Luc Deleaval

Christoph Kriegler

Dunkl spectral multipliers with values in UMD lattices

Keywords: Mathematics subject classification: 42A45, 42B25, 47A60 Spectral multiplier theorems, UMD valued L p spaces, Dunkl Laplacian

We show a Hörmander spectral multiplier theorem for A = A0 ⊗ IdY acting on the Bochner space L p (R d , h 2 κ ; Y ), where A0 is the Dunkl Laplacian, h 2 κ a weight function invariant under the action of a reflection group and Y is a UMD Banach lattice. We follow hereby a transference method developed by Bonami-Clerc and Dai-Xu, passing through a Marcinkiewicz multiplier theorem on the sphere. We hereby generalize works for A0 = -∆ acting on L p (R d , dx) by Girardi-Weis, Hytönen and others before. We apply our main result to maximal regularity for Cauchy problems involving A.

Introduction

Let f be a bounded function on (0, ∞) and u(f ) the operator on L p (R d ) defined by [f (-∆)g] ˆ= [u(f )g] ˆ= f ( ξ 2 )ĝ(ξ). Hörmander's theorem on Fourier multipliers [START_REF] Hörmander | Estimates for translation invariant operators in L p spaces[END_REF]Theorem 2.5] asserts that u(f ) : L p (R d ) → L p (R d ) is bounded for any p ∈ (1, ∞) provided that for some integer α strictly larger than d 2 , and q = 2,

(1.1)

f q H α q := max k=0,1,...,α sup R>0 1 R 2R R t k d k dt k f (t) q dt < ∞.
This theorem has many refinements and generalizations to various similar contexts. Namely, one can ask for different exponents q ∈ [1, ∞) in (1.1), and generalize to non-integer α there to get larger (for smaller q and for smaller α) admissible classes H α q = {f ∈ L 1 loc (0, ∞) : f H α q < ∞} of multiplier functions f (see Subsection 2.4). Moreover, it has been a deeply studied question over the last years to know to what extent one can replace the ordinary Laplacian subjacent to Hörmander's theorem by other operators A acting on some L p (Ω) space. A theorem of Hörmander type holds true for many elliptic differential operators A, including sub-laplacians on Lie groups of polynomial growth, Schrödinger operators and elliptic operators on Riemannian manifolds, see [START_REF]Spectral multipliers on Lie groups of polynomial growth[END_REF][START_REF] Christ | L p bounds for spectral multipliers on nilpotent groups[END_REF][START_REF] Duong | From the L 1 norms of the complex heat kernels to a Hörmander multiplier theorem for sub-Laplacians on nilpotent Lie groups[END_REF][START_REF] Duong | Plancherel-type estimates and sharp spectral multipliers[END_REF]. More recently, spectral multipliers have been studied for operators acting on L p (Ω) only for a strict subset of (1, ∞) of exponents [9,[START_REF] Chen | L p -bounds for Stein's square functions associated to operators and applications to spectral multipliers[END_REF]19,20,48,49], for abstract operators acting on Banach spaces [START_REF] Kriegler | Spectral multiplier theorems and averaged R-boundedness[END_REF], and for operators acting on product sets Ω 1 ×Ω 2 [START_REF] Sikora | Multivariable spectral multipliers and analysis of quasielliptic operators on fractals[END_REF][START_REF] Wróbel | Multivariate spectral multipliers for tensor product orthogonal expansions[END_REF]69]. A spectral multiplier theorem means then that the linear and multiplicative mapping (1.2)

H α q → B(X), f → f (A),
is bounded, where typically X = L p (Ω). One important consequence of a spectral multiplier theorem as in (1.2) is the boundedness of Bochner-Riesz means associated with A. Namely, we put for β, R > 0

f β R (t) = (1 -t/R) β 0 < t R 0 t > 0.
Then f β R belongs to H α q (with uniform norm bound for R > 0) if and only if β > α -1 q , for q ∈ [1, ∞) and α > 1 q , see [20, p. 11] and [START_REF] Kriegler | Spectral multiplier theorems via H ∞ calculus and R-bounds[END_REF]. Thus the boundedness of (1.2) yields boundedness of the Bochner-Riesz means for f β R (A) if β > α -1 q . For other applications of a Hörmander spectral multiplier theorem, see at the end of Subsection 3.2.

On the other hand, in the particular case of A = -∆, another direction of generalization of Hörmander's theorem is possible. Namely, [13,[START_REF] Girardi | Operator-valued Fourier multiplier theorems on L p (X) and geometry of Banach spaces[END_REF]39,41,[START_REF] Mcconnell | On Fourier multiplier transformations of Banach-valued functions[END_REF][START_REF] Štrkalj | On operator-valued Fourier multiplier theorems[END_REF][START_REF] Weis | Operator-valued Fourier multiplier theorems and maximal L p -regularity[END_REF][START_REF] Zimmermann | On vector-valued Fourier multiplier theorems[END_REF] have studied for which Banach spaces Y, the operator f (-∆) ⊗ Id Y , initially defined on L p (R d ) ⊗ Id Y extends to a bounded operator on L p (R d ; Y ) for any f belonging to some Hörmander class H α q (or some Mihlin class, which corresponds essentially to q = ∞ in H α q ). A necessary condition is that Y is a UMD space. Moreover, the Fourier type [START_REF] Girardi | Operator-valued Fourier multiplier theorems on L p (X) and geometry of Banach spaces[END_REF], and Rademacher type/cotype [41] of Y play a role when one strives for better or best possible derivation order α.

In this article, we extend this latter programme partly to Dunkl operators, in place of the pure Laplacian. "Partly" refers to the fact that we treat only radial multipliers (see [69] for multivariate Dunkl spectral multipliers in the case Y = C, but with nevertheless restriction on the underlying reflection group), we restrict to the subclass of Y being a UMD lattice, and we do not talk about operator valued spectral multipliers, i.e. f in (1.2) is a function with values in {T ∈ B(X) : T (λ -A) -1 = (λ -A) -1 T for all λ ∈ ρ(A)} (although this last part would be possible to some extent).

Roughly speaking, Dunkl operators are parameterized (with a continuous set of parameters κ) deformations of the partial derivatives and involve a reflection group W associated with a root system R (see Subsection 2.5.1. for their definition). A basic motivation for the study of these operators comes from the theory of spherical functions in analysis on Lie groups, which can be, in several situations, regarded as only the W -invariant part of a theory of Dunkl operators. Indeed, these operators play the role of derivatives for a generalized Laplacian ∆ κ (the so-called Dunkl Laplacian), whose restriction to W -invariant functions is given by

Res ∆ κ = ∆ + α∈R κ(α) ∂ α α,
• , and this formula coincides for particular root systems and particular values of κ to the radial part of the Laplace-Beltrami operator of a Riemannian symmetric space of Euclidean type (see [START_REF] Jeu | Paley-Wiener theorems for the Dunkl transform[END_REF]). More generally, Dunkl operators have significantly contributed to the development of harmonic analysis associated with a root system and to the theory of multivariable hypergeometric functions. They also naturally appear in various other areas of mathematics, which include for instance, the theory of stochastic processes with values in a Weyl chamber or the theory of integrable quantum many body systems of Calogero-Moser-Sutherland type. As regards the harmonic analysis of Dunkl operators and their related objects, the subjacent analytic structure has a rich analogy with the Fourier analysis. However, there are still many problems to be solved and the theory is still at its infancy. One of the main obstruction is the lack of an explicit formula for the operator V κ which intertwines the commutative algebra of Dunkl operators with the algebra of standard differential operators with constant coefficients. Apart from the case W = Z d 2 where the known formula for V k allows to tackle and bypass some difficulties, many tools of harmonic analysis are not accessible. However, in this paper, we do not restrict ourselves to this particular reflection group, and all our results on Dunkl spectral multipliers are stated and proven for a general reflection group.

Coming back in particular to our H α q Hörmander theorem, the order of derivation α and the integration parameter q that we get are

(1.3) α ∈ N, α > d 2 + γ κ + 1 2 , q = 1
where d + 2γ κ is the doubling dimension of the Dunkl weight h 2 κ on R d . Note that usually, one cannot expect to get a Hörmander H α 2 multiplier theorem for α < d 2 + γ κ (see e.g. [START_REF] Duong | Plancherel-type estimates and sharp spectral multipliers[END_REF]), and that H

d 2 +γκ+ 1 2 + 2 → H d 2 +γκ+ 1 2 + 1 → H d 2 +γκ 2
, for > 0, the exponents being sharp in these embeddings (see Lemma 2.8 2.). Our main Theorem, see Theorem 3.13 and the section Preliminaries for precisions, states as follows.

Theorem 1.1 Let 1 < p < ∞ and Y = Y (Ω ) be a UMD Banach lattice. Let A be the Dunkl Laplacian on R d , associated with both a general finite reflection group and a nonnegative multiplicity function κ. Let α be as above in (1.3)

. Assume that m : (0, ∞) → C belongs to H α 1 . Then m(A) ⊗ Id Y extends to a bounded operator on L p (R d , h 2 κ ; Y ).
One of the features of the vector valued character of this theorem is that an operator of the form B = Id L p ⊗ B 0 will commute with A (or powers of A) and therefore, spectral theory of a sum A β + B is at hand. Consequently, we apply Theorem 1.1 to existence, uniqueness and (maximal) regularity of solutions of Cauchy problems or time independent problems involving A β + B, where A is the Dunkl Laplacian, β > 0 is arbitrary and B is as above, see Section 4.

Theorem 1.1 has several ancestors and variants in the literature, where often the lattice property of Y is not needed. We mention here Girardi and Weis [START_REF] Girardi | Operator-valued Fourier multiplier theorems on L p (X) and geometry of Banach spaces[END_REF] resp. Hytönen [41] for operator valued resp. scalar valued Fourier multipliers on UMD valued L p spaces. Note that in these two works, the Fourier type resp. Rademacher type/cotype enters in the admissible choice of the derivation order α = d/type Y + 1, which is not necessary in our theorem, i.e. the value type Y in the derivation order α improves to the best value 2, when applied to the case of the pure Laplacian, i.e. κ = 0. However, we have to add 1/2 to α due to the value q = 1 as explained above, and need the lattice property of Y, whereas Hytönen only needs particular partial derivatives of the multiplier symbol, and Girardi's and Weis' result also applies to operator valued multiplier symbols. As a predecessor of all these results, we mention the UMD space valued singular integrals of convolution type [START_REF] Benedek | Convolution operators on Banach space valued functions[END_REF]12]. More recently, in [START_REF] Betancor | Square functions and spectral multipliers for Bessel operators in UMD spaces[END_REF]8] the boundedness of vector valued, but holomorphic, spectral multipliers associated with Hermite and Bessel operators is established, in the case that Y = [H, X] θ is a complex interpolation space between a Hilbert space and a UMD space. Note that according to [START_REF] Rubio De Francia | Martingale and integral transforms of Banach space valued functions[END_REF]Corollary p. 216], this case is an intermediate case between the bare UMD space case and the UMD lattice case.

The methods of proof that we use for Theorem 1.1 are:

• Maximal estimates for semigroups associated with the Dunkl operator on R d and on the sphere S d-1 [START_REF] Dai | Maximal function and multiplier theorem for weighted space on the unit sphere[END_REF];

• Square function estimates on L p (Ω; Y (Ω )), which is the same as R-boundedness for the space L p (Ω; Y ) as soon as Y is a UMD lattice;

• H ∞ functional calculus, in particular for vector valued diffusion semigroups [START_REF] Xu | H ∞ functional calculus and maximal inequalities for semigroups of contractions on vector-valued L p -spaces[END_REF];

• A reduction method to spherical harmonics, developed in [10, 24,[START_REF] Dai | Maximal function and multiplier theorem for weighted space on the unit sphere[END_REF][START_REF] Dai | Analysis on h-harmonics and Dunkl transforms[END_REF], which uses Cesàro means, that is, smoothed approximate identities similar to the Bochner-Riesz means f β R (A) above. The H ∞ functional calculus moreover is our starting point upon which we build the H α q functional calculus. This replaces the usually used selfadjoint calculus approach, which defines f (A) in (1.2) on L 2 (Ω)∩L p (Ω), and which by density gives an a priori meaning to f (A) as an operator acting on L p (Ω). Note that on L p (Ω; Y ), there is no selfadjoint calculus at hand, since L 2 (Ω; Y ) is not a Hilbert space in general.

