereA andB are norms in some spaces of some functional expressions, we mean that c≤ A/B ≤ C for two finite positive constantscandC which may depend on the spaces (or their parameters likep, n) in question, but never on the functions (such as f) or other objects appearing inside the norm. When dealing with functions which take values in a UMD Banach space X (i.e., a space with the unconditionality property of martingale differences; see [6] for more information), such estimates have to be given an appropriate meaning. This is done through a linearisation of the square function using randomisation, which gives (see [15])

  ˆ 0 t√Δe−t√Δf dW t√ t   L2(Ω;Lp(Rn;X)) fLp(Rn;X),

JOURNAL D’ANALYSE MATH ´EMATIQUE, Vol. 106 (2008) DOI 10.1007/s11854-008-0051-3

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where the integral is a Banach space-valued stochastic integral with respect to a standard Brownian motion W on a probability space (Ω,P) (see [26]), or, in a simpler discrete form,

(1.1)  k∈Z εk2k √ Δe−2k √ Δf L2(Ω;Lp(Rn;X)) fLp(Rn;X),

where (εk) is a sequence of independent Rademacher variables on (Ω,P). The latter was proven by Bourgain in [6], thereby starting the development of harmonic analysis for UMD-valued functions. In recent years, research in this field has accelerated, as it appeared that its tools, and in particular square function estimates, are of fundamental importance in the study of theH∞-functional calculus (see [21]) and in stochastic analysis in UMD Banach spaces (see [25]).

To some extent, even the scalar-valued theory (i.e.,X =C) has benefited from this probabilistic point of view (see, for instance, [17,23]). However, this fruitful linearisation has, so far, been limited to the above “vertical” square functions estimates, leaving aside “conical” estimates of the form

(1.2)  ˆ Rn  ¨ |y−x|<t t√Δe−t√Δf (y)2dy dt tn+1 p/2 dx 1/p  fLp(Rn), 1 < p≤ 2.

In the meantime, such estimates have attracted much attention, as it was realised that they could be used to extend the real variable theory of Hardy spaces in a way which is suitable to treat operators beyond the Calder´on-Zygmund class (see [3,10,14]). Indeed, elliptic operators of the form−divB∇, whereBis a matrix withL∞entries, are not, in general, sectorial onLpfor all1 < p <. Their study thus requires theLp-spaces to be replaced by appropriate Hardy spaces, on which they have good functional calculus properties (in the same way as L1 has to be replaced byH1when dealing with the Laplacian). To define such spaces, conical square functions have to be used, since the use of vertical ones would impose severe restrictions on the class of operators under consideration, viz.,Lp(R-)sectoriality. The present paper gives extensions of (1.2) to the UMD-valued context. This starts with the construction of appropriate tent spaces, which is carried out in Section 4 by reinterpreting and extending [12], using the methods of stochastic analysis in Banach spaces from [20,25,26]. Relevant notions and results from this theory are recalled in Section 2, while the crucial technical estimate is proven in Section3. Following ideas developed in [3], we then prove appropriate estimates for operators acting on these tent spaces in Section5. After collecting some basic results on bisectorial operator in Section6, this allows us in Section7 to define Hardy spaces associated with bisectorial operators of the formA⊗ IX, whereX is a UMD Banach space andAacts onL2(Rn,CN)and satisfies suitable off-diagonal

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estimates. We prove thatA⊗ IX always has anH∞-functional calculus on these Hardy spaces. Finally, in Section8, we specialise to differential operatorsAand, in particular, give a conical analogue to Bourgain’s square function estimate (1.1). If we specialise to the caseX =C, our approach allows to define Hardy spaces (associated with operators) using a class of functions which is wider than in [3]. This is possible since our estimates (see Proposition7.5) are directly obtained for a given value ofp(and actually depend on the type and cotype ofLp), instead of using interpolation.

To conclude this introduction, we point out some possible uses of our results. First, one can deduce the boundedness of the functional calculus of an operator A⊗ IX from conical square function estimates. For instance, with Theorem8.2, we recover the well-known fact that ifX is UMD and1 < p <∞,Δ⊗ IXadmits anH∞-calculus on Lp(X). Note that this characterises the UMD spaces among all Banach spaces and thus indicates that it cannot be expected that the results presented here extend beyond the UMD setting.

Another application is to deduce conical square function estimates for functions with limited decay from such estimates for functions with good decay properties. To formulate these, write

Sθ+={z ∈ C \ {0} : | arg(z)| < θ},

Ψβα(Sθ+) =f ∈ H∞(Sθ+) :∃C |f(z)| ≤ C min(|z|α,|z|−β)∀ z ∈ Sθ+. Letθ, ε > 0, and assume that either

ψ∈ Ψn/2+ε1 (Sθ+) and 1 < p < 2n n− 2 or ψ∈ Ψ1 n/2+ε(Sθ+) and 2n n + 2< p <∞. Theorem8.2together with Theorem7.10gives the estimate

ˆ Rn  ¨ |y−x|<t|ψ(t √ Δ)u(y)|2dy dt tn+1 p/2 dx upLp.

Acknowledgments. This paper was started while Hyt¨onen and van Neerven visited the Centre for Mathematics and its Applications (CMA) at the Australian National University (ANU), and it was finished during Portal’s visit to the De-partment of Mathematics and Statistics at the University of Helsinki. Hyt¨onen was supported by the Academy of Finland (SA) project 114374 “Vector-valued singular integrals”, and by the CMA while in Canberra. van Neerven was sup-ported by the VIDI subsidy 639.032.201 and VICI subsidy 639.033.604 of the

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Netherlands Organisation for Scientific Research (NWO). Portal was supported by the CMA and the Australian Research Council as a postdoctoral fellow, and by the above-mentioned SA project while in Helsinki. He thanks Alan McIntosh for his

guidance. The authors thank Alan McIntosh for his kind hospitality at ANU and

for many discussions which motivated and influenced this work.

2

Preliminaries

In this section, we establish some terminology and collect auxiliary results needed in the main body of the paper.

LetXandY be Banach spaces and letL (X, Y )denote the space of all bounded linear operators acting from X into Y. A family of bounded operators T ⊆ L (X, Y ) is calledγ-bounded if there is a constantC such that for all integers k 1and allT1, . . . , Tk ∈ T andξ1, . . . , ξj ∈ X we have

(2.1) E k  j=1 γjTjξj  2 C2E k  j=1 γjξj  2.

Hereγ1, . . . , γk are independent standard normal variables defined on some prob-ability space(Ω,F, P)andEdenotes the expectation with respect toP. The least admissible constant in (2.1) is denoted byγ(T ).

By the Kahane-Khintchine inequality, [9, Theorem 1.1] the exponent2may be replaced by any exponent1 p < ∞at the cost of a possibly different constant.

Upon replacing the standard normal variables by Rademacher variables in (2.1), one arrives at the notion ofR-boundedness. EveryR-bounded family isγ-bounded, and the converse holds ifY has finite cotype. Since we are primarily interested in UMD spacesY, which have finite cotype, the distinction betweenγ-boundedness andR-boundedness is immaterial. We prefer the former, since our techniques are Gaussian, and the use of Gaussian variables therefore seems more natural.

LetH be a Hilbert space. A linear operatorR : H → Xis calledγ-summing

if Rγ∞(H,X):= sup E k  j=1 γjRhj  21/2<∞,

where the supremum is taken over all integersk  1 and all finite orthonormal systems h1, . . . , hk in H. The space γ∞(H, X), endowed with the above norm, is a Banach space. The closed subspace ofγ∞(H, X)spanned by the finite rank operators is denoted by γ(H, X). A linear operator R : H → X is said to be γ-radonifying if it belongs toγ(H, X).

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A celebrated result of Hoffman-Jørgensen and Kwapie ´n [13,22] implies that γ∞(H, X) = γ(H, X)

for Banach spacesX not containing an isomorphic copy ofc0.

IfH is separable with orthonormal basis(hn)n1, then an operatorR : H → X is γ-radonifying if and only if the sum n1γnRhn converges in L2(Ω; X), in which case we have

Rγ(H,X)= E j1 γjRhj 21/2 .

The following criterion for membership ofγ(H, X)is referred to as covariance

domination.

Proposition 2.1. SupposeS∈ L (H, X)andT ∈ γ(H, X)satisfy

S∗ξ  CTξ, ξ∈ X,

withCindependent ofξ∗. ThenS∈ γ(H, X)andSγ(H,X) CT γ(H,X). For more details, we refer to [20,25] and the references therein.

