Dunkl spectral multipliers with values in UMD lattices

Transcription

1 Dukl spectral multipliers with values i UMD lattices Luc Deleaval - Christoph Kriegler April 6 Abstract We show a Hörmader spectral multiplier theorem for A = A Id Y actig o the Bocher space L p (R d, h κ; Y ), where A is the Dukl Laplacia, h κ a weight fuctio ivariat uder the actio of a reflectio group ad Y is a UMD Baach lattice We follow hereby a trasferece method developed by Boami-Clerc ad Dai-Xu, passig through a Marcikiewicz multiplier theorem o the sphere We hereby geeralize works for A = actig o L p (R d, dx) by Girardi-Weis, Hytöe ad others before We apply our mai result to maximal regularity for Cauchy problems ivolvig A Cotets Itroductio Prelimiaries 4 Symmetric cotractio semigroups 4 UMD lattices 5 3 R-boudedess ad square fuctios 5 4 Holomorphic (H ) ad Hörmader (Hp α ) fuctioal calculus 6 5 Dukl trasform, h-harmoic expasio, weighted space o the uit sphere 8 5 The Dukl trasform 8 5 h-harmoic expasios ad aalysis o the sphere 3 Spectral multipliers with values i UMD lattices 5 3 Marcikiewicz-type multiplier theorem for h-harmoic expasios 5 3 A multiplier theorem for the Dukl trasform 4 4 Applicatio to maximal regularity 8 Itroductio Let f be a bouded fuctio o (, ) ad u(f) the operator o L p (R d ) defied by [f( )g]ˆ = [u(f)g]ˆ = f( ξ )ĝ(ξ) Hörmader s theorem o Fourier multipliers [9, Theorem 5] asserts Mathematics subject classificatio: 4A45, 4B5, 47A6 Key words: Spectral multiplier theorems, UMD valued L p spaces, Dukl Laplacia

2 that u(f) : L p (R d ) L p (R d ) is bouded for ay ay p (, ) provided that for some iteger α strictly larger tha d, ad q =, () f q H α q := max sup k=,,,α R> R R t k dk R dt k f(t) q dt < This theorem has may refiemets ad geeralizatios to various similar cotexts Namely, oe ca ask for differet expoets q [, ) i (), ad geeralize to o-iteger α there to get larger (for smaller q ad for smaller α) admissible classes Hq α = {f L loc (, ) : f H α q < } of multiplier fuctios f (see Subsectio 4) Moreover, it has bee a deeply studied questio over the last years to kow to what extet oe ca replace the ordiary Laplacia subjacet to Hörmader s theorem by other operators A actig o some L p (Ω) space A theorem of Hörmader type holds true for may elliptic differetial operators A, icludig sublaplacias o Lie groups of polyomial growth, Schrödiger operators ad elliptic operators o Riemaia maifolds, see [3,, 5, 6] More recetly, spectral multipliers have bee studied for operators actig o L p (Ω) oly for a strict subset of (, ) of expoets [5, 9,,, 39, 4], for abstract operators actig o Baach spaces [36], ad for operators actig o product sets Ω Ω [5, 57, 58] A spectral multiplier theorem meas the that the liear ad multiplicative mappig () H α q B(X), f f(a), is bouded, where typically X = L p (Ω) Oe importat cosequece of a spectral multiplier theorem as i () is the boudedess of Bocher-Riesz meas associated with A Namely, we put for β, R > { f β R (t) = ( t/r) β < t R t > The f β R belogs to Hα q (with uiform orm boud for R > ) if ad oly if β > α q, for q [, ) ad α > q, see [, p ] ad [38] Thus the boudedess of () yields boudedess of the Bocher-Riesz meas for f β R (A) if β > α q For other applicatios of a Hörmader spectral multiplier theorem, see at the ed of Subsectio 3 O the other had, i the particular case of A =, aother directio of geeralizatio of Hörmader s theorem is possible Namely, [7, 8, 3, 3, 44, 5, 56, 63] have studied for which Baach spaces Y, the operator f( ) Id Y, iitially defied o L p (R d ) Id Y exteds to a bouded operator o L p (R d ; Y ) for ay f belogig to some Hörmader class Hq α (or some Mihli class, which correspods essetially to q = i Hq α ) A ecessary coditio is that Y is a UMD space Moreover, the Fourier type [8], ad Rademacher type/cotype [3] of Y play a role whe oe strives for better or best possible derivatio order α I this article, we exted this latter programme partly to Dukl operators, i place of the pure Laplacia Partly refers to the fact that we treat oly radial multipliers (see [58] for multivariate Dukl spectral multipliers i the case Y = C, but with evertheless restrictio o the uderlyig reflectio group), we restrict to the subclass of Y beig a UMD lattice, ad we do ot talk about operator valued spectral multipliers, ie f i () is a fuctio with values i {T B(X) : T (λ A) = (λ A) T for all λ ρ(a)} (although this last part would be possible to some extet) Roughly speakig, Dukl operators are parameterized (with a cotiuous set of parameters κ) deformatios of the partial derivatives ad ivolve a reflectio group W associated with a root system R (see Subsectio 5 for their defiitio) A basic motivatio for the study of

3 these operators comes from the theory of spherical fuctios i aalysis o Lie groups, which ca be, i several situatios, regarded as oly the W -ivariat part of a theory of Dukl operators Ideed, these operators play the role of derivatives for a geeralized Laplacia κ (the so-called Dukl Laplacia), whose restrictio to W -ivariat fuctios is give by Res κ = + α R κ(α) α α,, ad this formula coicides for particular root systems ad particular values of κ to the radial part of the Laplace-Beltrami operator of a Riemaia symmetric space of Euclidea type (see []) More geerally, Dukl operators have sigificatly cotributed to the developmet of harmoic aalysis associated with a root system ad to the theory of multivariable hypergeometric fuctios They also aturally appear i various other areas of mathematics, which iclude for istace, the theory of stochastic processes with values i a Weyl chamber or the theory of itegrable quatum may body systems of Calogero-Moser-Sutherlad type As regards the harmoic aalysis of Dukl operators ad their related objects, the subjacet aalytic structure has a rich aalogy with the Fourier aalysis However, there are still may problems to be solved ad the theory is still at its ifacy Oe of the mai obstructio is the lack of a explicit formula for the operator V κ which itertwies the commutative algebra of Dukl operators with the algebra of stadard differetial operators with costat coefficiets Apart from the case W = Z d where the kow formula for V k allows to tackle ad bypass some difficulties, may tools of harmoic aalysis are ot accessible However, i this paper, we do ot restrict ourselves to this particular reflectio group, ad all our result o Dukl spectral multipliers are stated ad prove for a geeral reflectio group Comig back i particular to our H α q Hörmader theorem, the order of derivatio α ad the itegratio parameter q that we get are (3) α N, α > d + γ κ +, q = where d + γ κ is the doublig dimesio of the Dukl weight h κ o R d Note that usually, oe caot expect to get a Hörmader H α multiplier theorem for α < d + γ κ (see eg [6]), ad that H d +γκ+ +ɛ H d +γκ+ +ɛ H d +γκ, for ɛ >, the expoets beig sharp i these embeddigs (see Lemma 7 ) Our mai Theorem, see Theorem 33 ad the sectio Prelimiaries for precisios, states as follows Theorem Let < p < ad Y = Y (Ω ) be a UMD Baach lattice Let A be the Dukl Laplacia o R d, associated with both a geeral fiite reflectio group ad a oegative multiplicity fuctio κ Let α be as above i (3) Assume that m : (, ) C belogs to H α The m(a) Id Y exteds to a bouded operator o L p (R d, h κ; Y ) Oe of the features of the vector valued character of this theorem is that a operator of the form B = Id L p B will commute with A (or powers of A) ad therefore, spectral theory of a sum A β + B is at had Cosequetly, we apply Theorem to existece, uiqueess ad (maximal) regularity of solutios of Cauchy problems or time idepedet problems ivolvig A β + B, where A is the Dukl Laplacia, β > is arbitrary ad B is as above, see Sectio 4 The methods of proof that we use for Theorem are: Maximal estimates for semigroups associated with the Dukl operator o R d ad o the sphere S d [6]; 3