We end this introduction with an overview of the following sections. In Section 2, we define and recall the central notions for this article, namely diffusion semigroups, UMD lattices, Rboundedness and square functions, functional calculus and Dunkl analysis. In Section 3, we then develop the proof of the spectral multiplier theorem. In a first place, in Subsection 3.1, we show a Marcinkiewicz multiplier Theorem 3.2, and then in Subsection 3.2, we deduce the Hörmander multiplier Theorem 3.13. We end the article with some illustrative applications to maximal regularity in Section 4.

Note that in the article, the symbol means an inequality up to a constant independent of the relevant variables.

Preliminaries

In this section, we define and recall the central notions of the article and we prove several lemmas which will be relevant for the sequel.

Symmetric contraction semigroups

Definition 2.1 Let (Ω, µ) be a σ-finite measure space. Let (T t ) t 0 be a family of operators which act boundedly on L p (Ω) for any 1 p < ∞. Then (T t ) t 0 is called symmetric contraction semigroup (on Ω), if 1. (T t ) t 0 is a strongly continuous semigroup on L p (Ω) for any 1 p < ∞; 2. T t is selfadjoint on L 2 (Ω) for any t 0; 3. T t L p (Ω)→L p (Ω) 1 for any t 0 and 1 p < ∞.

If in addition we have 1. T t is a positive operator for any t 0; 2. T t (1) = 1 (note that T t is bounded on L ∞ (Ω) by selfadjointness and boundedness on L 1 (Ω)),

then (T t ) t 0 is called a diffusion semigroup on Ω.
For a thorough study of diffusion semigroups, we refer the reader to [62] and [40] in the scalar and the vector valued case respectively.

UMD lattices

In this article, UMD lattices, i.e. Banach lattices which enjoy the UMD property, play a prevalent role. For a general treatment of Banach lattices and their geometric properties, we refer the reader to [52, Chapter 1]. We recall now definitions and some useful properties. A Banach space Y is called UMD space if the Hilbert transform

H : L p (R) → L p (R), Hf (x) = P V - R 1 x -y f (y)dy
extends to a bounded operator on L p (R; Y ), for some (equivalently for all) 1 < p < ∞ [42, Theorem 5.1]. The importance of the UMD property in harmonic analysis was recognized for the first time by Burkholder [START_REF] Burkholder | A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional[END_REF]15], see also his survey [START_REF] Burkholder | Martingales and singular integrals in Banach spaces[END_REF]. He settled a geometric characterization via a convex functional [START_REF] Burkholder | A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional[END_REF] and together with Bourgain [START_REF] Bourgain | Some remarks on Banach spaces in which martingale difference sequences are unconditional[END_REF], they showed that the UMD property can be expressed by boundedness of Y -valued martingale sequences. As a survey for UMD lattices and their properties in connection with results in harmonic analysis, we refer the reader to [START_REF] Rubio De Francia | Martingale and integral transforms of Banach space valued functions[END_REF].

We tacitly shall use several times the following almost trivial observation.

R-boundedness and square functions

Let X be a Banach space and τ ⊂ B(X). Then τ is called R-bounded if there is some C < ∞ such that for any n ∈ N, any x 1 , . . . , x n ∈ X and any T 1 , . . . , T n ∈ τ, we have

E n k=1 k T k x k X CE n k=1 k x k X ,
where the k are i.i.d. Rademacher variables on some probability space, that is, Prob( k = ±1) = 1 2 . The least admissible constant C is called R-bound of τ and is denoted by R(τ ). Note that trivially, we always have R({T }) = T for any T ∈ B(X).

Let Y = Y (Ω ) be a B-convex Banach lattice. Then we have the norm equivalence

(2.1) E n k=1 k y k Y ∼ = n k=1 |y k | 2 1 2
Y uniformly in n ∈ N [START_REF] Maurey | Type et cotype dans les espaces munis de structures locales inconditionnelles[END_REF]. In particular, this also applies to L p (Ω; Y ), 1 < p < ∞, since this will also be a B-convex Banach lattice. We deduce the following lemma.

Lemma 2.3 Let T be a bounded (linear) operator on a B-convex Banach lattice Y (Ω ). Then its tensor extension

T ⊗ Id 2 , initially defined on Y (Ω ) ⊗ 2 ⊂ Y (Ω ; 2 ) is again bounded, on Y (Ω ; 2 ). In particular, if Y (Ω ) is a UMD lattice, then Y (Ω ; 2
) is also a UMD lattice.

Proof : Let (e k ) k be the canonical basis of 2 . We have

(T ⊗ Id 2 ) n k=1 y k ⊗ e k Y (Ω ; 2 ) = n k=1 |T y k | 2 1 2 Y ∼ = E n k=1 k T y k Y R({T })E n k=1 k y k Y ∼ = T n k=1 |y k | 2

Holomorphic (H ∞ ) and Hörmander (H α p ) functional calculus

In this subsection, we recall the necessary background on functional calculus that we will treat in this article. Let -A be a generator of an analytic semigroup (T z ) z∈Σ δ on some Banach space X, that is, δ ∈ (0, π 2 ], Σ δ = {z ∈ C\{0} : | arg z| < δ}, the mapping z → T z from Σ δ to B(X) is analytic, T z+w = T z T w for any z, w ∈ Σ δ , and lim z∈Σ δ , |z|→0 T z x = x for any strict subsector Σ δ . We assume that (T z ) z∈Σ δ is a bounded analytic semigroup, which means sup z∈Σ δ T z < ∞ for any δ < δ.

It is well-known [36, Theorem 4.6, p. 101] that this is equivalent to A being pseudo-ωsectorial for ω = π 2 -δ, that is, 1. A is closed and densely defined on X;

2. The spectrum σ(A) is contained in Σ ω (in [0, ∞) if ω = 0); 3. For any ω > ω, we have sup λ∈C\Σ ω λ(λ -A) -1 < ∞.
We say that A is ω-sectorial if it is pseudo-ω-sectorial and has moreover dense range. In the sequel, we will always assume that A has dense range, to avoid technical difficulties. If A does not have dense range, but X is reflexive, which will always be the case in this article, then we may take the injective part A 0 of A on R(A) ⊆ X [50, Proposition 15.2], which then does have dense range. Here, R(A) stands for the range of A. Then -A generates an analytic semigroup on X if and only if so does -A 0 on R(A). This parallel will continue this section, i.e. the functional calculus for A 0 can be extended to A in an obvious way, see [START_REF] Kriegler | Spectral multipliers, R-bounded homomorphisms and analytic diffusion semigroups[END_REF]Illustration 4.87].

For θ ∈ (0, π), let

H ∞ (Σ θ ) = {f : Σ θ → C : f analytic and bounded} equipped with the uniform norm f ∞,θ . Let further H ∞ 0 (Σ θ ) = f ∈ H ∞ (Σ θ ) : ∃ C, > 0 : |f (z)| C min(|z| , |z| -) .
For an ω-sectorial operator A and θ ∈ (ω, π), one can define a functional calculus H ∞ 0 (Σ θ ) → B(X), f → f (A) extending the ad hoc rational calculus, by using a Cauchy integral formula. If moreover, there exists a constant C < ∞ such that f (A)

C f ∞,θ , then A is said to have bounded H ∞ (Σ θ ) calculus and the above functional calculus can be extended to a bounded Banach algebra homomorphism H ∞ (Σ θ ) → B(X). This calculus also has the property

f z (A) = T z for f z (λ) = exp(-zλ), z ∈ Σ π 2 -θ . Lemma 2.4 Let ω ∈ (0, π) and A be an ω-sectorial operator on X having an H ∞ (Σ θ ) calculus for some θ ∈ (ω, π). Let (f n ) n be a sequence in H ∞ (Σ θ ) such that f n (λ) → f (λ) for any λ ∈ Σ θ and sup n f n ∞,θ < ∞. Then for any x ∈ X, f (A)x = lim n f n (A)x.
Proof : See [50, Theorem 9.6] or [START_REF] Cowling | Banach space operators with a bounded H ∞ functional calculus[END_REF]Lemma 2.1].

The following classical lemma will be useful at several instances A proof for the case p = 1 can be found e.g. in [START_REF] Hytönen | Analysis in Banach spaces[END_REF]Proposition 2.1.1], and for a detailed study of the case p = ∞, see [START_REF] Lindemulder | Banach Space-valued Extensions of Linear Operators on L ∞ . Ordered Structures and Applications[END_REF]. Proof : Suppose first p = ∞. We have

T ⊗ Id Y k f k ⊗ y k L ∞ (Y ) = esssup x∈S d-1 k T f k (x)y k Y = esssup x∈S d-1 sup y * Y * 1 T k f k (x) y k , y * = sup y * Y * 1 esssup x∈S d-1 T k f k (x) y k , y * sup y * Y * 1 T L ∞ →L ∞ k f k (•) y k , y * L ∞ = T L ∞ →L ∞ k f k ⊗ y k L ∞ (Y )
, where the last equality follows by going the steps before backwards, with T replaced by Id L ∞ and T taken as a fixed factor. Now the claim for p = 1 follows essentially from duality. Namely, fix k f k ⊗ y k ∈ L 1 ⊗ Y, and take in the following the supremum over those l g l ⊗ z l ∈ L ∞ ⊗ Y * of norm less than 1. Then

(T ⊗ Id Y ) k f k ⊗ y k L 1 (Y ) sup (T ⊗ Id Y ) k f k ⊗ y k , l g l ⊗ z l = sup k,l T f k , g l y k , z l = sup k f k ⊗ y k , (T * ⊗ Id Y * ) l g l ⊗ z l sup k f k ⊗ y k L 1 (Y ) T * L ∞ →L ∞ l g l ⊗ z l = T L 1 →L 1 k f k ⊗ y k L 1 (Y )
.

We record the following proposition for later use.

Proposition 2.6

Let T t = exp(-tA) be a symmetric contraction semigroup on Ω.

Let Y be a UMD lattice. Then T t extends to a bounded analytic semigroup on L p (Ω; Y )

for any 1 < p < ∞. Moreover, its negative generator A has a bounded H ∞ (Σ θ ) calculus for some θ < π 2 . 2. Let Y be a general Banach space and 1 p < ∞. Then T t extends to a contractive strongly continuous semigroup on L p (Ω; Y ).

Proof : 1. See [START_REF] Xu | H ∞ functional calculus and maximal inequalities for semigroups of contractions on vector-valued L p -spaces[END_REF]Theorem 4].

2. According to Lemma 2.5, T t extends to a contraction on L 1 (Ω; Y ) and L ∞ (Ω) ⊗ Y , and thus, by complex interpolation [6, 5.1.2 Theorem part 2] also for any 1 < p < ∞. The semigroup property of T t is clear. To prove the strong continuity, we can restrict to a dense subspace of L p (Ω; Y ). Since we have p < ∞, L p (Ω) ⊗ Y is such a dense subspace, and for

f = k f k ⊗ y k , we clearly have (T t ⊗ Id Y )f = k T t f k ⊗ y k → k f k ⊗ y k as t → 0.
For further information on the H ∞ calculus, we refer e.g. to [50]. We now turn to Hörmander function classes and their calculi.

Definition 2.7 Let p ∈ [1, ∞) and α > 1 p .
We define the Hörmander class by

H α p = f : (0, ∞) → C bounded and continuous, sup R>0 φf (R •) W α p (R) := f H α p < ∞ .
Here φ is any C ∞ c (0, ∞) function different from the constant 0 function (different choices of functions φ resulting in equivalent norms) and W α p (R) is the classical Sobolev space.

The Hörmander classes have the following properties.