Let(A, Σ, μ)be aσ-finite measure space andXa Banach space. In the formula-tion of the next result, which is a multiplier result due to Kalton and Weis [20], we identifyX-valued functionsf⊗ ξ, wheref ∈ L2(A)andξ∈ X, with the operator Rf ⊗ξ∈ γ(L2(A), X)defined by

(2.2) Rf ⊗ξg := f, gξ, g ∈ L2(A). Here f, hdenotes the scalar product onL2(A).

Lemma 2.2. LetX be a Banach space, (A, Σ, μ)a σ-finite measure space, andM : A→ L (X)a function such thata→ M(a)ξis stronglyμ-measurable for allξ∈ X. If the set

M = {M(a) : a ∈ A}

isγ-bounded, then the mapping

f (·) ⊗ ξ → f(·) ⊗ M(·)ξ

extends to a bounded operatorM onγ(L2(A), X)of normM  γ(M ).

Let us also recall that for all1 p < ∞, the mappingf → [h → f(·)h]defines an isomorphism of Banach spaces

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This follows from a simple application of the Kahane-Khintchine inequality; we refer to [25, Proposition 2.6] for the details. Here, H and X are allowed to be arbitrary Hilbert spaces and Banach spaces, respectively; the norm constants in the isomorphism are independent ofH.

Letγ = (γn)n1be a sequence of independent standard normal variables on a probability space(Ω,F, P). Recall that a Banach spaceX is calledK-convex if

the mapping

πγ : f→ n1

γnE(γnf ), f ∈ L2(Ω; X),

defines a bounded operator on L2(Ω; X). This notion is well-defined: if π γ is bounded for some sequenceγ, then it is bounded for all sequencesγ. A celebrated result of Pisier [27] states thatX isK-convex if and only ifX isB-convex if and only ifX has nontrivial type.

IfH is a Hilbert space andX is aK-convex Banach space, then the isometry Iγ : γ(H, X)→ L2(Ω; X)defined by

IγR :=  n1

γnRhn

mapsγ(H, X) onto a complemented subspace of L2(Ω; X). Indeed, for all R γ(H, X), we have πγIγR = n1 γnn j1 γjRhj= n1 γnRhn= IγR.

Hence, the range ofIγ is contained in the range ofπγ. Since the range ofπγ is spanned by the functionsγn⊗ ξ = Iγ(hn⊗ ξ), the range isπγ is contained in the range ofIγ. We conclude that the ranges ofπγ andIγ coincide, and the claim is proved. As an application of this, we are able to describe complex interpolation spaces of the spacesγ(H, X).

Proposition 2.3. IfH is a Hilbert space and the Banach spacesX1andX2

areK-convex, then for all0 < θ < 1, we have

[γ(H, X1), γ(H, X2)]θ= γ(H, [X1, X2]θ) with equivalent norms.

Proof. In view of the preceding observations, this follows from general results

on interpolation of complemented subspaces [5, Chapter 5]. 

3

Main estimate

The main estimate of this paper is aγ-boundedness estimate for some averaging operators, which is proven below.

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We start by recalling some known results. The first is Bourgain’s extension to UMD spaces of Stein’s inequality [6] (see [7] for a complete proof).

Lemma 3.1. LetX be a UMD space and1 < p < ∞. Let (Fm)m∈Z be a

filtration on a probability space(Ω,F, P). Then the family of conditional expecta-tions

E = {E( · |Fm) : m∈ Z}

isγ-bounded onLp(Ω; X).

Let us agree that a cube inRn is any setQof the formx + [0, )n withx∈ Rn and > 0. We write (Q) := and call it the side-length of Q. A system of

dyadic cubes is a collectionΔ = k∈ZΔ2k, whereΔ2k is a disjoint cover ofRn

by cubes of side-length2k, and eachQ∈ Δ

2kis the union of2n cubesR∈ Δ2k−1.

We recall the following geometric lemma of Mei [24].

Lemma 3.2. There exist n + 1 systems of dyadic cubes Δ0, . . . , Δn and a

constant C < ∞ such that for any ballB ⊆ Rn, there is a Q n

k=0Δk which

satisfiesB⊆ Qand|Q| ≤ C |B|.

The following results can be found in [17].

Lemma 3.3. LetX be a UMD space and1 < p <∞. Letr∈ Zn\ {0}and xQ∈ X for allQ∈ Δ. Then

E k∈Z εk  Q∈Δ2k 1Q+r(Q)xQ p ≤ C(1 + log |r|)E k∈Z εk  Q∈Δ2k 1QxQ p .

Here1Qdenotes the indicator (or characteristic) function ofQ, whileεkstand for independent Rademacher variables on(Ω,F, P), i.e., random variables distributed by the lawP(εk = +1) =P(εk =−1) = 1/2. Later on,εj will stand for another

such sequence, independent of the first one.

Lemma 3.4. LetX be a UMD space, 1 < p <∞, andm ∈ Z+. For each

Q∈ Δ, letQ, Q∈ Δbe subcubes ofQof side-length2−mQ. Then for all ∈ Z

and allxQ ∈ X, E k≡ εk  Q∈Δ2k 1QxQ   p ≤ CE k≡ εk  Q∈Δ2k 1QxQ   p ,

wherek≡ is short-hand fork≡ mod (m + 1).

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Proposition 3.5. LetX be a UMD space,1 < p < ∞, and let Lp(X)have typeτ. Forα 1, letbe the family of operators

f → AαBf := 1αB B f dx := 1αB |B| ˆ B f dx,

whereB runs over all balls inRn. ThenA

αisγ-bounded onLp(X)withγ-bound

at mostC(1 + log α)αn/τ, whereCdepends only onX,p,τ andn.

Proof. We have to show that

Ek j=1 εj1αBj Bj fjdx   p ≤ CEk j=1 εjfj   p .

By splitting all the balls Bj into n + 1 subsets and considering each of them separately, we may assume by Mei’s lemma that there is a system of dyadic cubes ΔandQ1, . . . , Qk∈ Δsuch thatBj ⊆ Qj and|Qj| ≤ C |Bj|.

Letmbe the integer for which2m−1≤ α < 2m. LetQ∗j∈ Δbe the unique cube in the dyadic system which has side-length2m (Q

j)and containsQj. ThenαBj is contained in the union ofQ∗j and at most2n− 1of adjacent cubesR∈ Δof the same size. Writinggj= 1Bjfj, we observe that

Bj

fjdx = |Q|Bj| j| Qj

gjdx.

Since|Qj| / |Bj| ≤ C, it suffices by the contraction principle to show that E k  j=1 εj1Rj Qj gjdx p ≤ CE k  j=1 εjgj p ,

whereRj = Q∗j + r (Q∗j)for some |r| ≤ n. Thanks to Lemma 3.3, it suffices to considerr = 0.

We next write Q∗j as the union Mi=1Qji, where Qji ∈ Δare the M := 2nm subcubes ofQ∗j of side-length (Qj). Let us fix the enumeration so thatQj1= Qj. Writingxj :=

Qjgjdxfor short, we have

Ek j=1 εj1Q∗jxj   p =E M  i=1 k  j=1 εj1Qjixj   p ≤ CEEM i=1 εi k  j=1 εj1Qjixj   p ≤ C M i=1 E k  j=1 εj1Qjixj  τ p 1/τ ,

where the first estimate follows from the Khintchine–Kahane inequality and the disjointness of theQjifor each fixedj, and the second from the assumed type-τ property.

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If we assume, for the moment, that all the side-lengths2k(j) := (Qj) satisfy k(j)≡ k(j) mod (m + 1), we may apply Lemma3.4to continue the estimate with

≤ C M i=1 E k  j=1 εj1Qjxj  τ p 1/τ ≤ CM1/τE k  j=1 εj1Qj Qj gjdx   p ≤ CM1/τE k  j=1 εjgj p ,

where the last estimate applied Stein’s inequality Lemma3.1, observing that the op-erators

g → 1QjQ

jg dxare conditional expectations related to the dyadic filtration

in-duced byΔ. SinceM = 2nm ≤ 2nαn, we obtain the assertion, even without the logarithmic factor in this case.

In general, the above assumption may not be satisfied, but we can always split the indicesj intom + 1≤ c(1 + log α)subsets which verify the assumption, and

this concludes the proof. 

Remark 3.6. The proof simplifies considerably in the important special case

α = 1.

4

The vector-valued tent spaces

T

p,2

(X)

In order to motivate our approach, we begin with a simple characterisation of tent spaces in the scalar-valued case. We writeRn+1+ :=Rn× R

+and set Γ(x) ={(y, t) ∈ Rn+1+ : |x − y| < t}.