4 Square fuctio estimates o L p (Ω; Y (Ω )), which is the same as R-boudedess for the space L p (Ω; Y ) as soo as Y is a UMD lattice; H fuctioal calculus, i particular for vector valued diffusio semigroups [59]; A reductio method to spherical harmoics, developed i [6, 5, 6, 7], which uses Cesàro meas, that is, smoothed approximate idetities similar to the Bocher-Riesz meas f β R (A) above The H fuctioal calculus moreover is our startig poit upo which we build the Hq α fuctioal calculus This replaces the usually used selfadjoit calculus approach, which defies f(a) i () o L (Ω) L p (Ω), ad which by desity gives a a priori meaig to f(a) as a operator actig o L p (Ω) Note that o L p (Ω; Y ), there is o selfadjoit calculus at had, sice L (Ω; Y ) is ot a Hilbert space i geeral We ed this itroductio with a overview of the followig sectios I Sectio, we defie ad recall the cetral otios for this article, amely diffusio semigroups, UMD lattices, R- boudedess ad square fuctios, fuctioal calculus ad Dukl aalysis I Sectio 3, we the develop the proof of the spectral multiplier theorem I a first place, i Subsectio 3, we show a Marcikiewicz multiplier Theorem 3, ad the i Subsectio 3, we deduce the Hörmader multiplier Theorem 33 We ed the article with some illustrative applicatios to maximal regularity i Sectio 4 Note that i the article, the symbol meas a iequality up to a costat idepedet of the relevat variables Prelimiaries I this sectio, we defie ad recall the cetral otios of the article ad we prove several lemmas which will be relevat for the sequel Symmetric cotractio semigroups Defiitio Let (Ω, µ) be a σ-fiite measure space Let (T t ) t be a family of operators which act boudedly o L p (Ω) for ay p < The (T t ) t is called symmetric cotractio semigroup (o Ω), if (T t ) t is a strogly cotiuous semigroup o L p (Ω) for ay p < ; T t is selfadjoit o L (Ω) for ay t ; 3 T t Lp (Ω) L p (Ω) for ay t ad p < If i additio we have T t is a positive operator for ay t ; T t () = (ote that T t is bouded o L (Ω) by selfadjoitess ad boudedess o L (Ω)), the (T t ) t is called a diffusio semigroup o Ω For a thorough study of diffusio semigroups, we refer the reader to [5] ad [3] i the scalar ad the vector valued case respectively 4

5 UMD lattices I this article, UMD lattices, ie Baach lattices which ejoy the UMD property, play a prevalet role For a geeral treatmet of Baach lattices ad their geometric properties, we refer the reader to [4, Chapter ] We recall ow defiitios ad some useful properties A Baach space Y is called UMD space if the Hilbert trasform H : L p (R) L p (R), Hf(x) = P V x y f(y)dy exteds to a bouded operator o L p (R; Y ), for some (equivaletly for all < p < [33, Theorem 5] A UMD space is super-reflexive [], ad hece (almost by defiitio) B-covex Let i the followig Y be a UMD space which is also a Baach lattice By B-covexity, Y is order cotiuous ad therefore Y ad its dual Y ca be represeted o the same measure space (Ω, µ), ad moreover the duality is give simply by y, y = y(ω )y (ω )dµ(ω ), Ω see [4, a, b] It is ot difficult to show that if Y is UMD, the also its dual is UMD Hece the dual of a UMD lattice is agai a UMD lattice L p (Ω; Y ) is reflexive for (Ω, ν) a σ-fiite measure space, Y a UMD space ad < p <, sice Y is reflexive ad thus has the Rado-Nikodym property We tacitly shall use several times the followig almost trivial observatio Lemma Let p, (Ω, µ) be a measure space ad Y = Y (Ω ) a Baach fuctio lattice o (Ω, µ) Let M : L p (Ω; Y ) L p (Ω; Y ) be a subliear bouded operator o L p (Ω; Y ), ie M(f + g)(ω, ω ) M(f)(ω, ω ) + M(g)(ω, ω ) for almost all ω Ω ad ω Ω Let further T : D L p (Ω; Y ) L p (Ω; Y ) be a desely defied subliear operator If T f(ω, ω ) c Mf(ω, ω ) for f D ad almost all ω Ω ad ω Ω, the T f Lp (Ω;Y ) c M f Lp (Ω;Y ) for ay f D Proof : This follows immediately from the fact that L p (Ω; Y ) is a Baach fuctio lattice o Ω Ω, that f g is the give by f(ω, ω ) g(ω, ω ) almost everywhere, ad that f g implies f g i a Baach lattice 3 R-boudedess ad square fuctios Let X be a Baach space ad τ B(X) The τ is called R-bouded if there is some C < such that for ay N, ay x,, x X ad ay T,, T τ, we have X X E ɛ k T k x k CE ɛ k x k, k= where the ɛ k are iid Rademacher variables o some probability space, that is, Prob(ɛ k = ±) = The least admissible costat C is called R-boud of τ ad is deoted by R(τ) Note that trivially, we always have R({T }) = T for ay T B(X) Let Y = Y (Ω ) be a B-covex Baach lattice The we have the orm equivalece Y ( () E ɛ k y k = y k k= k= k= R Y 5

6 uiformly i N [43] I particular, this also applies to L p (Ω; Y ), < p <, sice this will also be a B-covex Baach lattice We deduce the followig lemma Lemma 3 Let T be a bouded (liear) operator o a B-covex Baach lattice Y (Ω ) The its tesor extesio T Id l, iitially defied o Y (Ω ) l Y (Ω ; l ) is agai bouded, o Y (Ω ; l ) I particular, if Y (Ω ) is a UMD lattice, the Y (Ω ; l ) is also a UMD lattice Proof : Let (e k ) k be the caoical basis of l We have ( ) Y ( (T Id l ) y k e k = (Ω T y k Y = E ɛ k T y k ;l ) Y k= k= Y R({T })E ɛ k y k k= ( = T y k Y This shows the first part For the secod part, we ote that if Y (Ω ) is UMD, the the Hilbert trasform H : L p (R; Y ) L p (R; Y ) is bouded for all < p < Sice L p (R; Y ) is agai a B-covex Baach lattice, by the first part, we have that H : L p (R; Y (Ω ; l )) L p (R; Y (Ω ; l )) is bouded Hece by defiitio, Y (Ω ; l ) is a UMD (lattice) k= k= 4 Holomorphic (H ) ad Hörmader (H α p ) fuctioal calculus I this subsectio, we recall the ecessary backgroud o fuctioal calculus that we will treat i this article Let A be a geerator of a aalytic semigroup (T z ) z Σδ o some Baach space X, that is, δ (, π ], Σ δ = {z C\{} : arg z < δ}, the mappig z T z from Σ δ to B(X) is aalytic, T z+w = T z T w for ay z, w Σ δ, ad lim z Σδ, z T z x = x for ay strict subsector Σ δ We assume that (T z ) z Σδ is a bouded aalytic semigroup, which meas sup T z Σδ z < for ay δ < δ It is well-kow [7, Theorem 46, p ] that this is equivalet to A beig pseudo-ωsectorial for ω = π δ, that is, A is closed ad desely defied o X; The spectrum σ(a) is cotaied i Σ ω (i [, ) if ω = ); 3 For ay ω > ω, we have sup λ C\Σω λ(λ A) < We say that A is ω-sectorial if it is pseudo-ω-sectorial ad has moreover dese rage I the sequel, we will always assume that A has dese rage, to avoid techical difficulties If A does ot have dese rage, but X is reflexive, which will always be the case i this article, the we may take the ijective part A of A o R(A) X [4, Propositio 5], which the does have dese rage Here, R(A) stads for the rage of A The A geerates a aalytic semigroup o X if ad oly if so does A o R(A) This parallel will cotiue this sectio, ie the fuctioal calculus for A ca be exteded to A i a obvious way, see [35, Illustratio 487] For θ (, π), let H (Σ θ ) = {f : Σ θ C : f aalytic ad bouded} 6