Lemma 2.8

1. Assume that α ∈ N and 1 p < ∞. Then a locally integrable function f : (0, ∞) → C belongs to the Hörmander class H α p if and only if

α k=0 sup R>0 2R R t k d k dt k f (t) p dt/t < ∞, if and only if max k=0 or k=α sup R>0 2R R t k d k dt k f (t) p dt/t < ∞,
and the above quantities are equivalent to f p

H α p . 2. We have the continuous embeddings H ∞ (Σ θ ) → H α q → H α p → H β q for θ ∈ (0, π), p < q and α β + 1 p -1 q . 3. H α
p is a Banach algebra for the pointwise multiplication.

The mapping

H α p → H α p , m → m((•) γ )
, is an isomorphism for any γ > 0.

Proof : See [44, Section 4.2.1] for everything except the second claimed equivalence in 1. For the latter, we note that for 0 l k, we have

2 1 d l dt l f (t) p dt 2 1 d k dt k f (t) p dt + 2 1 |f (t)| p dt according to [1, Theorem 5.2]. Now for a function g ∈ W k p (R, 2R), take f (t) = g(Rt)
, and substitute this in the above formula. One readily obtains that the first displayed term in 1. is dominated by the second displayed term. The converse estimate is trivial.

We can base a Hörmander functional calculus on the H ∞ calculus by the following procedure.

Definition 2.9

We say that a 0-sectorial operator has a bounded H α p calculus if for some θ ∈ (0, π) and any

f ∈ H ∞ (Σ θ ), f (A) C f H α p ( C f ∞,θ
). In this case, the H ∞ (Σ θ ) calculus can be extended to a bounded Banach algebra homomorphism H α p → B(X) in the following way. Let [START_REF] Kriegler | Spectral multiplier theorems via H ∞ calculus and R-bounds[END_REF]. Since W α p → H α p , by the above density, we get a bounded mapping W α p → B(X) extending the H ∞ calculus.

W α p = f : (0, ∞) → C : f • exp ∈ W α p (R) equipped with the norm f W α p = f • exp W α p (R) . Note that for any θ ∈ (0, π), the space H ∞ (Σ θ ) ∩ W α p is dense in W α p
Definition 2.10 Let (φ k ) k∈Z be a sequence of functions in C ∞ c (0, ∞) with the properties that suppφ k ⊂ [2 k-1 , 2 k+1 ] and k∈Z φ k (t) = 1 for all t > 0. Then (φ k ) k∈Z is called a dyadic partition of unity. Let (φ k ) k∈Z be a dyadic partition of unity. For f ∈ H α p , we have that φ k f ∈ W α p , hence (φ k f )(A) is well-defined. Then it can be shown that for any x ∈ X, n k=-n (φ k f )(A)x converges
as n → ∞ and that it is independent of the choice of (φ k ) k∈Z . This defines the operator f (A), which in turn yields a bounded Banach algebra homomorphism

H α p → B(X), f → f (A)
. This is the Hörmander functional calculus. For details of this procedure, we refer to [44, Sections 4.2.3 -4.2.6].

Dunkl transform, h-harmonic expansion, weighted space on the unit sphere

The Dunkl transform

We recall some basic concepts of Dunkl operators which will be needed in the article. For more details on Dunkl's analysis, the reader may especially consult [START_REF] Dunkl | Orthogonal polynomials of several variables[END_REF][START_REF] Rösler | Dunkl operators: theory and applications[END_REF] and the references therein. We can all the same point out that the theory was originally developed by Dunkl to bypass the classical approach to the construction of orthogonal polynomials as spherical functions on homogeneous spaces. The introduction of the differential-difference operators (and their related objects) now called Dunkl operators turned out to be a powerful tool in harmonic analysis associated with Coxeter groups, in the theory of multivariable special functions, in the theory of stochastic processes with values in Weyl chambers or in the theory of integrable quantum many body systems of Calogero-Moser-Sutherland type, for instance.

Let d ∈ N \ {0}. Let W ⊂ O(R d
) be a finite reflection group associated with a reduced root system R (not necessarily crystallographic) and let κ : R → [0, +∞[ be a multiplicity function, that is, a W -invariant function. The (rational) Dunkl operators D κ ξ on R d , introduced in [START_REF] Dunkl | Differential-difference operators associated to reflection groups[END_REF], are the following κ-deformations of directional derivatives ∂ ξ by reflections

D κ ξ f (x) = ∂ ξ f (x) + α∈R+ κ(α) f (x) -f (σ α (x)) x, α ξ, α , x ∈ R d ,
where •, • denotes the standard Euclidean inner product, σ α denotes the reflection with respect to the hyperplane orthogonal to α and R + denotes a positive subsystem of R. The definition is of course independent of the choice of the positive subsystem since κ is W -invariant. These operators map P d n to P d n-1 , where P d n is the space of homogeneous polynomials of degree n in d variables, and they mutually commute. The Dunkl Laplacian is

∆ κ f = d i=1 (D κ ei ) 2 f
, where (e i ) 1 i d is the canonical basis of R d , and can be written explicitly as follows (see [START_REF] Dunkl | Differential-difference operators associated to reflection groups[END_REF])

∆ κ f (x) = ∆f (x) + 2 α∈R+ κ(α) ∂ α f (x) α, x - α 2 2 f (x) -f (σ α (x)) α, x 2 .
It generates a semigroup H κ t on L p (R d , h 2 κ ), 1 p < ∞, which is a diffusion semigroup in the sense of Definition 2.1 [30, Theorem 2.6] (see also [START_REF] Rösler | Generalized Hermite polynomials and the heat equation for Dunkl operators[END_REF][START_REF] Rösler | One-parameter semigroups related to abstract quantum models of Calogero type[END_REF]), where the weight h 2 κ defined on

R d by h 2 κ (x) = α∈R+ | x, α | 2κ(α)
is invariant under the action of W and homogeneous of degree 2γ κ , with

γ κ = α∈R+ κ(α).
The Dunkl operators give rise to a rich analytic structure since they are also intertwined with the usual derivatives. Indeed, there exists a unique linear isomorphism V κ (called intertwining operator) on P = n 0 P d n such that

V κ (P d n ) = P d n , V κ| P d 0 = Id | P d 0 , D κ ξ V κ = V κ ∂ ξ ∀ξ ∈ R d .
Unfortunately, the intertwining operator is explicitly known only in some special cases but for a general reflection group, we all the same have the following significant Laplace-type representation due to Rösler (see [START_REF] Rösler | Positivity of Dunkl's intertwining operator[END_REF]): for every x ∈ R d , there exists a unique probability measure dµ κ x , compactly supported in the convex hull of the orbit of x under the action of W (among other properties) such that for any P ∈ P

V κ P (x) = R d P (ξ)dµ κ x (ξ),
and this formula allows to extend it to various larger function spaces. For y ∈ C d , let

E κ (x, y) = V κ e •,y (x), x ∈ R d ,
where •, • denotes the bilinear extension of the Euclidean inner product to

C d × C d . Then E κ (•, y)
is the unique real-analytic solution of the spectral problem

D κ ξ f = ξ, y f ∀ξ ∈ R d , f (0) = 1,
F κ f (x) = c κ R d E κ (-ix, y)f (y)h 2 κ (y)dy, x ∈ R d , where c -1 κ = R d e -x 2 /2 h 2 κ (x)
dx is a Mehta-type constant. We point out that the Dunkl transform coincides with the Euclidean Fourier transform when κ = 0 (since D 0 ξ = ∂ ξ and V 0 = Id) and that it is more or less a Hankel transform when d = 1 (and then W Z 2 ).

The Dunkl transform has the following properties, where for a given Banach lattice Y = Y (Ω ), we denote by

L p (R d , h 2 κ ; Y ) the Bochner space of classes of functions f : R d → Y such that f κ,p;Y = R d f (y) p Y h 2 κ (y)dy 1 p < ∞,
with the standard modification if p = ∞. If Y = C, we usually omit Y in the notations.

Lemma 2.11 1. If f ∈ L 1 (R d , h 2 κ ) then F κ f ∈ C 0 (R d ).
2. F κ is an isomorphism of the Schwartz class S(R d ) onto itself, and

F 2 κ f (x) = f (-x).

The Dunkl transform has a unique extension to an isometric isomorphism of

L 2 (R d , h 2 κ ). 4. Let f ∈ L 1 (R d , h 2 κ ). If F κ f is in L 1 (R d , h 2 κ
), then we have the inversion formula

f (x) = c κ R d E κ (ix, y)F κ f (y)h 2 κ (y)dy. 5. For f ∈ S(R d ), we have ∆ κ (f ) = F -1 κ [-ξ 2 F κ f (ξ)],
and the semigroup H κ t generated by the Dunkl Laplacian satisfies

H κ t (f ) = F -1 κ [e -t ξ 2 F κ f (ξ)]. 6. For m : (0, ∞) → C a bounded measurable function and Y a Banach space, T m (f ) = F -1 κ [m( ξ )F κ f (ξ)] is a well defined element of C 0 (R d ; Y ) for f ∈ S(R d ) ⊗ Y.
7. Let Y be a UMD lattice and 1 < p < ∞. Then -∆ κ is an ω-sectorial operator for some ω < π 2 on L p (R d , h 2 κ ; Y ), in particular injective with dense range.

8. Let Y be a UMD lattice and

1 < p < ∞. Let q ∈ [1, ∞) and α > 1 q . Let A = -∆ κ be the negative generator of the Dunkl heat semigroup H κ t on L p (R d , h 2 κ ; Y ). Let ω < θ ∈ (0, π) such that A is ω-sectorial on L p (R d , h 2 κ ; Y ).
(a) Suppose that for any m ∈ H ∞ 0 (Σ θ ), the above operator T m , initially defined on

S(R d ) ⊗ Y, extends to a bounded operator on L p (R d , h 2 κ ; Y ) and T m C m H α q . Then A has a Hörmander H α q calculus and m(A) = T m for m ∈ H α q and m(t) = m(t 2 ). (b) Suppose that there is a C < ∞ such that for any m ∈ H ∞ 0 (Σ θ ), m(A) L p (R d ,h 2 κ ;Y )→L p (R d ,h 2 κ ;Y ) C m H α q .
Then A has a H α q calculus, for any m ∈ H α q , T m defined above extends to a bounded operator on

L p (R d , h 2 κ ; Y ) and m(A) = T m.
Proof : For parts 1.,2.,3.,4.,5., we refer to [START_REF] Jeu | The Dunkl transform[END_REF], [START_REF] Thangavelu | Convolution operator and maximal function for the Dunkl transform[END_REF]. For 6., we note that

F κ f belongs again to S(R d ) ⊗ Y, so ξ → m( ξ )F κ f (ξ) belongs to L 1 (R d ) ⊗ Y. Now
(t + A) -1 f → 0 for any f ∈ L p (R d , h 2 κ ; Y ), as t → 0. Since sup t>0 t(t + A) -1 < ∞ by pseudo-sectoriality of A, it suffices to consider f ∈ S(R d ) ⊗ Y. For these f, we have t(t + A) -1 f = F -1 κ t t + ξ 2 F κ (f )(ξ) → 0 in C 0 (R d ; Y ) by dominated convergence, since | t t+ ξ 2 | 1 and F κ (f ) ∈ S(R d ) ⊗ Y according to part 2
. By [50, Proposition 15.2], we already know that t(t + A) -1 f converges in L p (R d , h 2 κ ; Y ), so by unicity of the limit, it converges to 0 in L p (R d , h 2 κ ; Y ). We turn to 8. It follows from 5. and the representation formula

(λ -A) -1 = - ∞ 0 e λt e -tA dt for λ < 0 that T m = m(A) for m(t) = (λ -t) -1
. This identity can be extended for λ ∈ C\Σ ω , where ω < θ and σ(A) ⊂ Σ ω , by analytic continuation. Then the identity follows for any m ∈ H ∞ 0 (Σ θ ) from the Cauchy formula defining the H ∞ 0 calculus. We now show step by step that T m = m(A) holds for m ∈ H ∞ (Σ θ ), for m ∈ W α q and for m ∈ H α q , under either the assumptions 8. (a) or 8. (b). Each time, it will suffice by linearity and density to show the identity applied to f ⊗ y with f ∈ S(R d ) and

y ∈ Y. So let m ∈ H ∞ (Σ θ ). Let ρ n (λ) = λ (1 + λ) 2 1 n ∈ H ∞ 0 (Σ θ ).
We have ρ n (λ) → 1 for any λ ∈ Σ θ and sup n ρ n ∞,θ = sup n ρ 1