Thus(y, t)∈ Γ(x) ⇔ y ∈ B(x, t), whereB(x, t) ={y ∈ Rn:|x − y| < t}. We write Lp= Lp(Rn), L2 dy dt tn+1  = L2  Rn+1 + , dy dt tn+1  , wheredyanddtdenote the Lebesgue measures onRnandR

+. Similar conventions apply to the vector-valued analogues. The dimensionn  1is considered to be fixed.

For1 p, q < ∞, the tent spaceTp,q= Tp,q(Rn+1

+ )consists of all (equivalence classes of) measurable functionsf :Rn+1+ → Csuch that

ˆ Rn  ˆ Γ(x) |f(y, t)|q dy dt tn+1 p/q dx <∞. With the norm

fTp,q:= ˆ Γ(·) |f(y, t)|qdy dt tn+1 1/q  Lp ,

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Tp,q is a Banach space. Tent spaces were introduced in the 1980’s by Coifman, Meyer and Stein [8]. Some of the principal results of that paper were simplified by Harboure, Torrea, and Viviani [12], who exploited the fact that

J : f → x→ [(y, t) → 1B(x,t)(y)f (y, t)]

maps Tp,q isometrically onto a complemented subspace of Lp(Lq(dy dttn+1)) for

1 < p, q <∞.

We now take q = 2 and extend the mapping J to functions in Cc⊗ X by J (g⊗ ξ) := Jg ⊗ ξand linearity. HereCcdenotes the space of continuous functions onRn+1+ with compact support. Note that by (2.2),J (g⊗ ξ)defines an element of Lp(γ(L2(dy dt

tn+1), X))in a natural way.

Definition 4.1. Let 1 ≤ p < ∞. The tent spaceTp,2(X)is defined as the completion ofCc⊗ Xwith respect to the norm

fTp,2(X):=JfLp(γ(L2(dy dt tn+1),X)).

It is immediate from this definition thatJ defines an isometry fromTp,2(X) onto a closed subspace ofLp(γ(L2(dy dt

tn+1), X)). In the sequel, we always identify Tp,2(X)with its image inLp(γ(L2(dy dt

tn+1), X).

Using the identificationγ(L2(dy dt

tn+1),C) = L2(dy dttn+1), we see that our definition

extends the definition of tent spaces in the scalar-valued case.

Our first objective is to prove that if X is a UMD space, then Tp,2(X) is complemented inLp(γ(L2(dy dt

tn+1), X)).

Proposition 4.2. LetXbe a UMD space and1 < p <∞. The mapping

N f (x, y, t) := 1B(y,t)(x) |B(y, t)|

ˆ B(y,t)

f (z, y, t) dz,

initially defined for operators of the form (2.2), extends to a bounded projection in

Lp(γ(L2(tdy dtn+1), X)) whose range isTp,2(X).

Proof. We follow the proof of Harboure, Torrea and Viviani [12, Theorem 2.1] for the scalar-valued case, the main difference being that the use of maximal functions is replaced by aγ-boundedness argument using averaging operators.

First we prove thatNis a bounded operator. In view of the isomorphism (2.3), it suffices to prove thatNacts as a bounded operator onγ(L2(tdy dtn+1), Lp(X)). This

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is achieved by identifyingN as a pointwise multiplier onLp(X)withγ-bounded range, and then applying Lemma2.2. In fact, putting

N (y, t) g := 1B(y,t) |B(y, t)|

ˆ

B(y,t)

g(z) dz, g∈ Lp(X),

andfy,t(x) := f (x, y, t) := f (y, t)⊗ g(x), we have

N f (·, y, t) = f (y, t)⊗ N(y, t)g = f (y, t)⊗ AB(y,t)g.

Theγ-boundedness of{N(y, t) : (y, t) ∈ Rn+1+ }now follows from Proposition3.5. Once it is known thatN is bounded onLp(γ(L2(dy dt

tn+1), X)), the fact that it is

a projection follows from the scalar case, since the linear span of the functions of the form 1B(x,t)⊗ (f ⊗ ξ), with f ∈ Cc, x ∈ Rn, and t > 0, is dense in

Lp(γ(L2(dy dttn+1), X)). 

Forα > 0, the vector-valued tent spaceTp,2

α (X) may be defined as above in terms of the norm

fTp,2

α (X):=JαfLp(γ(L2(tn+1dy dt),X)),

whereJαf := x→ [(y, t) → 1B(x,αt)(y)f (y, t)].

Theorem 4.3. Let1 < p <∞andX a UMD space such thatLp(X)has type τ. For allα > 0, a strongly measurable functionf :Rn+1+ → X belongs toTp,2(X)

if and only if it belongs toTp,2

α (X). Moreover, there exists a constantC = C(p, X)

such that

(4.1) fTp,2(X) fTp,2

α (X) C(1 + log α)αn/τfTp,2(X) forf ∈ Tp,2(X)andα > 1.

Proof. It suffices to prove the latter estimate in (4.1). OnLp(γ(L2(dy dt tn+1), X)),

we consider the operator

Nαf (x, y, t) :=1B(y,αt)(x) |B(y, t)|

ˆ

B(y,t)

f (z, y, t) dz.

Simple algebra shows thatNαJ f = Jαf, and hence fTp,2 α (X)=JαfLp(γ(L2(dy dt tn+1),X)) =NαJ f Lp(γ(L2(dy dt tn+1),X)) ≤ Nα L (Lp(γ(L2(dy dt tn+1),X)))JfLp(γ(L2(dy dttn+1),X)) .

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By the isomorphism (2.3), we may consider the boundedness ofNαon the space γ(L2(dy dt

tn+1), Lp(X))instead, and here this operator acts as the pointwise multiplier

Nα( f ⊗ g)(·, y, t) = f (y, t)⊗ AαB(y,t)g.

So its boundedness with the asserted estimate follows from Proposition3.5. 

Remark 4.4. If X = C, then one can take τ = min(2, p)in Theorem 4.3. Except possibly for the logarithmic factor, (4.1) gives the correct order of growth offTp,2

α in terms of the angleα 1.

To see this, consider functions of the formf (y, t) = 1[1,2](t)g(y). Then fTp,2

α =(ηα∗ |g|

2)1/2 p,

where theηαare functions having pointwise boundsc1B(0,α)≤ ηα≤ C1B(0,Cα)for some constantsC > 1 > c > 0depending only onn.

Let us take g =|g|2 = 1B(0,1). Then (ηα∗ |g|2)1/2 = ˜ηα, whereη˜α is another similar function, and hence

fTp,2 α =(˜ηα) 1/2 p  α n/p αn/pf Tp,2.

This proves the sharpness forp≤ 2.

Let us then chooseg = gα= 1B(0,α). Then ηα∗ |gα|2= αnη

α, η1∗ |gα|2= ηα, whereηα, η

αare yet more similar functions asηα. Writingfα(y, t) = 1[1,2](t)gα(y), we have fαTp,2 α =(α nη α)1/2p= α n/2 α)1/2p α n/2 α) 1/2 p= α n/2f αTp,2. This proves the sharpness forp 2.

In fact, forp = 2, a simple application of Fubini’s theorem shows that we have fT2,2

α = α

n/2f

T2,2 for all f ∈ T2,2 and α > 0, so the logarithmic factor is

unnecessary in this case.

Sometimes it is useful to use tent space norms defined with a smooth cut-off instead of the sharp cut-off 1B(x,t)(y). Given a function φ ∈ Cc∞(R) such that φ(w) = 1if|w| ≤ 1/2and φ(w) = 0if |w|  1, we are thus led to consider the mappingJφf := x→ [(y, t) → φ(|y−x|t )f (y, t)]and

fTp,2

φ (X):=JφfLp(γ(L2(dy dt

tn+1),X)) .

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Proposition 4.5. Let X be a UMD space and 1 < p < ∞. A strongly measurable functionf :Rn+1+ → X belongs toTp,2(X)if and only if it belongs to Tφp,2(X). Moreover,

fTp,2

φ (X) fTp,2(X) forf ∈ Tp,2(X).

Proof. The proof is the same as that of Theorem4.3. Consider the operators

Nφf (x, y, t) := φ( |y−x| t ) |B(y, t)| ˆ B(y,t) f (z, y, t) dz,  N1 2f (x, y, t) := 1B(x,t 2) B(y,2t) ˆ B(y,t2) f (z, y, t) dz. We haveJφ = NφJ andJ1

2 = N21Jφ. Moreover, the operatorsNφ and

 N1

2 act as

the pointwise multipliers

Nφ( f⊗ g)(·, y, t) = f (y, t)⊗ My,tφ A1B(y,t)g, 

N1

2( f⊗ g)(·, y, t) = f (y, t)⊗ A

1 B(y,t2)g,

where My,tφ g(x) := φ(|y−x|t )g(x). By Lemma 2.2 and Theorem 4.3, the result follows from Proposition3.5and Kahane’s contraction principle.  IfX is a UMD space and1 < p, q <∞satisfy1/p + 1/q = 1, we have natural isomorphisms

(Lp(γ(L2(dy dttn+1), X)))∗ Lq((γ(L2(tdy dtn+1), X))∗) Lq(γ(L2(tdy dtn+1), X∗))).