7 equipped with the uiform orm f,θ Let further H (Σ θ ) = { f H (Σ θ ) : C, ɛ > : f(z) C mi( z ɛ, z ɛ ) } For a ω-sectorial operator A ad θ (ω, π), oe ca defie a fuctioal calculus H (Σ θ ) B(X), f f(a) extedig the ad hoc ratioal calculus, by usig a Cauchy itegral formula If moreover, there exists a costat C < such that f(a) C f,θ, the A is said to have bouded H (Σ θ ) calculus ad the above fuctioal calculus ca be exteded to a bouded Baach 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 θ Lemma 4 Let ω (, π) ad A be a ω-sectorial operator o X havig a H (Σ θ ) calculus for some θ (ω, π) Let (f ) be a sequece i H (Σ θ ) such that f (λ) f(λ) for ay λ Σ θ ad sup f,θ < The for ay x X, f(a)x = lim f (A)x Proof : See [4, Theorem 96] or [4, Lemma ] We record the followig propositio for later use, see [59, Theorem 4] Propositio 5 Let Y be a UMD lattice A symmetric cotractio semigroup o Ω exteds to a bouded aalytic semigroup o L p (Ω; Y ) for ay < p < Moreover, its egative geerator A has a bouded H (Σ θ ) calculus for some θ < π For further iformatio o the H calculus, we refer eg to [4] We ow tur to Hörmader fuctio classes ad their calculi Defiitio 6 Let p [, ) ad α > p We defie the Hörmader class by Hp α = { f : (, ) C bouded ad cotiuous, sup φf(r ) W α p (R) < } R> }{{} := f H α p Here φ is ay Cc (, ) fuctio differet from the costat fuctio (differet choices of fuctios φ resultig i equivalet orms) ad Wp α (R) is the classical Sobolev space The Hörmader classes have the followig properties Lemma 7 Assume that α N ad p < The a locally itegrable fuctio f : (, ) C belogs to the Hörmader class Hp α if ad oly if if ad oly if α R sup t k dk R> k= R max sup k= or k=α R> dt k f(t) p dt/t <, R t k dk R ad the above quatities are equivalet to f p H α p dt k f(t) p dt/t <, We have the cotiuous embeddigs H (Σ θ ) H α q H α p H β q for θ (, π), p < q ad α β + p q 3 H α p is a Baach algebra for the poitwise multiplicatio 7

8 4 The mappig H α p H α p, m m(( ) γ ), is a isomorphism for ay γ > Proof : See [35, Sectio 4] for everythig except the secod claimed equivalece i For the latter, we ote that for l k, we have dl dt l f(t) p dt dk dt k f(t) p dt + f(t) p dt accordig to [, Theorem 5] Now for a fuctio g W k p (R, R), take f(t) = g(rt), ad substitute this i the above formula Oe readily obtais that the first displayed term i is domiated by the secod displayed term The coverse estimate is trivial We ca base a Hörmader fuctioal calculus o the H calculus by the followig procedure Defiitio 8 We say that a -sectorial operator has a bouded Hp α (, π) ad ay f H (Σ θ ), f(a) C f H α p ( C f,θ ) calculus if for some θ I this case, the H (Σ θ ) calculus ca be exteded to a bouded Baach algebra homomorphism H α p B(X) i the followig way Let W α p = { f : (, ) C : f exp W α p (R) } equipped with the orm f W α p = f exp W α p (R) Note that for ay θ (, π), the space H (Σ θ ) Wp α is dese i Wp α [38] Sice Wp α Hp α, by the above desity, we get a bouded mappig Wp α B(X) extedig the H calculus Defiitio 9 Let (φ k ) k Z be a sequece of fuctios i Cc (, ) with the properties that suppφ k [ k, k+ ] ad k Z φ k(t) = for all t > The (φ k ) k Z is called a dyadic partitio of uity Let (φ k ) k Z be a dyadic partitio of uity For f Hp α, we have that φ k f Wp α, hece (φ k f)(a) is well-defied The it ca be show that for ay x X, k= (φ kf)(a)x coverges as ad that it is idepedet of the choice of (φ k ) k Z This defies the operator f(a), which i tur yields a bouded Baach algebra homomorphism Hp α B(X), f f(a) This is the Hörmader fuctioal calculus For details of this procedure, we refer to [35, Sectios 43-46] 5 Dukl trasform, h-harmoic expasio, weighted space o the uit sphere 5 The Dukl trasform We recall some basic cocepts of Dukl operators which will be eeded i the article For more details o Dukl s aalysis, the reader may especially cosult [4, 49] ad the refereces therei Let d N \ {} Let W O(R d ) be a fiite reflectio group associated with a reduced root system R (ot ecessarily crystallographic) ad let κ : R [, + [ be a multiplicity fuctio, that is, a W -ivariat fuctio The (ratioal) Dukl operators D κ ξ o Rd, itroduced i [3], are the followig κ-deformatios of directioal derivatives ξ by reflectios Dξ κ f(x) = ξ f(x) + κ(α) f(x) f(σ α(x)) ξ, α, x R d, x, α α R + 8

9 where, deotes the stadard Euclidea ier product, σ α deotes the reflectio with respect to the hyperplae orthogoal to α ad R + deotes a positive subsystem of R The defiitio is of course idepedet of the choice of the positive subsystem sice κ is W -ivariat These operators map P d to P, d where P d is the space of homogeeous polyomials of degree i d variables, ad they mutually commute The Dukl Laplacia is κ f = d i= (Dκ e i ) f, where (e i i d is the caoical basis of R d, ad ca be writte explicitly as follows (see [3]) κ f(x) = f(x) + ( α f(x) κ(α) α, x α α R + ) f(x) f(σ α (x)) α, x It geerates a semigroup Ht κ o L p (R d, h κ), p <, which is a diffusio semigroup i the sese of Defiitio [, Theorem 6] (see also [46, 48]), where the weight h κ defied o R d by h κ(x) = α R + x, α κ(α) is ivariat uder the actio of W ad homogeeous of degree γ κ, with γ κ = α R + κ(α) The Dukl operators give rise to a rich aalytic structure sice they are also itertwied with the usual derivatives Ideed, there exists a uique liear isomorphism V κ (called itertwiig operator) o P = Pd such that V κ (P d ) = P d, V κ P d = Id P d, D κ ξ V κ = V κ ξ ξ R d Ufortuately, the itertwiig operator is explicitly kow oly i some special cases but for a geeral reflectio group, we all the same have the followig sigificat Laplace-type represetatio due to Rösler (see [47]): for every x R d, there exists a uique probability measure dµ κ x, compactly supported i the covex hull of the orbit of x uder the actio of W (amog other properties) such that for ay P P V κ P (x) = P (ξ)dµ κ x(ξ), R d ad this formula allows to exted it to various larger fuctio spaces For y C d, let E κ (x, y) = V κ ( e,y ) (x), x R d, where, deotes the biliear extesio of the Euclidea ier product to C d C d The E κ (, y) is the uique real-aalytic solutio of the spectral problem D κ ξ f = ξ, y f ξ R d, f() =, ad moreover, E κ exteds to a holomorphic fuctio o C d C d, see [45] This kerel, the socalled Dukl kerel, gives rise to a itegral trasform which geeralizes the Euclidea Fourier trasform For every f L (R d, h κ), the Dukl trasform of f, deoted by F κ f, is defied by F κ f(x) = c κ R d E κ ( ix, y)f(y)h κ(y)dy, x R d, 9