1 n ∞,θ < ∞.
The assumption 8. (b) readily implies that A has an H ∞ (Σ θ ) calculus by Lemma 2.8 2., whereas 8. (a) also implies it via the already provided identity T m = m(A) for m ∈ H ∞ 0 (Σ θ ). Thus, by the Convergence Lemma 2.4, we have with

m n := mρ n , m(A)(f ⊗ y) = lim n m n (A)(f ⊗ y) = lim n F -1 κ [m n ( ξ 2 )F κ f (ξ)] ⊗ y = F -1 κ [m( ξ 2 )F κ f (ξ)] ⊗ y = T m(f ⊗ y),
the first limit in L p (R d , h 2 κ ; Y ), the second limit in C 0 (R d ; Y ), by 1. and dominated convergence. It follows T m = m(A) for m ∈ H ∞ (Σ θ ), and that A has a H α q calculus, since

m(A) lim sup n m n (A) lim sup n m n H α q m H α q lim sup n ρ n H α q m H α q lim sup n ρ n θ,∞ m H α q .
Now let m ∈ W α q , and m n a sequence in H ∞ (Σ θ ) ∩ W α q approximating m in W α q . We have

m(A)(f ⊗ y) = lim n m n (A)(f ⊗ y) = lim n T mn (f ⊗ y) = lim n F -1 κ [m n ( ξ 2 )F κ f (ξ)] ⊗ y = F -1 κ [m( ξ 2 )F κ f ] ⊗ y,
where the last limit holds by W α q → L ∞ (0, ∞) and dominated convergence, plus part 1. Thus, m(A) = T m for m ∈ W α q . Let finally m ∈ H α q . Let (φ k ) k∈Z be a dyadic partition of unity as in Definition 2.10. Then

m(A)(f ⊗ y) = lim n n k=-n (φ k m)(A)(f ⊗ y) = lim n n k=-n T φ k m (f ⊗ y) = T m(f ⊗ y),
the second limit holding by almost the same argument as before in the case W α q . We now turn to h-harmonic expansions and analysis on the sphere.

h-harmonic expansions and analysis on the sphere.

For more details on h-harmonic expansions and analysis on the sphere, the reader may consult the expertly written book of Dai-Xu [START_REF] Dai | Analysis on h-harmonics and Dunkl transforms[END_REF]. For d 2, we let S d-1 = {x ∈ R d : x = 1}, and for 1 p < ∞ and Y a Banach lattice, we let L p (S d-1 , h 2 κ ; Y ) be the Bochner space of equivalence classes of measurable functions f :

S d-1 → Y such that f L p (S d-1 ,h 2 κ ;Y ) := a κ S d-1 f (y) p Y h 2 κ (y)dy 1 p < ∞,
with the standard modification if p = ∞. Here, the measure on S d-1 is surface measure and a -1 κ = S d-1 h 2 κ (y)dy. By abuse of notation, we also write

f L p (S d-1 ,h 2 κ ;Y ) = f κ,p;Y . Let P ∈ P d n .
Then P is called an h-harmonic polynomial of degree n if ∆ κ P = 0. It is wellknown (see [START_REF] Dunkl | Reflection groups and orthogonal polynomials on the sphere[END_REF]) that a homogeneous polynomial is an h-harmonic polynomial if it is orthogonal to all polynomials of lower degree with respect to the inner product of L 2 (S d-1 , h 2 κ ). For j 0, we let proj κ j :

L 2 (S d-1 , h 2 κ ) → L 2 (S d-1 , h 2 κ
) be the orthogonal projection with image the space of all h-harmonics of degree j. The projection proj κ j has the following integral representation

proj κ j f (x) = a κ S d-1 f (y)P κ j (x, y)h 2 κ (y)dy, x ∈ S d-1 ,
with (see [71, Theorem 3.2, (3.1)])

P κ j (x, y) = j + λ κ λ κ V κ C λκ j x, • (y), x, y ∈ S d-1 ,
where (2.2)

λ κ = d 2 + γ κ -1
and where C λκ j is the standard Gegenbauer (or ultraspherical) polynomial of degree j and index λ κ (see [64] for instance). Now, let

c -1 λκ = 1 -1 (1 -t 2 ) λκ-1 2 dt = √ π Γ(λ κ + 1 2 ) Γ(λ κ + 1)
.

Then, according to [START_REF] Xu | Weighted approximation of functions on the unit sphere[END_REF], we define for

f ∈ L 1 (S d-1 , h 2 κ ) and g ∈ L 1 ([-1, 1]
, ω λκ ), with ω λκ the weight for which the Gegenbauer polynomials are orthogonal, that is . We now state the following lemma, which will be useful in the sequel.

ω λκ (t) = (1 -t 2 ) λκ-1 2 , f * g(x) = a κ S d-1 f (y)V κ g x, • (y)
Lemma 2.12 For j 0, 1 < p < ∞ and any Banach space Y, the operator proj κ j extends boundedly to L p (S d-1 , h 2 κ ; Y ), and we have, for j large enough, the norm estimate

proj κ j L p (S d-1 ,h 2 κ ;Y )→L p (S d-1 ,h 2 κ ;Y ) j 2λκ . Proof : Let x ∈ S d-1 . Write for any j 0 proj κ j f (x) = a κ S d-1 f (y)P κ j (x, y)h 2 κ (y)dy = f * j + λ κ λ κ C λκ j (x).
Thus, by Young-type inequalities for the generalized convolution on the sphere, we have both the inequalities

proj κ j f L 1 (S d-1 ,h 2 κ ) f L 1 (S d-1 ,h 2 κ ) j + λ κ λ κ C λκ j L 1 ([-1,1],ω λκ ) proj κ j f L ∞ (S d-1 ,h 2 κ ) f L ∞ (S d-1 ,h 2 κ ) j + λ κ λ κ C λκ j L 1 ([-1,1],ω λκ )
.

Besides,

j + λ κ λ κ C λκ j L 1 ([-1,1],ω λκ ) = c λκ j + λ κ λ κ 1 -1 |C λκ j (t)|(1 -t 2 ) λκ-1 2 dt j + λ κ λ κ (2λ κ ) j j! ,
where we have used the inequality (see for instance [4, p. 350])

|C λκ j (t)| C λκ j (1) = (2λ κ ) j j! ,
with (x) n the so-called Pochhammer symbol. Moreover, since we can write (x

) n = Γ(x+n) Γ(x) , Stirling's formula Γ(a) √ 2πa a-1 2 e -a gives us (2.3) (2λ κ ) j j! = Γ(2λ κ + j) Γ(2λ κ )Γ(j + 1) C(λ κ )j 2λκ-1 .
We can conclude, for j large enough, that

j + λ κ λ κ C λκ j L 1 ([-1,1],ω λκ ) j 2λκ ,
and therefore, we have both the inequalities

proj κ j L 1 (S d-1 ,h 2 κ )→L 1 (S d-1 ,h 2 κ ) j 2λκ proj κ j L ∞ (S d-1 ,h 2 κ )→L ∞ (S d-1 ,h 2 κ ) j 2λκ .
We can tensorise these estimates to get estimates on Bochner spaces, namely according to Lemma 2.5 proj

κ j L p (S d-1 ,h 2 κ ;Y )→L p (S d-1 ,h 2 κ ;Y ) j 2λκ holds for p = 1, ∞, (L p (S d-1 , h 2 κ ; Y ) replaced by L ∞ (S d-1 , h 2 κ ) ⊗ Y L ∞ (S d-1 ,h 2 κ ;Y )
for p = ∞) which then implies by interpolation [6, 5.1.2 Theorem part 2] that the latter estimates holds also for p ∈ (1, ∞).

We close this section with some facts on both a generalized Poisson and heat semigroup on (S d-1 , h 2 κ (y)dy). We first recall their definition (see [25, p. 482]).

Definition 2.13

The Poisson semigroup T κ t on (S d-1 , h 2 κ (y)dy) is defined by

T κ t f = ∞ j=0 e -jt proj κ j f, t > 0.
The heat semigroup H κ t on (S d-1 , h 2 κ (y)dy) is defined by

H κ t f = ∞ j=0 e -j(j+2λκ)t proj κ j f, t > 0.
The following statement will be of particular interest.

Lemma 2.14 Both the Poisson and the heat semigroup are diffusion semigroups.

Proof : That they are semigroups is clear from the fact that the proj κ j are projections. The contractivity of H κ t on L p (S d-1 , h 2 κ ) for all 1 p ∞ and t 0 is proved in [25, Proof of Lemma 2.2]. This yields the strong continuity of both semigroups on L 2 . Indeed, strong continuity T κ t f → f and H κ t f → f (t → 0) is clear for elements of the form f = J j=0 proj κ j g (J finite), since e -tj → 1 and e -tj(j+2λκ) → 1 uniformly for j = 0, . . . , J as t → 0. Now use a 3 argument for general f ∈ L 2 . Then the strong continuity extrapolates on L p by contractivity. It is proved in [25, Proof of Lemma 2.2] that H κ t are positive operators, and [25, (2.10)] yields then that T κ t is also a positive operator. We finally show that T κ t (1) → 1 and H κ t (1) → 1 as t → ∞. This will imply that T κ t (1) = 1, and the same for H κ t . Indeed, T κ t+s (1) = T κ t T κ s (1), so 1 = lim s→∞ T κ t+s (1) = T κ t lim s→∞ T κ s (1) = T κ t [START_REF] Adams | Sobolev spaces[END_REF]. We have

T κ t (1) = proj κ 0 (1) + ∞ j=1
e -tj proj κ j [START_REF] Adams | Sobolev spaces[END_REF]. Now proj κ 0 (1) = 1 and by Lemma 2.12

∞ j=1 e -tj proj κ j (1) ∞ e -t ∞ j=1 e -t(j-1) proj κ j (1) ∞ e -t → 0 (t → ∞).
The same argument applies for H κ t .

Spectral multipliers with values in UMD lattices

Marcinkiewicz-type multiplier theorem for h-harmonic expansions

In this section, we take d ∈ N with d 2. Let us begin by recalling the definition of the usual difference operator.

Definition 3.1 Given a sequence (µ j ) j 0 of complex numbers, we define recursively

∆µ j = µ j -µ j+1 , ∆ n+1 µ j = ∆ n µ j -∆ n µ j+1 , j 0, n 1.
We now state the main result of this section.

Theorem 3.2 Let 1 < p < ∞ and Y = Y (Ω ) be a UMD Banach lattice. Let (µ j ) j 0 be a scalar sequence. Suppose that for some integer

n 0 > d 2 + γ κ = λ κ + 1, we have (C 0 ) sup j 0 |µ j | M < ∞, (C n0 ) sup j 0 2 j(n0-1) 2 j+1 l=2 j |∆ n0 µ l | M < ∞. Then (µ j ) j 0 defines an L p (S d-1 , h 2 κ ; Y ) multiplier, that is ∞ j=0 µ j proj κ j f κ,p;Y c p M f κ,p;Y ,
where the constant c p is independent of f and (µ j ) j 0 .

This theorem generalizes the scalar case proved by Dai and Xu in [START_REF] Dai | Maximal function and multiplier theorem for weighted space on the unit sphere[END_REF]. Therefore, if we specialize Theorem 3.2 to Y = C and κ = 0, then we recover the famous Marcinkiewicz type theorem for zonal multipliers due to .

The proof, which is divided into several lemmas and a proposition, follows the strategy of Bonami-Clerc adapted in the Dunkl setting by Dai-Xu. A crucial role in the proof will be played by several kinds of Littlewood-Paley type g-functions closely related to Cesàro means for h-harmonic expansions. Let us begin with the following notation. Let (T κ t ) t 0 be the generalized Poisson semigroup on L p (S d-1 , h 2 κ ; Y ). Then we set (3.1)

P κ r = T κ -log(r) , 0 < r < 1.
The first lemma will provide a new equivalent norm on L p (S d-1 , h 2 κ ; Y ), in terms of a well suited g-function.