The first of these follows from the fact thatX, and thereforeγ(L2(dy dt

tn+1), X), is reflexive; and the second follows from theK-convexity of UMD spaces. Denote byN the projection of Proposition4.2; it is easily verified that, under the above identification, the adjointN∗is given by the same formula. As a result, we obtain the following representation for the dual ofTp,2(X).

Theorem 4.6. IfX is a UMD space, and1 < p, q <∞satisfy1/p + 1/q = 1, we have a natural isomorphism

(Tp,2(X))∗ Tq,2(X∗).

As an immediate consequence of Proposition 2.3, we obtain the following result.

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Theorem 4.7. Let1 < p0  p1 < ∞, and let X0 andX1 be UMD spaces.

Then for all0 < θ < 1, we have

[Tp0,2(X

0), Tp1,2(X1)]θ= Tpθ,2([X0, X1]θ), 1/pθ= (1− θ)/p0+ θ/p1.

Proof. The result follows by combining (2.3) with the following facts: (i) if X is a UMD space, thenLp(X) is a UMD space for all 1 < p < ; (ii) UMD spaces are K-convex; (iii) for1  p0  p1 <∞, we have[Lp0(X

0), Lp1(X1)]θ =

Lpθ([X0, X1]θ), withpθas above. 

We conclude this section with a result showing that certain singular integral operators are bounded fromLp(X)toTp,2(X). This gives a Banach space-valued extension of [12, Section 4].

Theorem 4.8. LetX be a UMD space. Consider the singular integral oper-ator defined by

Sf (t, y) = ˆ

Rnkt(y, z)f (z) dz

for f ∈ Cc(Rn) and a measurable complex-valued function (t, y, z) → kt(y, z).

Assume that

(1) S∈ L (L2, T2,2);

(2) there existsα > 0such that for ally, z∈ Rn andt > 0, we have |kt(y, z)| 

(|y − z| + t)n+α;

(3) there exists β > 0 such that for all t > 0 and all y, z, z ∈ Rn satisfying |z − y| + t > 2|z − z|, we have

|kt(y, z)− kt(y, z)| 

|z − z| (|y − z| + t)n+1+β; (4) for allt > 0andy∈ Rn, we have

ˆ

Rnkt(y, z) dz = 0.

Let1 < p <∞. ThenS⊗IXextends to a bounded operator fromLp(X)toTp,2(X).

Proof. We consider the auxiliary operator T taking X-valued functions to functions with values inγ(L2(dy dttn+1), X)given by

T f (x) = ˆ

Rn

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whereK(x, z)is theL2(dy dttn+1)-valued kernel defined by

K(x, z) : (y, t)→ φ|y − x| t

 kt(y, z)

for some evenφ ∈ Cc∞(R)such thatφ(w) = 1 if|w| ≤ 1/2,φ(w) = 0if|w|  1, and´01φ(r)rn−1dr = 0. The claim of the theorem follows if we can show that T extends to a bounded operator from Lp(X) to Lp(γ(L2(dy dttn+1); X)). This is

proved by applying a version of theT (1)theorem for Hilbert space -valued kernels from [16] (which, in turn, is based on results from [18,19]). We first remark that the conditionT (1) = 0 follows directly from (4), whereas the vanishing integral assumption onφguarantees thatT(1) = 0, too. It remains to check the following L2(dy dt

tn+1)-valued versions of the standard estimates:

(4.2) sup x,z∈Rn|x − z| nK(x, z) L2(dy dt tn+1) 1, (4.3) sup x,x,z∈Rn |x−z|>2|x−x| |x − z|n+1 |x − x| K(x, z) − K(x, z)L2(dy dt tn+1)  1, (4.4) sup x,z,z∈Rn |x−z|>2|z−z| |x − z|n+1 |z − z| K(x, z) − K(x, z)L2(dy dt tn+1)  1,

and the weak boundedness property: for anyη,η ∈ Cc∞(B(0, 1))which satisfy the boundsη,η,∇η,∇η≤ 1, one should have

(4.5) sup (u,r)∈Rn×R+   ˆ Rn ˆ Rn K(x, z)ηx − u r  η(z− u r ) dz dx rn   L2(dy dt tn+1)  1.

Proof of (4.2). Using (2) and noting that φ|y−x|t  = 0fory ∈ B(x, t), we have ˆ 0 ˆ Rn  φ |y − x| t  kt(y, z)2dy dt tn+1  ˆ |x−z| 0 ˆ B(x,t)  (|x − z| + t − |y − x|)n+α2dy dt tn+1 + ˆ |x−z| ˆ B(x,t) dy dt t3n+1  ˆ |x−z| 0 t2α−1 |x − z|2n+2αdt + ˆ ∞ |x−z| dt t2n+1  |x − z|−2n. 

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Proof of (4.3). Using (2) and the mean value theorem and reasoning as above, we have forx, x, zsatisfying|x − z| > 2|x − x|

ˆ 0 ˆ Rn  φ |y − x| t  − φ|y − x| t  kt(y, z)2dy dt tn+1  ˆ 0 ˆ B(x,t)  |x − x|tα t(|y − z| + t)n+α 2dy dt tn+1 +similar  ˆ |x−z| 0 ˆ B(x,t)  |x − x|tα t(|x − z| + t − |y − x|)n+α 2dy dt tn+1 + ˆ |x−z||x − x |2 dt t2n+3 +similar  ˆ |x−z| 0 t2α−3|x − x|2 |x − z|2n+2α dt + |x − x|2 |x − z|2n+2 +similar  |x − z||x − x2n+2|2 ,

where the words “similar” above refer to a copy of the other terms appearing in the same step, with all the occurrences ofxandx interchanged. 

Proof of (4.4). Using (3), forx, z, zsatisfying|x − z| > 2|z − z|, we have ˆ 0 ˆ Rn  φ|y − x| t  kt(y, z)− kt(y, z)2dy dt tn+1  ˆ 0 ˆ B(x,t)  tβ|z − z| (|z − y| + t)n+1+β 2dy dt tn+1  ˆ |x−z| 0 ˆ B(x,t)  tβ|z − z| (|z − x| + t − |y − x|)n+1+β 2dy dt tn+1 + ˆ |x−z| |z − z|2 t2n+3 dt  ˆ |x−z| 0 t2β−1|z − z|2 |z − x|2n+2+2β dt + ˆ |x−z| |z − z|2 t2n+3 dt |z − z|2 |x − z|2n+2. 

Proof of (4.5). Using the Cauchy-Schwarz inequality and (1), we have ˆ 0 ˆ Rn   ˆ Rn ˆ Rn φ |y − x| t  kt(y, z)ηx − u r  ηz − u r  dz dx rn  2dy dt tn+1  1 rn ˆ 0 ˆ Rn ˆ Rn  φ|y − x| t  ˆ Rnkt(y, z)η z − u r  dz2dy dt dx tn+1  1 rnS  η· − u r  2 T2,2 η2L2  1. 

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5

Off-diagonal estimates and their consequences

We start by recalling some terminology.

Definition 5.1. LetM, t > 0. An operatorT ∈ L (L2) is said to have

off-diagonal estimates of orderM at the scale oftif there is a constantCsuch that

T fL2(E)≤ C d(E, F )/t−MfL2(F )

for all Borel setsE, F ⊆ Rnand allf ∈ L2(Rn)with support inF. Here, a = 1+|a| andd(E, F ) = inf{|x − y| : x ∈ E, y ∈ F }. The set of such operators is denoted by ODt(M ).

Note that a single operator belongs to ODt(M ) if and only if it belongs to ODs(M )whenevers, t > 0. However, the related constants are typically not the same. The scale of the off-diagonal estimates becomes very relevant when we want uniformity in the constants for a family of bounded operators. Thus we say that(Tz)z∈Σ⊆ L2, whereΣ⊆ C, satisfies off-diagonal estimates of orderM ifTz∈ OD|z|(M )for allz∈ Σwith the same constantC.

Theorem 5.2. Let1 < p <∞,Xbe a UMD Banach space, and letLp(X)have

typeτ. Let(Tt)t>0be a uniformly bounded family of operators onL2 satisfying

off-diagonal estimates of orderMfor someM > n/τ. Then the operatorTdefined onCc⊗ X by

T (g⊗ ξ)(y, t) := Tt(g(·, t))(y) ⊗ ξ

extends uniquely to a bounded linear operator onTp,2(X).