10 where c κ = e x / h R κ(x)dx is a Mehta-type costat We poit out that the Dukl d trasform coicides with the Euclidea Fourier trasform whe κ = (sice Dξ = ξ ad V = Id) ad that it is more or less a Hakel trasform whe d = (ad the W Z ) The Dukl trasform has the followig properties, where for a give Baach lattice Y = Y (Ω ), we deote by L p (R d, h κ; Y ) the Bocher space of classes of fuctios f : R d Y such that ( f = f(y) p p Y h κ(y)dy <, R d with the stadard modificatio if p = If Y = C, we usually omit Y i the otatios Lemma If f L (R d, h κ) the F κ f C (R d ) F κ is a isomorphism of the Schwartz class S(R d ) oto itself, ad F κf(x) = f( x) 3 The Dukl trasform has a uique extesio to a isometric isomorphism of L (R d, h κ) 4 Let f L (R d, h κ) If F κ f is i L (R d, h κ), the we have the iversio formula f(x) = c κ R d E κ (ix, y)f κ f(y)h κ(y)dy 5 For f S(R d ), we have κ (f) = Fκ [ ξ F κ f(ξ)], ad the semigroup Ht κ geerated by the Dukl Laplacia satisfies Ht κ (f) = Fκ [e t ξ F κ f(ξ)] 6 For m : (, ) C a bouded measurable fuctio ad Y a Baach space, T m (f) = F κ [m( ξ )F κ f(ξ)] is a well defied elemet of C (R d ; Y ) for f S(R d ) Y 7 Let Y be a UMD lattice ad < p < The κ is a ω-sectorial operator for some ω < π o Lp (R d, h κ; Y ), i particular ijective with dese rage 8 Let Y be a UMD lattice ad < p < Let q [, ) ad α > q Let A = κ be the egative geerator of the Dukl heat semigroup H κ t o L p (R d, h κ; Y ) Let ω < θ (, π) such that A is ω-sectorial o L p (R d, h κ; Y ) (a) Suppose that for ay m H (Σ θ ), the above operator T m, iitially defied o S(R d ) Y, exteds to a bouded operator o L p (R d, h κ; Y ) ad T m C m H α q The A has a Hörmader Hq α calculus ad m(a) = T m for m Hq α ad m(t) = m(t ) (b) Suppose that there is a C < such that for ay m H (Σ θ ), m(a) L p (R d,h κ ;Y ) Lp (R d,h κ ;Y ) C m H α q The A has a Hq α calculus, for ay m Hq α, T m defied above exteds to a bouded operator o L p (R d, h κ; Y ) ad m(a) = T m Proof : For parts,,3,4,5, we refer to [9], [55] For 6, we ote that F κ f belogs agai to S(R d ) Y, so ξ m( ξ )F κ f(ξ) belogs to L (R d ) Y Now apply part For 7, ote that sice κ geerates a diffusio semigroup, it is pseudo-ω-sectorial for some ω < π o Lp (R d, h κ; Y ) accordig to Propositio 5 The the fact that A = κ is ijective o L p (R d, h κ; Y ) (equivaletly, has dese rage, equivaletly is ω-sectorial) ca be see as

11 follows Accordig to [4, Propositio 5], it suffices to show that t(t + A) f for ay f L p (R d, h κ; Y ), as t Sice sup t> t(t + A) < by pseudo-sectoriality of A, it suffices to cosider f S(R d ) Y For these f, we have [ t(t + A) f = Fκ t ] t + ξ F κ(f)(ξ) i C (R d ; Y ) by domiated covergece, sice t+ ξ ad F κ (f) S(R d ) Y accordig to part By [4, Propositio 5], we already kow that t(t + A) f coverges i L p (R d, h κ; Y ), so by uicity of the limit, it coverges to i L p (R d, h κ; Y ) We tur to 8 It follows from 5 ad the represetatio formula (λ A) = t e λt e ta dt for Rλ < that T m = m(a) for m(t) = (λ t) This idetity ca be exteded for λ C\Σ ω, where ω < θ ad σ(a) Σ ω, by aalytic cotiuatio The the idetity follows for ay m H (Σ θ ) from the Cauchy formula defiig the H calculus We ow show step by step that T m = m(a) holds for m H (Σ θ ), for m Wq α ad for m Hq α, uder either the assumptios 8 (a) or 8 (b) Each time, it will suffice by liearity ad desity to show the idetity applied to f y with f S(R d ) ad y Y So let m H (Σ θ ) Let ( ρ (λ) = λ ( + λ) H (Σ θ ) We have ρ (λ) for ay λ Σ θ ad sup ρ,θ = sup ρ,θ < The assumptio 8 (b) readily implies that A has a H (Σ θ ) calculus by Lemma 7, whereas 8 (a) also implies it via the already provided idetity T m = m(a) for m H (Σ θ ) Thus, by the Covergece Lemma 4, we have with m := mρ, m(a)(f y) = lim m (A)(f y) = lim Fκ [m ( ξ )F κ f(ξ)] y = Fκ [m( ξ )F κ f(ξ)] y = T m (f y), the first limit i L p (R d, h κ; Y ), the secod limit i C (R d ; Y ), by ad domiated covergece It follows T m = m(a) for m H (Σ θ ), ad that A has a Hq α calculus, sice m(a) lim sup m (A) lim sup m H α q m H α q lim sup ρ H α q m H α q lim sup ρ θ, m H α q

12 Now let m W α q, ad m a sequece i H (Σ θ ) W α q approximatig m i W α q We have m(a)(f y) = lim m (A)(f y) = lim T m (f y) = lim Fκ [m ( ξ )F κ f(ξ)] y = F κ [m( ξ )F κ f] y, where the last limit holds by Wq α L (, ) ad domiated covergece, plus part Thus, m(a) = T m for m Wq α Let fially m Hq α Let (φ k ) k Z be a dyadic partitio of uity as i Defiitio 9 The m(a)(f y) = lim k= (φ k m)(a)(f y) = lim k= T φk m (f y) = T m(f y), the secod limit holdig by almost the same argumet as before i the case W α q We ow tur to h-harmoic expasios ad aalysis o the sphere 5 h-harmoic expasios ad aalysis o the sphere For more details o h-harmoic expasios ad aalysis o the sphere, the reader may cosult the expertly writte book of Dai-Xu [7] For d, we let S d = {x R d : x = }, ad for p < ad Y a Baach lattice, we let L p (S d, h κ; Y ) be the Bocher space of equivalece classes of measurable fuctios f : S d Y such that f Lp (S d,h κ ;Y ) := (a κ f(y) p p Y h κ(y)dy <, S d with the stadard modificatio if p = Here, the measure o S d is surface measure ad a κ = h S κ(y)dy By abuse of otatio, we also write f d Lp (S d,h κ ;Y ) = f Let P P d The P is called a h-harmoic polyomial of degree if κ P = It is wellkow (see []) that a homogeeous polyomial is a h-harmoic polyomial if it is orthogoal to all polyomials of lower degree with respect to the ier product of L (S d, h κ) For j, we let proj κ j : L (S d, h κ) L (S d, h κ) be the orthogoal projectio with image the space of all h-harmoics of degree j The projectio proj κ j has the followig itegral represetatio with (see [6, Theorem 3, (3)]) where proj κ j f(x) = a κ S d f(y)p κ j (x, y)h κ(y)dy, x S d, P κ j (x, y) = j + λ κ λ κ V κ [ C λκ j () λ κ = d + γ κ ( x, ) ] (y), x, y S d, ad where C λκ j is the stadard Gegebauer (or ultraspherical) polyomial of degree j ad idex λ κ (see [53] for istace) Now, let c λ κ = ( t ) λκ dt = π Γ(λ κ + ) Γ(λ κ + )