Lemma 3.3 Let 1 < p < ∞ and Y = Y (Ω ) be a UMD Banach lattice. Then, for any f ∈ L p (S d-1 , h 2 κ ; Y ), we have the two-sided estimate 1 c f κ,p;Y 1 0 (1 -r) ∂ ∂r P κ r f 2 dr 1 2 κ,p;Y c f κ,p;Y ,
where in the first inequality we assume that S d-1 f (y)h 2 κ (y)dy = 0. Proof : The semigroup (T κ t ) t 0 is a diffusion semigroup, so its vector-valued extension on L p (S d-1 , h 2 κ ; Y ) has an H ∞ (Σ ω ) calculus of some angle ω < π 2 , thanks to Proposition 2.6. According to [70, Proposition 9], we have, for any f ∈ L p (S d-1 , h 2 κ ; Y ), the following square function estimate

∞ 0 t ∂ ∂t T κ t f 2 dt 1 2 κ,p;Y C f κ,p;Y .
We next show that we also have the converse estimate to the previous one, under the additional assumption that S d-1 f (y)h 2 κ (y)dy = 0. Note that lim t→∞

T κ t f = lim t→∞ ∞ j=0 e -jt proj κ j f = proj κ 0 f
is the projection onto the kernel of the negative generator A p of (T κ t ) t 0 (its version on L p (S d-1 , h 2 κ ; Y )), according to the decomposition L p (S

f = ∞ 0 ψ(t Ãp )f dt t = ∞ 0 ψ(tA p )f dt t , for ψ ∈ H ∞ 0 (Σ π 2 -
) of dt t -integral 1, as an improper integral, under the additional assumption that f ∈ R(A p ) ∩ D(A p ). Here, D(A p ) stands for the domain of A p . Apply this partition to

ψ(t) = cte -t • te -t , we get f, g = c ∞ 0 tA p T κ t f, tA p * T κ t * g dt t , for any g ∈ L p * (S d-1 , h 2 κ ; Y * ). This implies f, g ∞ 0 tA p T κ t f, tA p * T κ t g dt t = ∞ 0 S d-1 Ω tA p T κ t (f )(y, ω ) • tA p * T κ t (g)(y, ω )dω h 2 κ (y)dy dt t S d-1 Ω ∞ 0 tA p T κ t (f )(y, ω ) • tA p * T κ t (g)(y, ω ) dt t dω h 2 κ (y)dy S d-1 Ω ∞ 0 tA p T κ t (f )(y, ω ) 2 dt t 1 2 • ∞ 0 tA p * T κ t (g)(y, ω ) 2 dt t 1 2 dω h 2 κ (y)dy ∞ 0 t ∂ ∂t T κ t f 2 dt 1 2 κ,p;Y ∞ 0 t ∂ ∂t T κ t g 2 dt 1 2 κ,p * ;Y * .
Applying now the upper estimate for g and on

L p * (S d-1 , h 2 κ ; Y * ), we get f, g ∞ 0 t ∂ ∂t T κ t f 2 dt 1 2 κ,p;Y g κ,p * ;Y * .
Taking the supremum over all g of norm 1 yields the desired estimate under the additional

assumption f ∈ R(A p ) ∩ D(A p ). Since R(A p ) ∩ D(A p ) is dense in R(A p ) [50, Proposition 9.4], we deduce (3.2) f κ,p;Y ∞ 0 t ∂ ∂t T κ t f 2 dt 1 2 κ,p;Y f κ,p;Y ,
the lower estimate under the assumption S d-1 f (y)h 2 κ (y)dy = 0. We next deduce from (3.2) the stated g-function norm equivalence of the lemma. To this end, we proceed by a modification of the proof in [26, p. 38-39]. By the change of variable r = e -t , we get

g 0 (f ) := ∞ 0 t ∂ ∂t P κ e -t f 2 dt 1 2 = 1 0 r| log(r)| ∂ ∂r P κ r f 2 dr 1 2
. Now, we write

g(f ) = 1 0 (1 -r) ∂ ∂r P κ r f 2 dr 1 2
.

We shall show that g(f ) κ,p;Y f κ,p;Y and g 0 (f ) κ,p;Y g(f ) κ,p;Y , which completes the proof.

Since (1 -r) r| log r| for 1 2 r < 1, we have

(3.3) g(f ) 1 2 0 (1 -r) ∂ ∂r P κ r f 2 dr 1 2 + 1 1 2 r| log r| ∂ ∂r P κ r f 2 1 2
.

The first term on the right hand side, we estimate by

sup 0 r 1 2 ∂ ∂r P κ r f sup 0 r 1 2 ∞ j=1 jr j-1 |proj κ j f | = ∞ j=1 j2 1-j |proj κ j f |.
By Lemma 2.12, we can sum over j 1 to get

sup 0 r 1 2 ∂ ∂r P κ r f κ,p;Y f κ,p;Y .
Use now (3.3) to deduce that

g(f ) κ,p;Y f κ,p;Y + g 0 (f ) κ,p;Y f κ,p;Y .
We have proved the upper estimate of the lemma. For the lower estimate, we simply use r| log r| (1 -r) on r ∈ [0, 1], to deduce g 0 (f ) g(f ), and thus,

f κ,p;Y g 0 (f ) κ,p;Y g(f ) κ,p;Y
when S d-1 f (y)h 2 κ (y)dy = 0. Now, we shall prove that the Cesàro means of h-harmonic expansions are R-bounded on L p (S d-1 , h 2 κ ; Y ). To this end, we recall some definitions.

Definition 3.4 For δ 0 and l ∈ N, we put

A δ l = l + δ l = (l + δ)(l + δ -1) . . . (δ + 1) l(l -1) . . . 1 = Γ(l + δ + 1) Γ(l + 1)Γ(δ + 1) = (δ + 1) l l! .
Then, for j 0 and n 0, we put

a δ,n j = 1 A δ n A δ n-j χ 0 j n .
We define the Cesàro means of order δ (from now on, just called Cesàro means) by the multiplier

S δ n f = ∞ j=0 a δ,n j proj κ j f.
We can state the R-boundedness of the Cesàro means of h-harmonic expansions. Recall that we have set

λ κ = d 2 + γ κ -1. Lemma 3.5 Let 1 < p < ∞, Y = Y (Ω ) be a UMD Banach lattice. Assume that δ > λ κ . Then the Cesàro means (S δ n ) n 0 are R-bounded on L p (S d-1 , h 2 κ ; Y ), that is, ∞ j=0 |S δ nj f j | 2 1 2 κ,p;Y c p,δ ∞ j=0 |f j | 2 1 2
κ,p;Y .

Proof : By [26, p. 37], we have the estimate

(3.4) sup n 0 |S δ n f (x, ω )| c M κ f (x, ω ) + M κ f (-x, ω ) , x ∈ S d-1 , ω ∈ Ω ,
where M κ is the following well-suited maximal operator for h-harmonic expansions [73, Proposition 2.3] or [25, (1.5)]

M κ f (x) = sup 0<θ π S d-1 |f (y)|V κ [χ B(x,θ) ](y)h 2 κ (y)dy S d-1 V κ [χ B(x,θ) ](y)h 2 κ (y)dy , x ∈ S d-1 , with B(x, θ) = {y ∈ R d : x, y cos θ} ∩ {y ∈ R d : y 1}. It is shown in [25, Proof of Theorem 2.1] that M κ f (x, ω ) c sup t>0 1 t t 0 H κ s |f (•, ω )|(x)ds,
where we recall that H κ s is the generalized heat semigroup on (S d-1 , h 2 κ (y)dy). Since it is a symmetric contraction semigroup, by [START_REF] Xu | H ∞ functional calculus and maximal inequalities for semigroups of contractions on vector-valued L p -spaces[END_REF]Theorem 1], the Hopf-Dunford-Schwartz maximal operator

M (H)f = sup t>0 1 t t 0 H κ s f ds is bounded on L p (S d-1 , h 2 κ ; Y (Ω ; 2 )), Y (Ω ;
2 ) being again a UMD-lattice according to Lemma 2.3. Therefore, we get We give some details. Firstly, according to [26, p. 41], we have for j 1, δ 0 and 0 < r < 1

∞ j=0 |S δ nj f j | 2 1 2 κ,p;Y ∞ j=0 |M κ f j | 2 1 2 κ,p;Y M (H)(f j ) j 0 L p (S d-1 ,h 2 κ ;Y (Ω ; 2 )) (f j ) j 0 L p (S d-1 ,h 2 κ ;Y (Ω ; 2 )) = ∞ j=0 |f j | 2
S δ nj P κ rj f j 2 c nj l=0 |b δ l,nj | |S δ l f j | 2 ,
where b δ l,n are scalars satisfying

n l=0 |b δ l,n | c δ . It follows from the R-boundedness of the 20 Cesàro means ∞ j=1 S δ nj P κ rj f j 2 1 2 κ,p;Y c ∞ j=1 nj l=0 |b δ l,nj | |S δ l f j | 2 1 2 κ,p;Y = c ∞ j=1 ∞ l=0 S δ l χ 0 l nj |b δ l,nj |f j 2 1 2 κ,p;Y ∞ j=1 ∞ l=0 χ 0 l nj |b δ l,nj |f j 2 1 2 κ,p;Y = ∞ j=1 nj l=0 |b δ l,nj | |f j | 2 1 2 κ,p;Y √ c δ ∞ j=1 |f j | 2 1 2
κ,p;Y .

Thus, we have shown

(3.5) ∞ j=1 S δ nj P κ rj f j 2 1 2 κ,p;Y ∞ j=1 |f j | 2 1 2
κ,p;Y . Now let for each j 1 and n 1, (r j,i ) 2 n i=0 ⊂ I j be a finite sequence such that r j,i -

r j,i-1 = 2 -n |I j | for all 1 i 2 n . Then for each n ∈ N, R j,n := 2 -n 2 n i=1 |P κ rj,i f j | 2 is a Riemann sum over Ω of the integral 1 |Ij | Ij |P κ r f j | 2 dr.
Thus, by dominated convergence, it follows

(3.6) ∞ j=1 1 |I j | Ij P κ r f j 2 dr 1 2 κ,p;Y = lim n→∞ 2 -n ∞ j=1 2 n i=1 |P κ rj,i f j | 2 1 2 κ,p;Y .
On the other hand, since for each n 1, r j < r j,i for all 1 i n and j 1, we have by (3.5)

∞ j=1 S δ nj P κ rj f j 2 1 2 κ,p;Y = 2 -n 2 n i=1 ∞ j=1 S δ nj P κ rj /rj,i (P κ rj,i f j ) 2 1 2 κ,p;Y 2 -n 2 n i=1 ∞ j=1 |P κ rj,i f j | 2 1 2 κ,p;Y .
We conclude with (3.6). Before stating a proposition which is a key step in the proof of Theorem 3.2, we need the following functional which is closely related to the Cesàro means of h-harmonic expansions. Definition 3.7 Let δ 0. We define the functional g δ (f ), for given

f ∈ L p (S d-1 , h 2 κ ; Y ), by g δ (f ) = ∞ n=1 S δ+1 n f -S δ n f 2 1 n 1 2
.

Moreover, let (ν k ) k 1 be a sequence of nonnegative numbers such that

sup n 1 1 n n k=1 ν k = M < ∞.
We define the functional g * δ (f ), for given

f ∈ L p (S d-1 , h 2 κ ; Y ), by g * δ (f ) = ∞ n=1 S δ+1 n f -S δ n f 2 ν n n 1 2 . Remark 3.8 Note that if in particular ν k = 1 for all k 1, then g * δ (f ) = g δ (f ).
The following proposition gives us two important norm inequalities involving the Littlewood-Paley functions g δ (f ) and g * δ (f ).