Proof. Let us consider a functionf = i

gi⊗ ξi∈ Cc⊗ X. We define the sets

C0(x, t) := B(x, 2t),

Cm(x, t) := B(x, 2m+1t)\ B(x, 2m, t), m = 1, 2, . . . so that there is a disjoint union ∞m=0Cm(x, t) =Rn. Let(u

m)∞m=0be the functions um: x→ (y, t)→ 1B(x,t)(y)Tt  1Cm(x,t)f (·, t)(y), where Tt1Cm(x,t)f (·, t)(y) := i Tt(1Cm(x,t)gi(·, t))(y) ⊗ ξi.

We then have the formal expansionJ (T f ) = ∞m=0um; and for a fixedx∈ Rn, we separately estimate theγ(L2(tdy dtn+1), X)-norms of eachum(x).

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Fixξ∗ ∈ X∗. Let us also write f(y, t), ξ∗ := i

gi(y, t) ξi, ξ∗. Form = 0, we estimate, using the uniform boundedness of the operatorsTtonL2,

u0(x)∗ξ∗2 L2(dy dt tn+1) = ˆ Rn+1 + 1B(x,t)(y)Tt  1B(x,2t) f(·, t), ξ∗(y)2 dy dt tn+1  ˆ Rn+1 + 1B(x,2t)(y)| f(y, t), ξ∗|2 dy dt tn+1. Hence, by covariance domination (Proposition2.1),

u0(x)γ(L2(dy dt tn+1),X) (y, t) → 1B(x,2t) (y)f (y, t) γ(L2(dy dt tn+1),X) ;

and we conclude that

u0Lp(γ(L2(dy dt

tn+1),X)) fT p,2

2 (X) fTp,2(X).

Form 1, the off-diagonal estimates of orderM imply um(x)∗ξ∗2 L2(dy dt tn+1) = ˆ Rn+1 + 1B(x,t)(y)Tt  1Cm(x,t) f(·, t), ξ∗(y)2 dy dt tn+1 ≤ 2−2mMˆ Rn+1 + 1B(x,2m+1t)(y)| f(y, t), ξ∗2 dy dt tn+1. Hence, by covariance domination,

um(x)γ(L2(dy dt tn+1),X) 2 −mM(y, t) → 1 B(x,2m+1t)(y)f (y, t) γ(L2(dy dt tn+1),X) ;

and from Theorem4.3, we conclude that umLp(γ(L2(dy dt tn+1),X)) 2 −mMf Tp,2 2m+1(X) 2 −mM· m · 2mn/τf Tp,2(X).

Keeping in mind that M > n/τ, we may sum over m to see that the formal expansionJ (T f ) = ∞m=0umconverges absolutely inLp(γ(L2(dy dt

tn+1), X)). Thus

we obtain the desired result. 

Remark 5.3. TheTp,2(X)-boundedness of the operatorTas considered above can be seen as a (pandXdependent) property of the (parametrised) operator family (Tt)t>0⊆ L (L2). Let us call this property tent-boundedness. A simple example of a tent-bounded family consists of the translationsTtf (x) = f (x + ty), whereyis some unit vector. Indeed, these are obviously uniformly bounded inL2(and inLp as well) and satisfy off-diagonal estimates of any order. In contrast to this, even whenX =C, it is well-known that this family is notγ-bounded inLpunlessp = 2.

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We next consider operators of the form (T f )t:= ˆ 0 Tt,sfs ds s , f ∈ Cc⊗ X,

whereTt,s ∈ L (L2). This is first done separately for upper and lower diagonal “kernels”Tt,s.

Proposition 5.4. Let1 < p < ∞, X be a UMD space, and letLp(X)have typeτ. Let(Ut,s)0<t≤s<∞be a uniformly bounded family of operators onL2such

that(Ut,s)st ∈ ODs(M )uniformly intfor someM > n/τ. Suppose thatα > n/2.

Then (U F )t= ˆ t (t/s)αUt,sFs ds s

extends to a bounded operator onTp,2(X).

Proof. LetF ∈ Cc⊗ Xbe arbitrary and fixed. It suffices to estimate the norm of the functionsuk∈ Lp(γ(L2(dy dttn+1), X))defined by

uk : x→  (y, t)→ 1B(x,t) ˆ t (t/s)αUt,s(1Ck(x,s)Fs)(y) ds s  , k = 0, 1, . . . ,

whereC0(x, s) := B(x, 2s), andCk(x, s) := B(x, 2k+1s)\ B(x, 2ks)fork 1. Letx∈ Rnbe fixed for the moment. To estimate the relevantγ(L2(dy dt

tn+1), X)

-norm at this point, we wish to use covariance domination. Hence let ξ∗ ∈ X∗, writefs:= Fs(·), ξ∗ ∈ L2for short, and consider the quantity

(uk(x))(y, t), ξ∗ = 1B(x,t) ˆ t  t/sαUt,s(1Ck(x,s)fs)(y) ds s . Its norm inL2(dy dttn+1)is dominated by

ˆ 0  ˆ t (t/s)α1B(x,t)Ut,s(1Ck(x,s)fs)L2 ds s 2 dt tn+1 1/2 ≤ ˆ ∞ 0  ˆ ∞ t (t/s)2 ds s  ˆ ∞ t (t/s)2(α− )1B(x,t)Ut,s(1Ck(x,s)fs)2L2 ds s  dt tn+1 1/2  ˆ ∞ 0 ˆ t (t/s)2(α− )2−kM1B(x,2k+1s)fsL2 2ds s dt tn+1 1/2  2−kM ˆ ∞ 0 1B(x,2k+1s)fs2L2 ds sn+1 1/2 ,

where in the last step we have exchanged the order of integration and integrated out thet variable; the convergence required that 2(α− ) > n, which holds for sufficiently small > 0, sinceα > n/2.

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The right-hand side of our computation is2−kM times the L2(dy dttn+1)-norm of

1B(x,2k+1s) Fs(y), ξ∗, so that covariance domination gives us

uk(x) γ(L2(dy dt tn+1),X)  2−kN(J 2k+1F )(x) γ(L2(dy dt tn+1),X) . TakingLp-norms and using Theorem4.3yields

uk Lp(γ(L2(dy dt tn+1),X))  2−kMF  Tp,2 2k+1(X)  2−kM(1 + k)2kn/τF  Tp,2(X).

Recalling thatM > n/τ, we find that the formal expansionJ (U F ) = ∞k=0uk converges absolutely inLp(γ(L2(dy dttn+1), X)); and we obtain the desired estimate

UF Tp,2(X) F Tp,2(X). 

Proposition 5.5. Let1 < p <∞,Xbe a UMD space, and letLp(X)have type τ. Let(Lt,s)0<s≤t<∞be a uniformly bounded family of operators onL2such that (Lt,s)ts∈ ODt(N )uniformly insfor someN > n/τ. Supposeβ > n(1/τ− 1/2). Then (LF )t= ˆ t 0 (s/t)βLt,sFs ds s

extends to a bounded operator onTp,2(X).

Proof. The proof follows an approach similar to the previous one. This time,

we expandJ (LF )in a double series ∞k,m=0vk,m, where

vk,m: x→  (y, t)→ ˆ 2−mt 2−(m+1)t (s/t)β1B(x,t)(y)Lt,s(1Ck(x,t)Fs)(y) ds s  . Again, we wish to estimate the γ(L2(dy dttn+1), X)-norm of vk,m(x) by covariance domination, for which purpose we take ξ∗ ∈ X∗, write fs := Fs(·), ξ∗, and compute  vk,m(x), ξ∗ L2(dy dt tn+1) ≤ ˆ 0  ˆ 2−mt 2−(m+1)t 2−mβ1B(x,t)Lt,s(1Ck(x,t)Fs)L2 ds s 2 dt tn+1 1/2  2−mβ ˆ ∞ 0 ˆ 2−mt 2−(m+1)t  2−kN1B(x,2k+1t)FsL2 2ds s dt tn+1 1/2  2−m(β+n/2)2−kN ˆ ∞ 0 1B(x,2k+m+2s)Fs2L2 ds sn+1 1/2 . This is 2−m(β+n/2)2−kN times the L2(dy dt

tn+1)-norm of 1B(x,2k+m+2s)(y) Fs(y), ξ∗; hence, by covariance domination,

vk,m(x) γ(L2(dy dt tn+1),X) 2 −m(β+n/2)2−kN(J 2k+m+2F )(x) γ(L2(dy dt tn+1),X) .