13 The, accordig to [6], we defie for f L (S d, h κ) ad g L ([, ], ω λκ ), with ω λκ the weight for which the Gegebauer polyomials are orthogoal, that is ω λκ (t) = ( t ) λκ, [ f g(x) = a κ f(y)v κ g S ( x, )] (y)h κdy, d ad this geeralized covolutio, which reduces whe κ = to the spherical covolutio [8], satisfies Youg-type iequalities [6, Propositio ] We ow state the followig lemma, which will be useful i the sequel Lemma For j, p ad ay Baach space Y, the operator proj κ j boudedly to L p (S d, h κ; Y ), ad we have, for j large eough, the orm estimate exteds Proof : Let x S d Write for ay j proj κ j L p (S d,h κ ;Y ) Lp (S d,h κ ;Y ) j λκ proj κ j f(x) = a κ S d f(y)p κ j (x, y)h κ(y)dy = ( f [ j + λκ λ κ ] ) C λκ j (x) Thus, by Youg-type iequalities for the geeralized covolutio o the sphere, we have both the iequalities proj κ j f L (S d,h κ ) f L (S d,h κ ) j + λ κ λ κ proj κ j f L (S d,h κ ) f L (S d,h κ ) j + λ κ λ κ C λκ j C λκ j L ([,],ω λκ ) L ([,],ω λκ ) Besides, j + λ κ λ κ C λκ j L ([,],ω λκ ) = c λκ j + λ κ λ κ j + λ κ λ κ (λ κ ) j, j! C λκ j (t) ( t ) λκ dt where we have used the iequality (see for istace [4, p 35]) C λκ j (t) C λκ j with (x) the so-called Pochhammer symbol Stirlig s formula Γ(a) πa a e a gives us () = (λ κ) j, j! Moreover, sice we ca write (x) = Γ(x+) Γ(x), (3) (λ κ ) j j! = Γ(λ κ + j) Γ(λ κ )Γ(j + ) C(λ κ)j λκ We ca coclude, for j large eough, that j + λ κ λ κ C λκ j j λκ, L ([,],ω λκ ) 3

14 ad therefore, we have both the iequalities proj κ j L (S d,h κ ) L (S d,h κ ) j λκ proj κ j L (S d,h κ ) L (S d,h κ ) j λκ We ca tesorise these estimates to get estimates o Bocher spaces, amely proj κ j L p (S d,h κ ;Y ) Lp (S d,h κ ;Y ) j λκ holds for p =,, ad the by iterpolatio also for p [, ] We close this sectio with some facts o both a geeralized Poisso ad heat semigroup o (S d, h κ(y)dy) We first recall their defiitio (see [6, p 48]) Defiitio The Poisso semigroup T κ t o (S d, h κ(y)dy) is defied by Tt κ f = e jt proj κ j f, t > j= The heat semigroup H κ t o (S d, h κ(y)dy) is defied by H κ t f = e j(j+λκ)t proj κ j f, t > j= The followig statemet will be of particular iterest Lemma 3 Both the Poisso ad the heat semigroup are diffusio semigroups Proof : That they are semigroups is clear from the fact that the proj κ j are projectios The cotractivity of Ht κ o L p (S d, h κ) for all p ad t is proved i [6, Proof of Lemma ] This yields the strog cotiuity of both semigroups o L Ideed, strog cotiuity Tt κ f f ad Ht κ f f (t ) is clear for elemets of the form f = J j= projκ j g (J fiite), sice e tj ad e tj(j+λκ) uiformly for j =,, J as t Now use a 3ɛ argumet for geeral f L The the strog cotiuity extrapolates o L p by cotractivity It is proved i [6, Proof of Lemma ] that Ht κ are positive operators, ad [6, ()] yields the that Tt κ is also a positive operator We fially show that Tt κ () ad Ht κ () as t This will imply that Tt κ () =, ad the same for Ht κ Ideed, Tt+s() κ = Tt κ Ts κ (), so = lim s Tt+s() κ = Tt κ lim s Ts κ () = Tt κ () We have Now proj κ () = ad by Lemma T κ t () = proj κ () + e tj proj κ j () j= e tj proj κ j () e t j= j= e t(j ) proj κ j () e t (t ) The same argumet applies for H κ t 4

15 3 Spectral multipliers with values i UMD lattices 3 Marcikiewicz-type multiplier theorem for h-harmoic expasios I this sectio, we take d N with d Let us begi by recallig the defiitio of the usual differece operator Defiitio 3 Give a sequece (µ j ) j of complex umbers, we defie recursively µ j = µ j µ j+, + µ j = µ j µ j+, j, We ow state the mai result of this sectio Theorem 3 Let < p < ad Y = Y (Ω ) be a UMD Baach lattice Let (µ j ) j be a scalar sequece Suppose that for some iteger > d + γ κ = λ κ +, we have (C ) sup j µ j M <, (C ) sup j j( ) j+ l= j µ l M < The (µ j ) j defies a L p (S d, h κ; Y ) multiplier, that is µ j proj κ j f c p M f, j= where the costat c p is idepedet of f ad (µ j ) j This theorem geeralizes the scalar case proved by Dai ad Xu i [6] Therefore, if we specialize Theorem 3 to Y = C ad κ =, the we recover the famous Marcikiewicz type theorem for zoal multipliers due to Boami-Clerc [6] The proof, which is divided ito several lemmas ad a propositio, follows the strategy of Boami-Clerc adapted i the Dukl settig by Dai-Xu A crucial role i the proof will be played by several kids of Littlewood-Paley type g-fuctios closely related to Cesàro meas for h-harmoic expasios Let us begi with the followig otatio Let (Tt κ ) t be the geeralized Poisso semigroup o L p (S d, h κ; Y ) The we set (3) P κ r = T κ log(r), < r < The first lemma will provide a ew equivalet orm o L p (S d, h κ; Y ), i terms of a well suited g-fuctio Lemma 33 Let < p < ad Y = Y (Ω ) be a UMD Baach lattice The, for ay f L p (S d, h κ; Y ), we have the two-sided estimate ( c f ( r) ) r P r κ f dr c f, where i the first iequality we assume that S d f(y)h κ(y)dy = 5

16 Proof : The semigroup (Tt κ ) t is a diffusio semigroup, so its vector-valued extesio o L p (S d, h κ; Y ) has a H (Σ ω ) calculus of some agle ω < π, thaks to Propositio 5 Accordig to [59, Propositio 9], we have, for ay f L p (S d, h κ; Y ), the followig square fuctio estimate ( t ) t T t κ f dt C f We ext show that we also have the coverse estimate to the previous oe, uder the additioal assumptio that S d f(y)h κ(y)dy = Note that lim T t κ f = lim t t e jt proj κ j f = proj κ f j= is the projectio oto the kerel of the egative geerator A p of (T κ t ) t (its versio o L p (S d, h κ; Y )), accordig to the decompositio L p (S d, h κ; Y ) = Ker(A p ) R(A p ), valid for ay egative geerator of a aalytic semigroup o a reflexive space [4, Propositio 5] Sice proj κ f = a κ S d f(y)h κ(y)dy =, our particular f must lie i R(A p ) Cosider ow the part Ãp of A p o R(A p ) [4, Propositio 5], which has dese rage The accordig to [4, Lemma 93], we have the partitio f = ψ(tãp)f dt t = ψ(ta p )f dt t, for ψ H (Σ π ɛ ) of dt t -itegral, as a improper itegral, uder the additioal assumptio that f R(A p ) D(A p ) Here, D(A p ) stads for the domai of A p Apply this partitio to ψ(t) = cte t te t, we get f, g = c for ay g L p (S d, h κ; Y ) This implies f, g ta p T κ t f, ta p T κ t g dt t, ta p Tt κ f, ta p Tt κ g dt t = ta p Tt κ (f)(y, ω ) ta p Tt κ (g)(y, ω )dω h κ(y)dy dt S d Ω t ta p Tt κ (f)(y, ω ) ta p Tt κ (g)(y, ω ) dt S d Ω t dω h κ(y)dy ( ) ta p Tt κ (f)(y, ω ) ( dt ta p Tt κ (g)(y, ω ) dt S d Ω t t ( t ) t T t κ f ( dt t ) t T t κ g dt κ,p ;Y Applyig ow the upper estimate for g ad o L p (S d, h κ; Y ), we get ( f, g t ) t T t κ f dt g κ,p ;Y dω h κ(y)dy 6