Proposition 3.9 Let 1 < p < ∞ and Y = Y (Ω ) be a UMD Banach lattice. Let δ 0. If f ∈ L p (S d-1 , h 2 κ ; Y ) satisfies S d-1 f (y)h 2 κ (y)dy = 0, then f κ,p;Y c p,δ g δ (f ) κ,p;Y . Conversely, if the Cesàro means (S δ n ) n 0 are R-bounded on L p (S d-1 , h 2 κ ; Y ), then g * δ (f ) κ,p;Y c p,δ M f κ,p;Y ,
where

M = sup n 1 1 n n k=1 ν k .
Proof : First, recall that we have set in the proof of Lemma 3.3

g(f ) = 1 0 (1 -r) ∂ ∂r P κ r f 2 dr 1 2
.

It is shown in [26, Section 4.3.2] that g(f )(x) c δ g δ (f )(x). Therefore, it follows immediately with Lemma 3.3 f κ,p;Y c p g(f ) κ,p;Y c p c δ g δ (f ) κ,p;Y ,
in the case S d-1 f (y)h 2 κ (y)dy = 0. We proceed to the second stated inequality and suppose that the Cesàro means to an order index δ are R-bounded. We follow closely the lines of [26, Section 4.3.2] but, for the readers' convenience, we present the proof all the same. First we may assume that n n j=1 ν j 2n, since the desired conclusion for general (ν j ) j 1 can be deduced from this case applied to the two sequences νj = 1 and νj = M -1 ν j + 1. Now let µ 1 = 1 and µ n = 1 + n-1 i=1 ν i for n 2. Let further r n = 1 -1 µn and f n = P κ rn f. It is shown in [26, p. 44] that

S δ+1 n f -S δ n f 2 c S δ+1 n f n -S δ n f n 2 + cn -3 n-1 j=1 j 2 S δ+1 j f n -S δ j f n 2 .
Therefore, in view of Lemma 3.3, we are left with the task of establishing the following inequalities

(3.7) ∞ n=1 n -1 S δ+1 n f n -S δ n f n 2 ν n 1 2 κ,p;Y c g(f ) κ,p;Y and (3.8) ∞ n=1 ν n n 4 n-1 j=1 j 2 S δ+1 j f n -S δ j f n 2 1 2 κ,p;Y c g(f ) κ,p;Y .
We start by showing (3.7). To this end, let η ∈ C ∞ (R) with η(t) = 1 for |t| 1 and η(t) = 0 for |t| 2. Moreover, for n 1, let L κ n and Lκ n be the following multipliers

L κ n f = ∞ j=0 η j n proj κ j f, Lκ n f = - ∞ j=0 jη j n proj κ j f.
Comparing symbols of multipliers yields [26, p. 44] for 1 j n N (3.9) S δ+1 j f n -S δ j f n = (j + δ + 1) -1 P κ rn S δ j ( Lκ N f ) . Using this last equality specializing to j = n, we obtain by Lemma 3.6

N n=1 n -1 S δ+1 n f n -S δ n f n 2 ν n 1 2 κ,p;Y N n=1 ν n n 3 P κ rn S δ n Lκ N (f ) 2 1 2 κ,p;Y N n=1 ν n n 3 1 r n+1 -r n rn+1 rn P κ r Lκ N (f ) 2 dr 1 2 κ,p;Y . Obviously, we have |P κ r ( Lκ N (f ))| = r|L κ N ( ∂ ∂r P κ r (f ))|. Moreover, the operators L κ N are uniformly bounded in N ∈ N on L p (S d-1 , h 2 κ ; Y )
. Indeed, a straightforward computation gives us

L κ N f = 2N j=0 ∆ l+1 η j N A l j S l j f. Since we have |∆ l+1 η j N | N -l-1 , then L κ N f 1 N l+1 2N j=0 |A l j ||S l j f | sup j 0 |S l j f | N l+1 2N j=0 A l j . But 2N j=0 A l j = A l+1 2N , then we claim (see (2.3)) that N -l-1 A l+1 2N = N -l-1 (l + 2) 2N (2N )! c,
and in view of (3.4), choosing l > λ κ , the operators L κ N are uniformly bounded in N ∈ N on L p (S d-1 , h 2 κ ; Y ). Now, since they are linear, by Lemma 2.3, a single operator

L κ N is also bounded on L p (S d-1 , h 2 κ ; Y (Ω ; L 2 ([0, 1]; dr))). Therefore, N n=1 ν n n 3 1 r n+1 -r n rn+1 rn P κ r Lκ N (f ) 2 dr 1 2 κ,p;Y N n=1 ν n n 3 1 r n+1 -r n rn+1 rn ∂ ∂r P κ r f 2 dr 1 2 κ,p;Y . Since r n+1 -r n = νn µnµn+1 νn n 2 and 1 -r 1 n for all r ∈ [r n , r n+1 ], it follows that N n=1 ν n n 3 1 r n+1 -r n rn+1 rn ∂ ∂r P κ r f 2 dr 1 2 κ,p;Y ∞ n=1 1 n rn+1 rn ∂ ∂r P κ r f 2 dr 1 2 κ,p;Y g(f ) κ,p;Y .
Thus, we have proved that

N n=1 n -1 S δ+1 n f n -S δ n f n 2 ν n 1 2 κ,p;Y g(f ) κ,p;Y
and letting N → ∞ yields (3.7). We now turn to the proof of (3.8), which is similar to the previous one. Indeed, using Lemma 3.6 and (3.9), we have

N n=1 ν n n 4 n-1 j=1 j 2 S δ+1 j f n -S δ j f n 2 1 2 κ,p;Y N n=1 ν n n 4 n-1 j=1 P κ rn S δ j Lκ N (f ) 2 1 2 κ,p;Y ∞ n=1 ν n n 4 n-1 j=1 1 r n+1 -r n rn+1 rn ∂ ∂r P κ r f 2 dr 1 2 κ,p;Y ∞ n=1 ν n n 3 1 r n+1 -r n rn+1 rn ∂ ∂r P κ r f 2 dr 1 2 κ,p;Y g(f ) κ,p;Y .
We obtain (3.8) by letting N → ∞. The proof is complete. In view of Remark 3.8, we immediately obtain the following corollary.

Corollary 3.10 Let 1 < p < ∞ and Y = Y (Ω ) be a UMD Banach lattice. If δ > λ κ , then 1 c p f κ,p;Y g δ (f ) κ,p;Y c p f κ,p;Y ,
where in the first inequality we assume that S d-1 f (y)h 2 κ (y)dy = 0.

We now state the last lemma we shall need for the proof of Theorem 3.2.

Lemma 3.11 Let δ to be the smallest integer strictly larger than λ κ and let n 0 = δ + 1. Let (µ j ) j 0 be a sequence as in the hypotheses of Theorem 3.2, i.e. satisfying (C 0 ) and (C n0 ), with bound M. Write

M µ f = ∞ j=0 µ j proj κ j f
the associated multiplier. Then we have

g δ (M µ f ) κ,p;Y C g * δ (f ) κ,p;Y ,
where the sequence

(ν k ) k 1 is ν k = 1 + δ+1 j=1 |∆ j µ k |k j and satisfies sup n 1 1 n n j=1 ν j cM. Proof : It is shown in [26, (4.4.2)] that g δ (M µ f ) Cg * δ (f )
holds pointwise, with the sequence (ν k ) k 1 given in the lemma. By Lemma 2.2, we immediately deduce [26, p. 47].

g δ (M µ f ) p,κ;Y C g * δ (f ) κ,p;Y . The statement on (ν k ) k 1 is shown in
We are now in a position to prove Theorem 3.2. Proof of Theorem 3.2 : We can assume that µ 0 = 0. Indeed,

∞ j=0 µ j proj κ j f κ,p;Y |µ 0 | proj κ 0 f κ,p;Y + ∞ j=1 µ j proj κ j f κ,p;Y , and proj 0 κ is bounded on L p (S d-1 , h 2 κ ; Y ) by Lemma 2.12. Let δ = n 0 -1 > λ κ . Note that a κ S d-1 M µ (f )(y)h 2 κ (y)dy = proj κ 0 (M µ (f )) = 0 if µ 0 = 0.
According to Lemma 3.5, the Cesàro means (S δ n ) n 0 are R-bounded. Hence by Proposition 3.9 in conjunction with Lemma 3.11,

M µ (f ) κ,p;Y g δ (M µ (f )) κ,p;Y g * δ (f ) κ,p;Y M f κ,p;Y .

A multiplier theorem for the Dunkl transform

Bounded vector-valued multipliers on the sphere S d yield bounded vector-valued spectral multipliers of the Dunkl Laplacian on R d by a transference principle, presented in Theorem 3.12 below. In the scalar case, a transference principle from zonal multipliers on S d to radial multipliers on R d was oberved and proved by Bonami-Clerc in [10], and their strategy has been recently adapted by Dai-Wang [24] to obtain bounded multipliers for the Dunkl transform on R d from bounded multipliers for h-harmonic expansions on the unit sphere S d . Let W ⊂ O(d) be a finite reflection group associated with a reduced root system R and let κ : R → [0, +∞[ be a multiplicity function with associated weight function h 2 κ . We transfer this to S d ⊆ R d+1 . Namely, for g ∈ W, there exists a unique orthogonal transformation on R d+1 , denoted by g and determined by

g x = (gx, x d+1 ), x = (x, x d+1 ) ∈ R d × R.
Then W = {g : g ∈ W } is a finite reflection group on R d+1 associated with the reduced root system R = {(α, 0) : α ∈ R}. Finally, we let

κ : R → R + , (α, 0) → κ(α)
and associate with it the weight h 2 κ . We can now state the following transference principle for the Dunkl transform. Theorem 3.12 Let Y be a Banach space. Let m : (0, ∞) → R be a continuous and bounded function. For > 0 and n 0, let µ n = m( n). Let further M = M µ be the multiplier

M (f ) = ∞ n=0 m( n)proj κ n f.
Assume that for some 1 < p < ∞ and any

f ∈ L p (S d , h 2 κ ; Y ), sup >0 M f L p (S d ,h 2 κ ;Y ) A f L p (S d ,h 2 κ ;Y ) .
Then m is a radial Dunkl spectral multiplier on

L p (R d , h 2 κ ; Y ), that is, for any f ∈ L p (R d , h 2 κ ; Y ), T m f L p (R d ,h 2 κ ;Y ) c d,κ A f L p (R d ,h 2 κ ;Y ) ,
where T m is a priori defined on S(R d ) ⊗ Y by

T m (f ) = F -1 κ [m( ξ )F κ (f )(ξ)].
This theorem generalizes the scalar case proved by Dai and Wang in [24]. Therefore, if we specialize Theorem 3.12 to Y = C and κ = 0, then we recover the standard result due to . Proof : We follow closely the strategy of [24,Section 3]. Note that continuity of m in 0 is not needed there. We first assume that for some c 1 , c 2 > 0

|m(t)| c 1 e -c2t , t > 0.
In [24,Lemma 3.5], it is shown that the operator T m has the following integral representation

T m f (x) = R d f (y)K(x,
R d R d K k=1 f k (y)y k , L l=1 g l (x)z l K(x, y)h 2 κ (x)h 2 κ (y)dxdy cA holds whenever f = k f k ⊗ y k ∈ S(R d ) ⊗ Y and g = l g l ⊗ z l ∈ S(R d ) ⊗ Y * have compact support and satisfy f L p (R d ,h 2 κ ;Y ) 1, g L p * (R d ,h 2 κ ;Y * ) 1.
Denote the above double integral by I and let ψ : R d → S d be the mapping

ψ(x) = ξ sin x , cos x , for x = x ξ ∈ R d with ξ ∈ S d-1 .
For N 1, let moreover

ψ N : R d → N S d = {x ∈ R d+1 : x = N } x → N ψ( x N ).
It is shown in [24,Remark 3.1] that given a function h :

B(0, N ) = {x ∈ R d : x N } → R, there exists a unique function h N supported in {x ∈ N S d : arccos(N -1 x d+1 ) 1} such that h N (ψ N x) = h(x), x ∈ B(0, N ).
Moreover, it is also shown there that (3.11)