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TakingLp-norms and using Theorem4.3, we get vk,mLp(γ(L2(dy dt tn+1),X)) 2 −m(β+n/2)2−kNF  Tp,2 2k+m+2(X)  2−m(β+n/2)2−kN(1 + k + m)2(k+m)n/τF  Tp,2(X);

and we can sum up the series overkandmsinceβ + n/2 > n/τ andN > n/τ.  Combining the previous two propositions with a duality argument, we finally obtain

Theorem 5.6. Let1 < p <∞,Xbe a UMD space, and letLp(X)have typeτ

and cotypeγ. Let(Tt,s)0<t,s<∞be a uniformly bounded family of operators onL2

such that (i) (Tt,s)s>t∈ ODs(M )uniformly int, (ii) (Tt,s)t>s∈ ODt(N )uniformly ins. Then (T F )t= ˆ 0 min(t/s)α, (s/t)βTt,sFsds s

extends to a bounded operator on Tp,2(X) if at least one of the following four

conditions is satisfied:

(a) M > n/τ,α > n/2,N > n/τ, andβ > n(1/τ− 1/2);

(b) M > n/τ,α > n/2,N > n(1− 1/γ), andβ > n/2;

(c) M > n(1− 1/γ),α > n(1/2− 1/γ),N > n/τ, andβ > n(1/τ− 1/2);

(d) M > n(1− 1/γ),α > n(1/2− 1/γ),N > n(1− 1/γ), andβ > n/2.

Proof. We split T into a sumU + L of upper and lower triangular parts as considered in the previous two propositions. Part (a) is an immediate consequence, since the conditions onM andαguarantee the boundedness ofU and those onN andβ that ofL.

For part (b), the boundedness ofUfollows as before. As forL, we observe that its (formal) adjoint onTp,2(X)is the upper triangular operator

(L∗G)t= ˆ t (t/s)βTs,t∗ Gsds s , whereTs,t∗ ∈ ODs(N )andLp 

(X∗) = (Lp(X))has typeγ= γ/(γ− 1). We know that this operator is bounded onTp,2(X∗)under the conditions thatN > n/γ = n(1− 1/γ)andβ > n/2.

Parts (c) and (d) are proved similarly, by consideringU∗andL, andU∗andL∗,

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The most important case for us is when N = M, and we record this as a corollary for later reference. In this situation, the condition (b) of Theorem5.6

becomes redundant, since it is always contained in condition (a).

Corollary 5.1. Let1 < p <∞,Xbe a UMD space, and letLp(X)have typeτ

and cotypeγ. Let(Tt,s)0<t,s<∞be a uniformly bounded family of operators onL2

such thatTt,s∈ ODmax{t,s}(M )uniformly intands. Then

(5.1) (T F )t= ˆ 0 min(t/s)α, (s/t)βTt,sFs ds s

extends to a bounded operator onTp,2(X) if at least one of the following three conditions is satisfied:

(a) M > n/τ,α > n/2, andβ > n(1/τ− 1/2);

(c) M > n· max{1/τ, 1 − 1/γ},α > n(1/2− 1/γ), andβ > n(1/τ− 1/2);

(d) M > n(1− 1/γ),α > n(1/2− 1/γ), andβ > n/2.

Remark 5.7. IfX =C(or, more generally, a Hilbert space), then one can take τ = min(2, p)andγ = max(2, p)in Corollary5.1. Forp∈ [2, ∞)(so thatτ = 2), part (a) provides the following sufficient condition for theTp,2-boundedness of (5.1): M, α > n/2, and β > 0. For p ∈ (1, 2](so that γ = 2), part (d) in turn gives M, β > n/2, and α > 0. This recovers the corresponding result in [3] in the Euclidean case forp∈ (1, ∞). Note that in [3], the end-pointsp∈ {1, ∞}are also considered; in fact, the proof forp∈ (1, 2)goes via interpolating between estimates available in the atomic spaceT1,2and the Hilbert spaceT2,2. See also [1], where a weak type(1, 1)estimate is obtained.

6

Bisectorial operators and functional calculus

In this section, we collect some generalities concerning bisectorial operators and their H∞-calculus. We denote by Sθ the (open) bisector of angle θ, i.e., Sθ = Sθ+∪ Sθ−withSθ+={z ∈ C \ {0} : | arg(z)| < θ}andSθ−=−Sθ+. We denote byΓθ the boundary ofSθ, which is parametrised by arc-length and oriented anticlockwise aroundSθ.

A closed, densely defined, linear operator A acting in a Banach spaceY is called bisectorial (of angleω, where0 < ω < (1/2)π) if the spectrum of A is contained inSωand for allω < θ < (1/2)π, there exists a constantCθsuch that for all nonzeroz∈ C \ Sθ,

(I + zA)−1  C θ |z|

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Forα, β > 0, we set

Ψα(Sθ) =f ∈ H∞(Sθ) : ∃C |f(z)| ≤ C min(|z|α, 1)for allz∈ Sθ, Ψβ(Sθ) =



f ∈ H∞(Sθ) : ∃C |f(z)| ≤ C min(1, |z|−β)for allz∈ Sθ 

, Ψβα(Sθ) =f ∈ H∞(Sθ) : ∃C |f(z)| ≤ C min(|z|α,|z|−β)for allz∈ Sθ andΨ(Sθ) =

α,β>0Ψβα(Sθ).

Letω < θ < (1/2)πbe fixed. Forψ∈ Ψ(Sθ), we define ψ(A) = 1

2πi ˆ

Γθ

ψ(z)(z− A)−1dz.

The resolvent bounds forAimply that this integral converges absolutely inL (Y ). If one has, in addition, the quantitative estimate

ψ(A)L (Y ) ψ∞, thenAis said to haveH∞(Sθ)-calculus onY.

Lemma 6.1. LetAbe bisectorial of angleωand letθ > ω.

(1) Forφ1, φ2 ∈ Ψ(Sθ), we haveφ1(A)φ2(A) = (φ1· φ2)(A); this is also true if φ2 ∈ H∞(Sθ) is a rational function, in which caseφ2(A)is defined in the usual way by using the resolvents ofA.

(2) For allψ1∈ Ψ(Sθ),ψ2∈ H∞(Sθ),ψ3∈ Ψ(Sθ), we have

ψ1(A)(ψ2ψ3)(A) = (ψ1ψ2)(A)ψ3(A).

Proof. The first claim is the well-known homomorphism property, which in

both cases can be proved by writing out the definition ofφ1(A)φ2(A), performing a partial fraction expansion, and using Cauchy’s theorem. The second claim follows from the homomorphism property for ψ2 ∈ Ψ(Sθ), and the general case can be obtained from this by approximation (cf. [21, Theorem 9.2(i)]). 

Lemma 6.2. LetA be bisectorial of angle ω and let θ > ω. LetD(A) and

R(A)denote the domain and range ofA, respectively. Then

R(A) = R(A) ∩ D(A) = R(A(I + A)−2) =  ψ∈Ψ(Sθ)

R(ψ(A)).

Proof. Iff = ψ(A)g∈ R(ψ(A)), letfε:= A(ε + A)−1f ∈ R(A). Then f − fε= ε(ε + A)−1ψ(A)g = 1 2πi ˆ Γ ε ε + zψ(z)(z− A) −1g dz.

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The integrand is bounded byψ(z)z−1 ∈L1(Γ,| dz|)and tends pointwise to zero asε→ 0. Hencefε→ f by dominated convergence.

Next we observe thatfε = (I + εA)−1f → f asε→ 0. Indeed, if f ∈ D(A), thenf− fε= ε· (I + εA)−1Af has norm at most, since the second factor stays uniformly bounded. Since the operators(I + εA)−1 are uniformly bounded and D(A)is dense, the convergence remains true for all f. If now f ∈ R(A), then fε∈ R(A) ∩ D(A).

To complete the chain, letf ∈ R(A) ∩ D(A). Then for someg ∈ D(A2), we havef = Ag = A(I + A)−2(I + A)2g = ψ(A)h, whereψ(z) = z/(1 + z)2∈ Ψand

h = (I + A)2g. This completes the proof. 

We say thatψ ∈ Ψβ

α(Sθ)is degenerate if (at least) one of the restrictionsψ|S± θ

vanishes identically; otherwise, it is called non-degenerate. The following two lemmas go back to Calder ´on; cf. [28, Section IV.6.19]. For the convenience of the reader, we include simple proofs.

Lemma 6.3 (Calder ´on’s reproducing formula, I). Let ψ ∈ Ψβ

α(Sθ) be

non-degenerate. Ifα αandβ β, there existsψ∈ Ψβα(Sθ)such that

(6.1)

ˆ 0

ψ(tz) ψ(tz)dt

t = 1, z∈ Sθ.