17 Takig the supremum over all g of orm yields the desired estimate uder the additioal assumptio f R(A p ) D(A p ) Sice R(A p ) D(A p ) is dese i R(A p ) [4, Propositio 94], we deduce ( (3) f t ) t T t κ f dt f, the lower estimate uder the assumptio S d f(y)h κ(y)dy = We ext deduce from (3) the stated g-fuctio orm equivalece of the lemma To this ed, we proceed by a modificatio of the proof i [7, p 38-39] By the chage of variable r = e t, we get Now, we write ( g (f) := t ( t P e κ tf dt = r log(r) r P r κ f dr ( g(f) = ( r) r P r κ f dr We shall show that g(f) f ad g (f) g(f), which completes the proof Sice ( r) r log r for r <, we have (33) g(f) ( ( r) r P r κ f dr + ( The first term o the right had side, we estimate by sup r P r κ f r sup r j= r log r r P r κ f jr j proj κ j f = j j proj κ j f By Lemma, we ca sum over j to get sup r P r κ f f Use ow (33) to deduce that r j= g(f) f + g (f) f We have proved the upper estimate of the lemma For the lower estimate, we simply use r log r ( r) o r [, ], to deduce g (f) g(f), ad thus, whe S d f(y)h κ(y)dy = f g (f) g(f) Now, we shall prove that the Cesàro meas of h-harmoic expasios are R-bouded o L p (S d, h κ; Y ) To this ed, we recall some defiitios 7

18 Defiitio 34 For δ ad l N, we put ( ) l + δ A δ (l + δ)(l + δ ) (δ + ) Γ(l + δ + ) l = = = l l(l ) Γ(l + )Γ(δ + ) = (δ + ) l l! The, for j ad, we put a δ, j = A δ A δ jχ j We defie the Cesàro meas of order δ (from ow o, just called Cesàro meas) by the multiplier S δ f = j= a δ, j proj κ j f We ca state the R-boudedess of the Cesàro meas of h-harmoic expasios Recall that we have set λ κ = d + γ κ Lemma 35 Let < p <, Y = Y (Ω ) be a UMD Baach lattice Assume that δ > λ κ The the Cesàro meas (S) δ are R-bouded o L p (S d, h κ; Y ), that is, ( S δ j f j ( c p,δ f j j= Proof : By [7, p 37], we have the estimate [ ] (34) sup Sf(x, δ ω ) c M κ f(x, ω ) + M κ f( x, ω ), x S d, ω Ω, where M κ is the followig well-suited maximal operator for h-harmoic expasios [6, Propositio 3] or [6, (5)] f(y) V S M κ f(x) = sup d κ [χ B(x,θ) ](y)h κ(y)dy, x S d, <θ π V S d κ [χ B(x,θ) ](y)h κ(y)dy with B(x, θ) = {y R d : x, y cos θ} {y R d : y } It is show i [6, Proof of Theorem ] that M κ f(x, ω t ) c sup Hs κ f(, ω ) (x)ds, t> t where we recall that Hs κ is the geeralized heat semigroup o (S d, h κ(y)dy) Sice it is a symmetric cotractio semigroup, by [59, Theorem ], the Hopf-Duford-Schwartz maximal operator t M(H)f = sup t> t Hs κ fds j= 8

19 is bouded o L p (S d, h κ; Y (Ω ; l )), Y (Ω ; l ) beig agai a UMD-lattice accordig to Lemma 3 Therefore, we get ( S δ j f j ( M κ f j j= j= M(H)(fj ) j Lp (S d,h κ ;Y (Ω ;l )) (fj ) Lp j (S d,h κ ;Y (Ω ;l )) ( = f j j= The secod lemma we eed, uder the assumptio that the Cesàro meas are R-bouded, provides us a crucial orm iequality ivolvig both the Cesàro meas ad the geeralized Poisso semigroup (P κ r i fact) o L p (S d, h κ; Y ) Lemma 36 Let < p < ad Y = Y (Ω ) be a UMD Baach lattice Let δ ad assume that the Cesàro meas (S δ ) are R-bouded o L p (S d, h κ; Y ) If for j, r j (, ) ad I j is a subiterval of [r j, ), the ( ( ) S δ j P κ rj f j j= ( c p j= P κ I j r f j dr I j Proof : The proof follows closely the lies of [7, Lemma 435] which itself follows [6] We give some details Firstly, accordig to [7, p 4], we have for j, δ ad < r < S δ j Pr κ j f j j c b δ l, j Sl δ f j, l= where b δ l, are scalars satisfyig l= bδ l, c δ It follows from the R-boudedess of the Cesàro meas ( S δ ( ) ( j P κ rj f j j c b δ l, j Sl δ f j j= j= l= ( ) = c Sl (χ δ l j b ) δl,j f j j= l= ( ) χ l j b δ l, j f j j= l= ( j = b δ l, j f j j= l= ( c δ f j j= 9

20 Thus, we have show ( ( ) (35) S δ j P κ rj f j j= ( f j Now let for each j ad, (r j,i ) i= I j be a fiite sequece such that r j,i r j,i = I j for all i The for each N, R j, := i= P r κ j,i f j is a Riema sum over Ω of the itegral I j I j Pr κ f j dr Thus, by domiated covergece, it follows (36) ( j= P κ I j r f j dr I j j= ( = lim j= i= Pr κ j,i f j O the other had, sice for each, r j < r j,i for all i ad j, we have by (35) ( ( ) S δ j P κ rj f j = ( S δ j Pr κ j/r j,i (Pr κ j,i f j ) j= i= j= ( Pr κ j,i f j i= j= We coclude with (36) Before statig a propositio which is a key step i the proof of Theorem 3, we eed the followig fuctioal which is closely related to the Cesàro meas of h-harmoic expasios Defiitio 37 Let δ We defie the fuctioal g δ (f), for give f L p (S d, h κ; Y ), by ( g δ (f) = S δ+ f Sf δ Moreover, let (ν k ) k be a sequece of oegative umbers such that sup ν k = M < = We defie the fuctioal gδ (f), for give f Lp (S d, h κ; Y ), by ( gδ (f) = S δ+ f Sf δ ν = Remark 38 Note that if i particular ν k = for all k, the g δ (f) = g δ(f) k= The followig propositio gives us two importat orm iequalities ivolvig the Littlewood- Paley fuctios g δ (f) ad g δ (f) Propositio 39 Let < p < ad Y = Y (Ω ) be a UMD Baach lattice Let δ If f L p (S d, h κ; Y ) satisfies S d f(y)h κ(y)dy =, the f c p,δ g δ (f) Coversely, if the Cesàro meas (S δ ) are R-bouded o L p (S d, h κ; Y ), the where M = sup k= ν k g δ (f) c p,δ M f,