S d h N (N x)h 2 κ (x)dx = N -2λ κ -1 B(0,N ) h(x)h 2 κ (x) sin( x /N ) x /N 2λ κ dx with λ κ = λ κ + 1 2 = d 2 + γ κ -1 2
. Let now N be so large that both f and g are supported in B(0, N ). Let h(x) := f (x, •) Y and f N : N S d → Y be the function defined by

f N (ψ N (x)) = f (x), so that we have h N (x) = f N (x, •) Y .
As mentioned in [24, p. 4064], it follows from (3.11) that

h N (N •) L p (S d ,h 2 κ ) = f N (N •) L p (S d ,h 2 κ ;Y ) N -2λ κ +1 p .
Similarly, with h(x) := g(x, •) Y * , it follows that

g N (N •) L p * (S d ,h 2 κ ;Y * ) N -2λ κ +1 p * .
Recall in the following that P κ n (x, y) is the kernel of proj κ n . We write f k N (ψ N (y)) = f k (y), g l N (ψ N (x)) = g l (x) and deduce that . We now prove the theorem removing this assumption on m. Let thus m be a general multiplier function satisfying the hypotheses of Theorem 3.12. For δ > 0, put m δ (t) = m(t)e -δt . Then, for any f ∈ S(R d ) ⊗ Y, we have

N 2λ κ +1 S d S d ∞ n=0 m(N -1 n)P κ n (x, y) k f k N (N y)y k , l g l N (N x)z l h 2 κ (x)h 2 κ (y)dxdydµ(ω ) = N 2λ κ +1 S d S d ∞ n=0 m(N -1 n)P κ n (x, y) k f k N (N y)y k h 2 κ (y)dy, l g l N (N x)z l h 2 κ (x)dx N 2λ κ +1 S d S d ∞ n=0 m(N -1 n)P κ n (x, y) f N (N y, •)h 2 κ (y)dy Y g N (N x, •) Y * h 2 κ (x)dx = N 2λ κ +1 S d M 1/N f N (N •, •)(x) Y g N (N x, •) Y * h 2 κ (x)dx N 2λ κ +1 M 1/N f N (N •, •)(•) κ ,p;Y g N (N •, •) κ ,p * ;Y * N 2λ κ +1 AN -2λ κ +1 p N -2λ κ +1 p * = A.
T m δ (f ) = P κ δ T m (f ),
where (P κ t ) t 0 is the Dunkl Poisson symmetric contraction semigroup on L p (R d , h 2 κ ; Y ), and we have P κ δ f → f as δ → 0, because according to Proposition 2.6, (P κ t ) t is strongly continuous on L p (R d , h 2 κ ; Y ). On the one hand, we have by the first part of the proof

T m δ L p (R d ,h 2 κ ;Y )→L p (R d ,h 2 κ ;Y ) sup >0 M (m( n)e -δn ) n 0 L p (S d ,h 2 κ ;Y )→L p (S d ,h 2 κ ;Y )
On the other hand, we write

T m (f ) L p (R d ,h 2 κ ;Y ) = lim δ→0 P κ δ T m (f ) L p (R d ,h 2 κ ;Y ) = lim δ→0 T m δ (f ) L p (R d ,h 2 κ ;Y ) A f L p (R d ,h 2 κ ;Y ) .
The proof of the theorem is complete.

We shall now prove several important consequences of this theorem. The first one provides us a Hörmander type multiplier theorem for the Dunkl transform. Theorem 3.13 Let 1 < p < ∞ and Y = Y (Ω ) be a UMD Banach lattice. Let n 0 be an integer

> λ κ + 1 = λ κ + 3 2 = d 2 + γ κ + 1 2 .
Assume that the multiplier function m : (0, ∞) → R belongs to H n0 1 , that means, is bounded with m ∞ A and satisfies the following Hörmander condition

sup R>0 1 R 2R R t n0 d n0 dt n0 m(t) dt A.
Then the spectral multiplier T m , initially defined for

f ∈ S(R d ) ⊗ Y by T m (f ) = F -1 κ m ξ F κ f (ξ) extends to a bounded operator on L p (R d , h 2 κ ; Y ) with T m L p (R d ,h 2 κ ;Y )→L p (R d ,h 2 κ ;Y ) c p,n0,d A.
This theorem, which generalizes the scalar case proved by Dai-Wang [24, Theorem 4.1], only concerns radial multipliers. For scalar valued multivariate (i.e. not necessarily radial) spectral multipliers for Dunkl operators, but only in the particular case where the reflection group W is (Z/2Z) d , see [69]. Proof : It is shown in [24, Proof of Theorem 4.1] that the above condition on m yields that the sequence (µ n ) n 0 defined by µ n = m( n) satisfies the hypotheses (C 0 ) and (C n0 ) of Theorem 3.2 with a bound M cA uniformly in > 0. Note that in Theorem 3.2, we take S d in place of S d-1 as it is stated there verbatim. Then Theorem 3.2 yields that the hypotheses of Theorem 3.12 are satisfied and we apply it to get the desired conclusion. Remark 3.14 Let us compare Theorem 3.13 with two results on (pure Laplacian) vector-valued Fourier multiplier theorems. Namely, in [START_REF] Girardi | Operator-valued Fourier multiplier theorems on L p (X) and geometry of Banach spaces[END_REF]Corollary 4.4], it is shown that -∆ has a H α ∞ calculus on L p (R d ; Y ) with 1 < p < ∞ and α = d Fourier type Y + 1 as soon as Y is a UMD space (not necessarily a lattice). In fact, even more is shown, i.e. non-radial multiplier symbols are admitted, and even operator valued ones, in which case the correct version of the multiplier space H α ∞ involves R-boundedness. Secondly, in [41, Theorem 3.1], it is shown that -∆ has a H α ∞ calculus on L p (R d ; Y ) with 1 < p < ∞ and α = d max(t,q ) + 1 as soon as Y is a UMD space. Here, t denotes the Rademacher type of Y and q the Rademacher cotype of Y . Even more is shown, non-radial multipliers are allowed and Hytönen had discovered that the partial derivations taken on the symbol need only to be of order 0 or 1 in each fixed direction.

Let us compare these two results with our Theorem 3.13, when restricted to the trivial multiplicity function κ = 0. Note that the best Fourier type, the best Rademacher type and cotype all equal 2, so that the derivation order in [START_REF] Girardi | Operator-valued Fourier multiplier theorems on L p (X) and geometry of Banach spaces[END_REF]41]

(x) = (1 -x) δ + on L p (R d ) for all p ∈ (1, ∞) for the order δ < d-1
2 and the fact that [20, p. 11] shows already in the scalar case Y = C that in Theorem 3.13, in general, α cannot be chosen less than d+1 2 . Note also that

m δ ∈ H α 1 iff δ > α - 1
H α ∞ → H α 1 → H α-1-

∞

for > 0, the exponents being optimal in these two strict embeddings, so that the multiplier classes from [START_REF] Girardi | Operator-valued Fourier multiplier theorems on L p (X) and geometry of Banach spaces[END_REF]41] on the one hand and Theorem 3.13 on the other hand complement each other when applicable to a common setting. Finally, let us emphasize that we need the UMD space to be a lattice, which is not the case in [START_REF] Girardi | Operator-valued Fourier multiplier theorems on L p (X) and geometry of Banach spaces[END_REF]41] as well as in many other vector valued spectral multiplier results in the literature (see the introduction).

From Theorem 3.13, we can also immediately deduce the following corollary on Bochner-Riesz means.

Corollary 3.15 Let 1 < p < ∞ and Y = Y (Ω ) be a UMD Banach lattice. Let α > λ κ + 1. For R > 0, let f α R (t) = (1 -t/R) α 0 < t R 0 t > R.
: f H α+1 1 1} is an R-bounded subset of B(L p (R d , h 2 κ ; Y )))
, according to [START_REF] Kriegler | Spectral multiplier theorems via H ∞ calculus and R-bounds[END_REF]. Another application of Theorem 3.13 is the following spectral decomposition of Paley-Littlewood type. We refer e.g. to [46] for applications of this decomposition to the description of complex and real interpolation spaces associated with an abstract operator (the Dunkl Laplacian in our case). ). We deduce the following.

f κ,p;Y ∼ = n∈Z φ n (A)f 2 1 2 κ,p;Y ∼ = ∞ n=0 ψ n (A)f 2
Corollary 3.17 Let 1 < p < ∞ and Y = Y (Ω ) be a UMD Banach lattice. Let ψ ∈ (0, π 2 ). For f ∈ L p (R d , h 2 κ ; Y ), define the maximal function M ψ (f )(x) = sup z∈Σ ψ exp(-zA)f (x, •) Y . Then M ψ (f ) κ,p C ψ f κ,p;Y .
Moreover, exp(-zA)f (x) converges to f (x) in Y for almost every x ∈ R d , as z tends to 0 within the sector Σ ψ .

has maximal regularity. This means, for any given T ∈ (0, ∞], q ∈ (1, ∞) and f ∈ L q ([0, T ), L p (R d , h 2 κ ; Y )), the solution u of (4. In the above Corollary 4.2, the domain of A β is given by general theory of (analytic) semigroups. We give in the next lemma some supplementary information on the domain. ) in X, we can choose a sequence (g n ) n in D(A n-β ) such that g n → h in X. Then f n = (1 + A) -β g n belongs to D(A n ). Moreover,

A β f -A β f n X = A β (1 + A) -β h -A β (1 + A) -β g n X
A β (1 + A) -β h -g n X → 0. Similarly, f -f n X → 0. We have shown the density D(A n ) ⊂ D(A β ) and thus, D ⊂ D(A β ) is dense.

As an illustration, we apply Corollary 4.2 to two partial differential equations involving the Dunkl Laplacian, one involving a first order time derivative, and one stationary. For simplicity, we restrict to the one dimensional case d = 1 (that is W = Z 2 ) and β = 1 for the Dunkl Laplacian. In this case, A takes the form + for given f ∈ L r ((0, T ); L p (R, |s| 2κ ds; L q (R n+1 + ))) has a unique solution u ∈ L r loc ((0, T ); L p (R, |s| 2κ ds; L q (R n+1 + ))) which is almost everywhere differentiable in t, and there exists a constant C < ∞ such that

d dt u + d ds + κ Id -σ s 2 u + |α|=2 a α D α u C f ,
where the four norms here are all in L r ((0, T ); L p (R, |s| 2κ ds; L q (R n+1 + ))).

2. Assume that ω 0 < π, and that B is invertible, e.g. to given B as above change a 0 to a 0 + for some > 0. Then the problem

-d ds + κ Id-σ s 2 u(s, x) + |α|=2 a α D α u(s, x) = f (s, x) (s ∈ R, x ∈ R n+1 + ) |β|=1 b β D β u(s, x) = 0 (s ∈ R, x ∈ ∂R n+1 + )
for given f ∈ L p (R, |s| 2κ ds; L q (R n+1 + )) has a unique solution u ∈ L p (R, |s| 2κ ds; L q (R n+1 + )) and there exists a constant C < ∞ such that

u + d ds + κ Id -σ s 2 u + |α|=2 a α D α u C f ,
where the four norms here are all in L p (R, |s| 2κ ds; L q (R n+1 + )).

Proof : In [50, 7.9 Corollary], it is shown that B is R-sectorial and that ω R (B) ω 0 .

1. This follows then directly from Corollary 4.2 part 3. Note hereby that L q (R n+1 + ) indeed has property (α) [50, p. 128].

2. This follows from Corollary 4.2 part 1. Note that since B is assumed to be invertible, A 1 2 +B is also invertible according to [START_REF] Kalton | The H ∞ -calculus and sums of closed operators[END_REF]Theorem 6.3], and we get that u = (A 1 2 +B) -1 f C f . Remark 4.5 Proposition 4.4 holds as it stands for other powers β > 0 than 1 and dimensions d ∈ N. In part 1. of Proposition 4.4, for the chosen exponent β = 1, (but not for β > 1), we do not need the Hörmander calculus of A but merely the R-sectoriality of it for some angle < π 2 , which follows from Proposition 2.6.