Proof. Letψ(z) := ψ(z). Letm max(α− α, β− β)and set

c± := ˆ 0 (±t)m (1 + t2)mψ(±t)ψ(±t) dt t .

By non-degeneracy,c±> 0. Hence the functionψ(z) = c ±−1zm(1 + z2)−mψ(z)for

z∈ Sθ± has the desired properties. 

Lemma 6.4 (Calder ´on’s reproducing formula, II). Letψ, ψ ∈ Ψ(Sθ)satisfy (6.1). Then

ˆ

0

ψ(tA) ψ(tA)fdt

t = f, f ∈ R(A),

where the left side is defined as anL2-valued indefinite integral, i.e., the limit of

the finite integrals´abasa→ 0andb→ ∞.

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integral in the claim converges absolutely and ˆ 0 ψ(tA) ψ(tA)f dt t = ˆ 0 (ψ(t·) ψ(t·)φ(·))(A)gdt t = ˆ 0 1 2πi ˆ Γθ ψ(tz) ψ(tz)φ(z)(z− A)−1g dzdt t = 1 2πi ˆ Γθ ˆ 0 ψ(tz) ψ(tz)dt t φ(z)(z− A) −1g dz = 1 2πi ˆ Γθ φ(z)(z− A)−1g dz = φ(A)g = f

by Lemma 6.1, absolute convergence and Fubini’s theorem. To conclude, we recall from Lemma6.2that functions as above are dense inR(A)and observe that ´b

aψ(sz) ψ(sz) ds/sare uniformly in H∞(Sθ)so that the corresponding operators obtained by the formal substitutionz := Aare uniformly bounded by the functional calculus. From this, the convergence of the indefinite integral involving a general

f∈ R(A)to the asserted limit follows easily. 

7

Hardy spaces associated with bisectorial operators

We now move on to more specific spaces and operators. We are concerned with systems of N2 operators on L2, which we view as single operators on (L2)N. Off-diagonal estimates for families of such operators can be defined by the same formal requirement as in the caseN = 1above. It is easy to see that off-diagonal estimates for a family of operators in(L2)Nare equivalent to off-diagonal estimates for theN2 families of operators in L2 corresponding to the entries in the matrix representation of the original operators.

In the vector-valued context, we frequently use the natural identifications (L2)N⊗ X = (L2⊗ CN)⊗ X = L2⊗ (CN ⊗ X) ⊂ L2(XN).

Under these identifications, the elementary tensor(0, . . . , 0, f, 0, . . . , 0)⊗ x(withf at then-th entry) is identified with elementary tensorf⊗ (0, . . . , 0, x, 0, . . . , 0)(with xat then-th entry).

Throughout this section, we fix a UMD Banach spaceX and an exponent 1 < p <∞and suppose that the following assumptions are satisfied.

Assumption 7.1. The numbers1 τ  2and2 γ  ∞are fixed in such a way thatLp(X)has typeτand cotypeγ.

Assumption 7.2. The operatorAin(L2)Nis bisectorial of angle0 < ω < π/2.

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the family((I + ζA)−1)ζ∈C\Sθ satisfies off-diagonal estimates of orderM, where

M > n· min{1/τ, 1 − 1/γ}.

With only the above assumptions at hand, A may fail to be bisectorial even for N = 1 and, in particular, may fail to have an H∞-calculus, in Lp for some values ofp= 2. The tensor extensionA⊗ IX may already fail these properties in L2(X). To study problems involving operatorsf (A) in such spaces, we are thus led to define an appropriate scale of Hardy spaces associated withA. When Ais the Hodge–Dirac operator or the Hodge–de Rham Laplacian on a complete Riemannian manifold, this has been done in [3]. We build on the ideas of this paper.

Lemma 7.3. For ω < θ < π/2 and ε > 0, let g ∈ H∞(Sθ), and let ψ ∈ Ψε

M+ε(Sθ). Then {(g · ψ(t·))(A)}t>0 satisfies off-diagonal estimates of orderM,

and the off-diagonal constant has an upper bound which depends linearly ong∞.

Proof. Let us denote byδ := d(E, F )the ‘distance’ of two Borel setsE and F as defined previously. Then, using the fact that (I− z−1A)−1 ∈ OD1/|z|(M ) uniformly inz∈ Sθ, we have 1E(g· ψ(t·))(A)1Ff = 1 2πi ˆ Γθ g(z)ψ(tz)1E  I−1 zA −1 1Ff dz z    ˆ Γθ min(t|z|)M+ε, (t|z|)−ε(δ|z|)−Mf| dz| |z|  ˆ 1/t 0 tM+εrM+ε· δ−Mr−Mfdr r + ˆ 1/t t−εr−ε· δ−Mr−Mfdr r  tMδ−Mf;

and this proves the claim. 

Lemma 7.4. Letα, β, ε > 0, and

ψ∈ Ψβ+εmax{M−β,α}+ε(Sθ), ψ∈ Ψα+εmax{M−α,β}+ε(Sθ), φ∈ C1 ⊕ Ψ(Sθ).

Then

ψ(tA)φ(A) ψ(sA) = min(t/s)α, (s/t)βSt,s,

where(St,s)t,s>0is a uniformly bounded family of operators acting on(L2)N such

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Proof. We have

ψ(tA)φ(A) ψ(sA) = (t/s)αψ

0(tA)φ(A) ψ0(sA) = (s/t)βψ1(tA)φ(A) ψ1(sA), where

ψ0(z) := z−αψ(z)∈ Ψα+β+εε , ψ0(z) := zαψ(z) ∈ ΨεM+ε, ψ1(z) := zβψ(z)∈ Ψε

M+ε, ψ1(z) := z−αψ(z) ∈ Ψα+β+εε .

The cases tof the claim follows from Lemma7.3(withsplaying the role oft in that Lemma) withg(z) = ψ0(tz)φ(z)andψ0 in place ofψ, while for the other case we takeg(z) = φ(z) ψ1(sz)andψ1in place ofψ.  It is immediate to check that Tp,2(XN)  (Tp,2(X))N. Just as we extended the action of some operators onL2toTp,2(X), we may use this isomorphism to extend operators on(L2)N toTp,2(XN)by using their matrix representation and the extension procedure already discussed whenN = 1.

Proposition 7.5. Letψ, ˜ψ∈ Ψ(Sθ)andφ∈ C1 ⊕ Ψ(Sθ). Then (T F )t= ˆ 0 ψ(tA)φ(A)ψ(sA)Fs ds s

extends to a bounded operator onTp,2(XN)if at least one of the following

condi-tions is satisfied: (a) M > n/τ,ψ∈ Ψn(1/τ −1/2)+εn/2+ε , andψ˜∈ Ψn/2+εn(1/τ −1/2)+ε; (c) M > max{n/τ, n(1 − 1/γ)},ψ∈ Ψn(1/τ −1/2)+εn/2+n max{1/γ−1/τ,0}+ε, andψ˜∈ Ψn(1/2−1/γ)+εn/2+n max{1/τ −1/γ,0}+ε; (d) M > n(1− 1/γ),ψ∈ Ψn/2+εn(1/2−1/γ)+ε, andψ˜∈ Ψn(1/2−1/γ)+εn/2+ε , whereε > 0is arbitrary.

Proof. This is directly, if slightly tediously, verified as a corollary of

Lemma 7.4and Corollary 5.1, so that the different conditions of Proposition7.5

correspond to those of Corollary5.1. As for the application of Corollary5.1, we simply apply this to theN2matrix elements of the full operator, each of which acts

onTp,2(X). 

Definition 7.6. We say that a pair of functions(ψ, ˜ψ) ∈ Ψ(Sθ)× Ψ(Sθ) has

sufficient decay if they verify at least one of the conditions (a), (c), or (d) of

Proposition7.5.

Remark 7.7. (i) Note that the notion of sufficient decay as defined above

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fixed. Also observe that if the parameters are such that, for instance,n(1− 1/γ) < M ≤ n/τ, then only the condition(d)above is applicable.

(ii) If(ψ, 0)∈ Ψ(Sθ)× Ψ(Sθ)has sufficient decay, by Calder ´on’s reproducing formula, there exists a ψ˜ ∈ Ψ(Sθ)which satisfies (6.1) and decays as rapidly as desired; in particular, we may arrange that the pair(ψ, ˜ψ)also has sufficient decay. A similar remark applies if we start fromψ˜∈ Ψ(Sθ)such that(0, ˜ψ)has sufficient decay. Forf = igi⊗ ξi∈ (L2)N ⊗ X andψ∈ Ψ(S θ), we write (Qψf )(y, t) :=  i

ψ(tA)gi(y)⊗ ξi := ψ(tA)f (y).