21 Proof : First, recall that we have set i the proof of Lemma 33 ( g(f) = ( r) r P r κ f dr It is show i [7, Sectio 43] that g(f)(x) c δ g δ (f)(x) Therefore, it follows immediately with Lemma 33 f c p g(f) c p c δ g δ (f), i the case f(y)h S κ(y)dy = d We proceed to the secod stated iequality ad suppose that the Cesàro meas to a order idex δ are R-bouded We follow closely the lies of [7, Sectio 43] but, for the readers coveiece, we preset the proof all the same First we may assume that j= ν j, sice the desired coclusio for geeral (ν j ) j ca be deduced from this case applied to the two sequeces ν j = ad ν j = M ν j + Now let µ = ad µ = + i= ν i for Let further r = µ ad f = Pr κ f It is show i [7, p 44] that S δ+ f Sf δ c S δ+ f Sf δ + c 3 j= j S δ+ j f S δ j f Therefore, i view of Lemma 33, we are left with the task of establishig the followig iequalities ( (37) S δ+ f Sf δ ν c g(f) = ad (38) ( = ν 4 j= j S δ+ j f S δ j f c g(f) We start by showig (37) To this ed, let η C (R) with η(t) = for t ad η(t) = for t Moreover, for, let L κ ad L κ be the followig multipliers L κ f = j= ( j η proj ) κ j f, Lκ f = j= ( j jη proj ) κ j f Comparig symbols of multipliers yields [7, p 44] for j N (39) S δ+ j f S δ j f = (j + δ + ) P κ r ( S δ j ( L κ Nf) ) Usig this last equality specializig to j =, we obtai by Lemma 36 ( N S δ+ f S δ ( f N ν ( P κ ν 3 r S δ ( Lκ N (f) )) ) = = ( N ν r+ 3 Pr κ ( Lκ r + r N (f) ) dr = r

22 Obviously, we have Pr κ ( L κ N (f)) = r Lκ N ( r P r κ (f)) Moreover, the operators L κ N are uiformly bouded i N N o L p (S d, h κ; Y ) Ideed, a straightforward computatio gives us L κ Nf = Sice we have l+ η ( j N ) N l, the But N j= Al j = Al+ N L κ N f N l+ N j= N j=, the we claim (see (3)) that N l A l+ N ( j ) l+ η A l N jsjf l A l j S l jf sup j S l j f N l+ = N l (l + ) N (N)! ad i view of (34), choosig l > λ κ, the operators L κ N are uiformly bouded i N N o L p (S d, h κ; Y ) Now, sice they are liear, by Lemma 3, a sigle operator L κ N is also bouded o L p (S d, h κ; Y (Ω ; L ([, ]; dr))) Therefore, ( N ν r+ 3 Pr κ ( Lκ r = + r N (f) ) dr r ( N ν r+ 3 ) r + r r P r κ f dr Sice r + r = ( N = ν 3 r + r ν µ µ + ν r+ r = c, N j= A l j ad r for all r [r, r + ], it follows that ) r P r κ f ( r+ dr r P r κ f dr = r r g(f) Thus, we have proved that ( N S δ+ f Sf δ ν g(f) = ad lettig N yields (37) We ow tur to the proof of (38), which is similar to the previous oe Ideed, usig Lemma 36 ad (39), we have ( N ν 4 j S δ+ j f Sj δ ( f N ν 4 Pr κ (Sj δ ( Lκ N (f) )) ) = j= = j= ( ν r+ 4 ) r = j= + r r r P r κ f dr ( ν r+ 3 ) r + r r P r κ f dr = g(f) r

23 We obtai (38) by lettig N The proof is complete I view of Remark 38, we immediately obtai the followig corollary Corollary 3 Let < p < ad Y = Y (Ω ) be a UMD Baach lattice If δ > λ κ, the c p f g δ (f) c p f, where i the first iequality we assume that S d f(y)h κ(y)dy = We ow state the last lemma we shall eed for the proof of Theorem 3 Lemma 3 Let δ to be the smallest iteger strictly larger tha λ κ ad let = δ + Let (µ j ) j be a sequece as i the hypotheses of Theorem 3, ie satisfyig (C ) ad (C ), with boud M Write M µ f = µ j proj κ j f the associated multiplier The we have j= g δ (M µ f) C g δ (f), where the sequece (ν k ) k is ν k = + δ+ j= j µ k k j ad satisfies sup j= ν j cm Proof : It is show i [7, (44)] that g δ (M µ f) Cg δ (f) holds poitwise, with the sequece (ν k ) k give i the lemma By Lemma, we immediately deduce g δ (M µ f) p,κ;y C g δ (f) The statemet o (ν k ) k is show i [7, p 47] We are ow i a positio to prove Theorem 3 Proof of Theorem 3 : We ca assume that µ = Ideed, µ j proj κ j f µ proj κ f + µ j proj κ j f, j= ad proj κ is bouded o L p (S d, h κ; Y ) by Lemma Let δ = > λ κ Note that j= a κ S d M µ (f)(y)h κ(y)dy = proj κ (M µ (f)) = if µ = Accordig to Lemma 35, the Cesàro meas (S δ ) are R-bouded Hece by Propositio 39 i cojuctio with Lemma 3, M µ (f) g δ (M µ (f)) g δ (f) M f 3

24 3 A multiplier theorem for the Dukl trasform Bouded vector-valued multipliers o the sphere S d yield bouded vector-valued spectral multipliers of the Dukl Laplacia o R d by a trasferece priciple, preseted i Theorem 3 below I the scalar case, a trasferece priciple from zoal multipliers o S d to radial multipliers o R d was oberved ad proved by Boami-Clerc i [6], ad their strategy has bee recetly adapted by Dai-Wag [5] to obtai bouded multipliers for the Dukl trasform o R d from bouded multipliers for h-harmoic expasios o the uit sphere S d Let W O(d) be a fiite reflectio group associated with a reduced root system R ad let κ : R [, + [ be a multiplicity fuctio with associated weight fuctio h κ We trasfer this to S d R d+ Namely, for g W, there exists a uique orthogoal trasformatio o R d+, deoted by g ad determied by g x = (gx, x d+ ), x = (x, x d+ ) R d R The W = {g : g W } is a fiite reflectio group o R d+ associated with the reduced root system R = {(α, ) : α R} Fially, we let { κ R R +, : (α, ) κ(α) ad associate with it the weight h κ We ca ow state the followig trasferece priciple for the Dukl trasform Theorem 3 Let Y = Y (Ω ) be a UMD Baach lattice Let m : (, ) R be a cotiuous ad bouded fuctio For ɛ > ad, let µ ɛ = m(ɛ) Let further M ɛ = M µ ɛ be the multiplier M ɛ (f) = m(ɛ)proj κ f = Assume that for some p ad ay f L p (S d, h κ ; Y ), sup M ɛ f L p (S d,h κ ɛ> ;Y ) A f L p (S d,h κ ;Y ) The m is a radial Dukl spectral multiplier o L p (R d, h κ; Y ), that is, for ay f L p (R d, h κ; Y ), where T m is a priori defied o S(R d ) Y by T m f Lp (R d,h κ ;Y ) c d,κ A f Lp (R d,h κ ;Y ), T m (f) = Fκ [m( ξ )F κ (f)(ξ)] This theorem geeralizes the scalar case proved by Dai ad Wag i [5] Therefore, if we specialize Theorem 3 to Y = C ad κ =, the we recover the stadard result due to Boami-Clerc [6] Proof : We follow closely the strategy of [5, Sectio 3] Note that cotiuity of m i is ot eeded there We first assume that for some c, c > m(t) c e ct, t > 4