Lemma 2 . 5

 25 Let p = 1 or p = ∞ and T : L p (Ω) → L p (Ω) be a bounded operator. Let Y be any Banach space. Then T ⊗ Id Y : L p (Ω) ⊗ Id Y → L p (Ω; Y ) extends to a bounded operator on L p (Ω) ⊗ Y ⊆ L p (Ω; Y ) (equality here if p = 1) with norm T L p (Ω)→L p (Ω) .

  h 2 κ dy, and this generalized convolution, which reduces when κ = 0 to the spherical convolution [17], satisfies Young-type inequalities [72, Proposition 2.2]

.Lemma 3 . 6 .

 36 The second lemma we need, under the assumption that the Cesàro means are R-bounded, provides us a crucial norm inequality involving both the Cesàro means and the generalized Poisson semigroup (P κ r in fact) onL p (S d-1 , h 2 κ ; Y ). Let 1 < p < ∞ and Y = Y (Ω )be a UMD Banach lattice. Let δ 0 and assume that the Cesàro means (S δ n ) n 0 are R-bounded on L p (S d-1 , h 2 κ ; Y ). If for j 1, r j ∈ (0, 1) and I j is a subinterval of [r j , 1), then Proof : The proof follows closely the lines of [26, Lemma 4.3.5] which itself follows [10].

  Denote the expression under the modulus in the first line of the above estimate by I N . It is shown in [24, Inspection of the Proof of Theorem 3.1] that lim N →∞ I N = c δ,κ I holds in the case Y = Y * = C. Then it also holds for a general Banach space Y, if we take f ∈ S(R d ) ⊗ Y, g ∈ S(R d ) ⊗ Y * , take consequently a double sum k,l , and use Y, Y * -duality both in the definition of I and I N . Then line (3.10) follows immediately, and we have proved Theorem 3.12 in the case where |m(t)| c 1 e -c2t

Corollary 3 . 16

 316 Let 1 < p < ∞ and Y = Y (Ω ) be a UMD Banach lattice. Let (φ n ) n∈Z be a dyadic partition of unity (see Definition 2.10). Further let ψ n = φ n for n 1 and ψ 0 = 0 n=-∞ φ n , so that ∞ n=0 ψ n (t) = 1 for all t > 0. Denote A = -∆ κ the Dunkl Laplacian. Then, for any f ∈ L p (R d , h 2 κ ; Y ), we have the norm description

Lemma 4. 3 F - 1 κ

 31 Let A be a Dunkl Laplacian on L p (R d , h 2 κ ; Y ) as in Corollary 4.2. Then D = S(R d ) ⊗ Y is a core of A β for any β > 0.Proof : Assume first that β = n belongs to N. According to [27, Theorem 1.9], since D is dense in L p (R d , h κ ; Y ), it suffices to show that D is invariant under the action of exp(-tA n ). We clearly have for f= K k=1 f k ⊗ y k ∈ D, exp(-tA n )f = K k=1 [exp(-t ξ 2n )F κ (f k )] ⊗ y k ,and exp(-t ξ 2n ) belongs to S(R d ) for any t > 0. Thus, also exp(-t ξ 2n )F κ (f k ) ∈ S(R d ) according to Lemma 2.11 2., and, again by Lemma 2.11 2., exp(-tA n )f ∈ D. Now consider the general case β > 0. Choose n ∈ N with β < n. Then D ⊂ D(A n ) ⊂ D(A β ), where the first embedding is dense by the above and the second embedding holds according to [50, Theorem 15.15]. It is also dense. Indeed, let f ∈ D(A β ). Then f = (1 + A) -β h for some h ∈ X := L p (R d , h 2 κ ; Y ). By density of D(A n-β

2 u 1 .. 1 M ξ 2 1 2 2 .

 211212 , u ∈ L p (R, |s| 2κ ds), where κ 0 is the only value (still denoted by κ) taken by the multiplicity function (since there is only one class of conjugation), σ(u)(x) = u(-x), and (R d , h 2 κ (s)ds) = (R, |s| 2κ ds). Let us introduce B being the operator A B from [50, p. 156]. That is, we takeY = L q (R n+1 + ), 1 < q < ∞, where R n+1 + = R × [0, ∞). Further, we define the homogeneous second order differential operator acting on R n+1+ , A = |α|=2 a α D α ,and the homogeneous first order boundary operatorB = |β|=1 b β D β ,where a α , b β ∈ C and for α = (α 1 , . . . , α n+1 ) ∈ N n+1 0 , we letD α = (-i) α1+...+αn+1 ∂ α1 ∂x α1Let ω 0 ∈ [0, π) and assume thatA(ξ) := |α|=2 a α (-i) α1+...+αn+1 ξ α1 1 • . . . • ξ αn+1 n+1 ∈ Σ ω0 (ξ ∈ R n+1 ),and A(ξ) for some M > 0. For the operator B, we assume the Lopatinskij-Shapiro condition [50,(7.4)]. That is, we set b 0 = b (0,0,...,0,1) , a k (ξ ) = |β|=k a (β,2-k) (ξ ) β for k = 0, 1, 2 and ξ ∈ R n . We assume b 0 = 0 for simplicity, and that the characteristic polynomiala 0 µ 2 + a 1 (ξ )µ + a 2 (ξ ) + λ = 0has two distinct roots µ ± with µ + > 0 > µ -, for any ξ ∈ R n and λ ∈ Σ ω0 . We eventually define the operator B byD(B) = u ∈ W 2 p (R n+1 + ) : B(u) = 0 on ∂R n+1 + , Bu = Au.Now we obtain the following existence, unicity and regularity results on partial differential equations involving A from (4.2) and the differential operator B. Proposition 4.4 Let 1 < p, q, r < ∞ and 0 < T ∞. Let A acting on L p (R, |s| 2κ ds) and B acting on L q (R n+1 + ) be as above. 1. Assume that ω 0 < π Then the Cauchy problem     d dt u(t, s, x) -d ds + κ Id-σ s 2 u(t, s, x) + |α|=2 a α D α u(t, s, x) = f (t, s, x) t > 0, s ∈ R, x ∈ R n+1 + u(0, s, x) = 0 (s ∈ R, x ∈ R n+1 + ) |β|=1 b β D β u(t, s, x) = 0 t > 0, s ∈ R, x ∈ ∂R n+1

  and moreover, E κ extends to a holomorphic function on C d × C d , see[55]. This kernel, the socalled Dunkl kernel, gives rise to an integral transform which generalizes the Euclidean Fourier transform. For every f ∈ L 1 (R d , h 2 κ ), the Dunkl transform of f , denoted by F κ f , is defined by

  y)h 2 κ (y)dy for a certain K : R d × R d → C, the formula holding for f ∈ S(R d ) and a.e. x ∈ R d . Then it also holds for f ∈ S(R d ) ⊗ Y. It is now sufficient to prove that

	(3.10)

  becomes then α = d 2 + 1. In our case, we have the (for odd d bigger) number α = d 2 + 1 2 + 1. The failure of the boundedness of the Bochner-Riesz multiplier m δ

  Then the Bochner-Riesz means f α R (A) associated with the Dunkl Laplacian A are uniformly bounded in R > 0 on L p (R d , h 2 κ ; Y ). Proof : Note that by Lemma 2.11 8. and Theorem 3.13, A has a H n0 1 calculus on L p (R d , h 2 κ ; Y ) for n 0 an integer > λ κ + 1. It suffices to note that f α

	1 R H α+1-	< ∞ for > 0 (see e.g. [47]) and
	to apply Theorem 3.13.
	Note that there is a partial converse of Corollary 3.15. More precisely, if the Bochner-Riesz
	means {f α

R (A) : R > 0} of a sectorial operator A with H ∞ calculus are R-bounded, then A must have a H α+1 1 calculus (even R-bounded, i.e. {f (A)

  Hörmander calculus implies by Lemma 2.8 2. that the Dunkl Laplacian A has an H ∞ (Σ ω ) calculus for any ω ∈ (0, π 2

	1
	2
	κ,p;Y

. Proof : Once a Hörmander calculus of A on L p (R d , h κ ; Y ) is guaranteed by Theorem 3.13, the corollary follows from [46, Theorem 4.1] resp. (2.1), to decompose the norm in Rademacher sums resp. square sums. A

  1) exists, is almost everywhere differentiable, has values in D(A β ) ∩ D(B), and there exists C < ∞ such that+ A β u(t) L q ([0,T ),L p (R d ,h 2 κ ;Y )) + Bu(t) L q ([0,T ),L p (R d ,h 2 κ ;Y )) C f L q ([0,T ),L p (R d ,h 2 κ ;Y )) .Proof : 1. We can apply[START_REF] Kalton | The H ∞ -calculus and sums of closed operators[END_REF] Theorem 6.3] since A β has an H ∞ (Σ θ ) calculus such that θ + ω < π. 2. We note that if Y has property (α), then also L p (Ω; Y ) has. In our situation, (Ω, µ) = (R d , h 2 κ (x)dx). Now apply the second part of [43, Theorem 6.3]. 3. This follows from part 2. together with [67, Theorem 4.2].

	d dt	u	L q ([0,T ),L p (R d ,h 2 κ ;Y ))

Acknowledgments. The authors wish to thank the referee for her/his careful reading of the manuscript and helpful comments and remarks which improved the quality of the paper.

Proof : Once an H ∞ (Σ ω ) calculus is guaranteed for ω < π 2 -ψ according to the above, the maximal function boundedness follows from [START_REF] Taggart | Evolution equations and vector-valued L p spaces. Strichartz estimates and symmetric diffusion semigroups[END_REF]Theorem 2.4.1]. Then the pointwise convergence follows from the proof in [65, Section 2.6 and Discussion after Corollary 1.3.4], see also [22, Corollary 2]. Remark 3.18 Note that Theorem 3.13 in the scalar case Y = C, i.e. [24,Theorem 4.1] autoimproves by interpolation with self-adjoint calculus. That is, since the Dunkl Laplacian is self-adjoint, T m will be bounded on L 2 (R d , h 2 κ ) for any m : (0, ∞) → C measurable and bounded. Note that

This interpolation improvement works partially in the UMD-lattice-valued case:

According to [START_REF] Rubio De Francia | Martingale and integral transforms of Banach space valued functions[END_REF], a UMD lattice Y is a complex interpolation space Y = [H, Z] θ , between a Hilbert space H = L 2 (Ω) and another UMD lattice Z over Ω. But then t above is coupled to the fixed number θ ∈ (0, 1) and one cannot shrink β below n 0 θ.

Application to maximal regularity

In this section we apply Theorem 3.13 to abstract Cauchy problems involving the Dunkl Laplacian, and show existence, unicity and regularity results for the solutions. Definition 4.1 Let B be an ω-sectorial operator for some ω ∈ (0, π), acting on some Banach space, and let θ ∈ [ω, π). We say that

In this case, we denote by ω R (B) the infimum over all θ such that B is R-θ-sectorial.

According to Lemma 2.8 2. and Lemma 2.11 8., Theorem 3.13 implies that the Dunkl Laplacian A has an H ∞ (Σ θ ) calculus for any θ ∈ (0, π). Thus, also any fractional power A β has an H ∞ (Σ θ ) calculus for any θ ∈ (0, π), for any β > 0. We then deduce Corollary 4.2 below on maximal regularity. To this end, we let B be an ω-sectorial operator for some ω ∈ (0, π), acting on L p (R d , h 2 κ ; Y ). We further impose that resolvents of B commute with resolvents of A. For example B = Id L p ⊗ B 0 , where B 0 is an ω-sectorial operator acting on Y. In that case, B is (R-)ω-sectorial on L p (R d , h 2 κ ; Y ) if and only if B 0 is (R-)ω-sectorial for a given angle ω ∈ (0, π). Moreover, it is easily checked that

2. Assume that for some ω ∈ (0, π), B is R-ω-sectorial. If Y has property (α) (see e.g. [50, 4.9]), then A β + B is again R-ω-sectorial and ω R (A β + B) ω(B).