Definition 7.8. For1≤ p < ∞and a non-degenerateψ∈ Ψ(Sθ), the Hardy

spaceHA,ψp (XN)associated withAandψis the completion of the space {f ∈ R(A) ⊗ X ⊆ (L2)N⊗ X : Q

ψf ∈ Tp,2(XN)} with respect to the norm

fHA,ψp (XN):=QψfTp,2(XN).

It is clear that · Hp

A,ψ(XN) is a seminorm onR(A) ⊗ X; that it is actually a

norm will be seen shortly. By definition, the operator

(Qψf )(·, t) := ψ(tA)f

embeds the Hardy spaceHA,ψp (XN)isometrically into the tent spaceTp,2(XN). Of importance also is another operator acting to the opposite direction. Forψ∈ Ψ(Sθ), we defineSψf ∈ (L2)N ⊗ Xby (7.1) SψF := ˆ 0  ψ(sA)F (s,·)ds s for those functionsF∈ L1

loc(R+; (L2)N)⊗ Xfor which the integral exists as a limit in(L2)N of the finite integrals´b

a, wherea→ 0andb→ ∞.

By Calder ´on’s reproducing formula, for a given ψ ∈ Ψ(Sθ), there exists ˜

ψ∈ Ψ(Sθ)such that the defining formula (7.1) makes sense for all F ∈ Qψ(R(A) ⊗ X),

and we have

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Hence, iffHp

A,ψ(XN)= 0for somef ∈ R(A) ⊗ X, it follows from the definition

of the norm thatQψf = 0, and the identity (7.2) yields immediatelyf = 0. Thus  · HA,ψp (XN)is indeed a norm.

Proposition 7.9. Let(ψ, ψ)∈ Ψ(Sθ)×Ψ(Sθ)be a pair with sufficient decay. If f ∈ Tp,2(XN) is such that the defining formula (7.1) is valid, then S

 ψf ∈ HA,ψp (XN), and the mappingf → S



ψf extends uniquely to a bounded operator

fromTp,2(XN)toHp

A,ψ(XN).

Proof. Writeg := Sψf. First we check thatg∈ R(A) ⊗ X: this is clear from the defining formula, since ψ(sA)f (·, s) ∈ R(A) for each s > 0 by Lemma 6.2, and Bochner integration in the Banach space(L2)N preserves the closed subspace R(A).

By Proposition7.5,

(y, t)→ ψ(tA)g(y) = ˆ

0

ψ(tA) ψ(sA)f (y, s)ds s defines an elementψ(·A)gofTp,2(XN); and we have

SψfHA,ψp (XN)=ψ(·A)gTp,2(XN) fTp,2(XN).

The subspace ofTp,2(XN)where the defining formula (7.1) is valid contains, e.g., (Cc)N⊗Xand is therefore dense inTp,2(XN). Hence the mappingS



ψhas a unique extension to a bounded operator fromTp,2(XN)toHp

A,ψ(XN). 

Next we show thatHA,ψp (XN)is independent ofψ∈ Ψ(S

θ), provided(ψ, 0)has sufficient decay. A typical function with this property is

ψ(z) = (√z2)n(1/2−1/γ)+1e−√z2

,

whereγ denotes the cotype of Lp(X). This gives the classical definition by the Poisson kernel whenX =Cand1 < p≤ 2, takingγ = 2.

Theorem 7.10. Letψ, ψ∈ Ψ(Sθ)be two functions such that(ψ, 0)and(ψ, 0)

have sufficient decay. Then

(i) HA,ψp (XN) = Hp

A,ψ(XN) =: H p A(XN);

(ii) Ahas anH∞-functional calculus onHAp(XN).

Proof. Letφ ∈ C1 ⊕ Ψ(Sθ) be arbitrary and fixed. Let f ∈ R(A) ⊗ X. By Calder ´on’s reproducing formula, there exists ψ ∈ Ψ(Sθ) (with any prescribed decay) such that

ψ(tA)φ(A)f = ˆ

0

ψ(tA)φ(A) ψ(sA)ψ(sA)f ds s .

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Thus

φ(A)fHpA,ψ(XN)=T QψfTp,2(XN),

whereT is the operator onTp,2(XN)given by T F (y, t) =

ˆ

0

ψ(tA)φ(A) ψ(sA)F (y, s)ds s . From Proposition7.5, we deduce that

φ(A)fHpA,ψ(XN) QψfTp,2(XN)=fHp A,ψ(XN).

Takingφ = 1gives (i). Takingφ∈ Ψ(Sθ), we obtain (ii).  The following, by now quite simple result has some useful consequences.

Proposition 7.11. If (0, ψ)has sufficient decay, then the bounded mapping

Sψ: Tp,2(XN)→ Hp

A(XN)is surjective.

Proof. By Remark7.7, we can findψ∈ Ψ(Sθ)such that (7.2) is satisfied and (ψ, ψ)has sufficient decay. Now let f ∈ HAp(XN) = HA,ψp (XN)be arbitrary and let limn→∞fn = f inHA,ψp (XN)withf

n ∈ R(A) ⊗ X. The functionsgn:= Qψfn belong toTp,2(XN)andgn− gmTp,2(XN)=fn− fmHp

A,ψ(XN) for allm, n. It

follows that the sequence(fn)is Cauchy inTp,2(XN)and therefore converges to some f ∈ Tp,2(XN). From fn = Sψgn and the continuity of Sψ, it follows that

f = Sψg. 

Corollary 7.1. Let(0, ψ)have sufficient decay. An equivalent description of the Hardy space is

HAp(XN) = Hp A, ψ(X

N) :={S 

ψF : F ∈ Tp,2(XN)},

and an equivalent norm is given by

fHA, pψ(XN):= inf{F Tp,2(XN): f = SψF}.

As a further consequence, we deduce an interpolation result for Hardy spaces from the following general principle (see Theorem 1.2.4 in [29]). LetX0, X1and Y0, Y1be two interpolation couples such that there exist operatorsS∈ L (Yi, Xi)and Q ∈ L (Xi, Yi) with SQx = x for all x ∈ Xi and i = 0, 1. Then [X0, X1]θ = S[Y0, Y1]θ. Here we take (ψ, ψ) as in the Calder ´on reproducing formula with sufficient decay,S = SψandQ = Qψ.

Corollary 7.2. LetX be a UMD space. For all1 < p0< p1<∞,0 < θ < 1,

andN  1, we have

[Hp0

A(XN), H p1

A(XN)]θ= HApθ(XN)

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8

Hardy spaces associated with differential operators

The construction described in Section7is particularly relevant when dealing with differential operatorsA = DB inL2⊕ (L2)n, where

DB = 0 −divB

∇ 0

 ,

with B a multiplication operator on(L2)n given by an(n× n)-matrix with L∞ entries. Such operators have been considered in connection with the celebrated square root problem of Kato, which was originally solved in [2]. A new proof based on first order methods was devised in [4], where it was shown thatDB is bisectorial onL2(C ⊕ Cn)and satisfies off-diagonal estimates of any order.

In [17], the H∞-functional calculus of DB ⊗ IX in Lp(X) ⊕ (Lp(X))n = Lp(X⊕ Xn) is described in terms of R-boundedness of the resolvents. Al-though these resolvent conditions, and hence the functional calculus, may fail onLp(X⊕Xn)in general, it follows from Section7that these operators do have an H∞-functional calculus onHDp

B(X⊕ X

n), which in particular implies Kato type estimates in this space.

To express these estimates, observe first thatR(DB) =R(divB) ⊕ R(∇). Let us therefore write a functionf ∈ R(DB)⊗ X as(f0, f1), where

f0∈ R(divB) ⊗ X ⊆ L2⊗ X, f1∈ R(∇) ⊗ X ⊆ (L2)n⊗ X = L2⊗ Xn denote theX-valued andXn-valued parts off, respectively. Defining

HDp

B(X⊕ X

n) := Hp

DB,ψ(X⊕ Xn) by means of the (even!) functionψ(z) = (√z2)Ne−√z2

with N large enough, we note thatψ(tDB) = φ(t2D2B), whereφ(z) =

zNe−√zand the operator

D2B= −divB∇ 0

0 −∇divB

 ,

and hence φ(t2D2

B), is diagonal with respect to the splitting f = (f0, f1). In particular, this shows that

(f0, f1)HpDB(X⊕Xn) (f0, 0)HDBp (X⊕Xn)+(0, f1)HDBp (X⊕Xn).

Hence also the full space HDp

B(X ⊕ X

n) (constructed as the completion of R(DB)⊗ Xwith respect to the above-given norm) has a natural direct sum splitting

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