25 I [5, Lemma 35], it is show that the operator T m has the followig itegral represetatio T m f(x) = f(y)k(x, y)h κ(y)dy R d for a certai K : R d R d C, the formula holdig for f S(R d ) ad ae x R d The it also holds for f S(R d ) Y Recall that Y = Y (Ω ) is a fuctio lattice o Ω, ad its dual Y = Y (Ω ) is a fuctio lattice o the same measure space Ω, ad the duality is give by y, y = y(ω )y (ω )dµ(ω ) Ω for a certai measure µ o Ω The it is sufficiet to prove that (3) f(y, ω )g(x, ω )K(x, y)h κ(x)h κ(y)dxdydµ(ω ) ca Ω R d R d holds wheever f S(R d ) Y ad g S(R d ) Y have compact support ad satisfy f L p (R d,h κ ;Y ), g L p (R d,h κ ;Y ) Deote the above triple itegral by I ad let ψ : R d S d be the mappig For N, let moreover ψ(x) = ( ξ si x, cos x ), for x = x ξ R d with ξ S d ψ N : { R d NS d = {x R d+ : x = N} x Nψ( x N ) It is show i [5, Remark 3] that give a fuctio h : B(, N) = {x R d : x N} R, there exists a uique fuctio h N supported i {x NS d : arccos(n x d+ ) } such that h N (ψ N x) = h(x), x B(, N) Moreover, it is also show there that (3) h N (Nx)h λ κ (x)dx = N S d κ B(,N) ( ) λ si( x /N) h(x)h κ κ(x) dx x /N with λ κ = λ κ + = d + γ κ Let ow N be so large that both f ad g are supported i B(, N) Let h(x) := f(x, ) Y (Ω ) ad f N : NS d Y be the fuctio defied by f N (ψ N (x), ω ) = f(x, ω ), so that we have h N (x) = f N (x, ) Y (Ω ) As metioed i [5, p 464], it follows from (3) that h N (N ) Lp (S d,h κ ) = f N (N ) Lp (S d,h κ ;Y ) N λ κ + p Similarly, with h(x) := g(x, ) Y (Ω ), it follows that g N (N ) L p (S d,h κ ;Y ) N λ κ + p 5

26 Recall i the followig that P κ (x, y) is the kerel of proj κ We deduce that [ ] N λ κ + m(n )P κ (x, y) f N (Ny, ω )g N (Nx, ω )h κ (x)h κ (y)dxdydµ(ω ) Ω S d S d = N λ κ + M /N f N (Nx, ω ) g N (Nx, ω ) dxdµ(ω ) Ω S d N λ κ + M /N f N (N, ) κ,p;y g N (N, ) κ,p ;Y N λ κ + can λ κ + p N λ κ + p = ca Deote the expressio uder the modulus i the first lie of the above estimate by I N It is show i [5, Proof of Theorem 3] that lim N I N = c δ,κ I holds i the case Y = Y = C The it also holds poitwise o Ω, ad thus, sice we take itegrals over Ω both i the defiitio of I ad I N, also for geeral Baach lattices Y (Ω ) ad Y (Ω ) The lie (3) follows immediately, ad we have proved Theorem 3 i the case where m(t) c e ct We ow prove the theorem removig this assumptio o m Let thus m be a geeral multiplier fuctio satisfyig the hypotheses of Theorem 3 For δ >, put m δ (t) = m(t)e δt The, for ay f S(R d ) Y, we have T mδ (f) = P κ δ T m (f), where (P κ t ) t is the Dukl Poisso symmetric cotractio semigroup o L p (R d, h κ; Y ), ad we have P κ δ f f as δ, because accordig to Propositio 5, (Pκ t ) t is strogly cotiuous o L p (R d, h κ; Y ) O the oe had, we have by the first part of the proof T mδ L p (R d,h κ ;Y ) Lp (R d,h κ ;Y ) sup M (m(ɛ)e ɛδ ) Lp (S d,h κ ɛ> ;Y ) Lp (S d,h κ ;Y ) O the other had, we write A T m (f) Lp (R d,h κ ;Y ) = lim δ P κ δ T m (f) Lp (R d,h κ ;Y ) = lim δ T mδ (f) Lp (R d,h κ ;Y ) A f Lp (R d,h κ ;Y ) The proof of the theorem is complete We shall ow prove several importat cosequeces of this theorem The first oe provides us a Hörmader type multiplier theorem for the Dukl trasform Theorem 33 Let < p < ad Y = Y (Ω ) be a UMD Baach lattice Let be a iteger > λ κ + = λ κ + 3 = d + γ κ + Assume that the multiplier fuctio m : (, ) R belogs to H, that meas, is bouded with m A ad satisfies the followig Hörmader coditio sup R> R R R t d dt m(t) dt A The the spectral multiplier T m, iitially defied for f S(R d ) Y by [ T m (f) = Fκ m ( ξ ) ] F κ f(ξ) exteds to a bouded operator o L p (R d, h κ; Y ) with T m Lp (R d,h κ ;Y ) Lp (R d,h κ ;Y ) c p,,da 6

27 This theorem, which geeralizes the scalar case proved by Dai-Wag [5, Theorem 4], oly cocers radial multipliers For scalar valued multivariate (ie ot ecessarily radial) spectral multipliers for Dukl operators, but oly i the particular case where the reflectio group W is (Z/Z) d, see [58] Proof : It is show i [5, Proof of Theorem 4] that the above coditio o m yields that the sequece (µ ) defied by µ = m(ɛ) satisfies the hypotheses (C ) ad (C ) of Theorem 3 with a boud M ca uiformly i ɛ > Note that i Theorem 3, we take S d i place of S d as it is stated there verbatim The Theorem 3 yields that the hypotheses of Theorem 3 are satisfied ad we apply it to get the desired coclusio From Theorem 33, we ca also immediately deduce the followig corollary o Bocher- Riesz meas Corollary 34 Let < p < ad Y = Y (Ω ) be a UMD Baach lattice Let α > λ κ + For R >, let { fr(t) α ( t/r) α < t R = t > R The the Bocher-Riesz meas fr α (A) associated with the Dukl Laplacia A are uiformly bouded i R > o L p (R d, h κ; Y ) Proof : Note that by Lemma 8 ad Theorem 33, A has a H calculus o L p (R d, h κ; Y ) for a iteger > λ κ + It suffices to ote that fr α H α+ ɛ < for ɛ > (see eg [38]) ad to apply Theorem 33 Note that there is a partial coverse of Corollary 34 More precisely, if the Bocher-Riesz meas {fr α(a) : R > } of a sectorial operator A with H calculus are R-bouded, the A must have a H α+ calculus (eve R-bouded, ie {f(a) : f H α+ } is a R-bouded subset of B(L p (R d, h κ; Y ))), accordig to [38] Aother applicatio of Theorem 33 is the followig spectral decompositio of Paley- Littlewood type We refer eg to [37] for applicatios of this decompositio to the descriptio of complex ad real iterpolatio spaces associated with a abstract operator (the Dukl Laplacia i our case) Corollary 35 Let < p < ad Y = Y (Ω ) be a UMD Baach lattice Let (φ ) Z be a dyadic partitio of uity (see Defiitio 9) Further let ψ = φ for ad ψ = = φ, so that = ψ (t) = for all t > Deote A = κ the Dukl Laplacia The, for ay f L p (R d, h κ; Y ), we have the orm descriptio ( f = φ (A)f ( = ψ (A)f Z = Proof : Oce a Hörmader calculus of A o L p (R d, h κ ; Y ) is guarateed by Theorem 33, the corollary follows from [37, Theorem 4] resp (), to decompose the orm i Rademacher sums resp square sums A Hörmader calculus implies by Lemma 7 that the Dukl Laplacia A has a H (Σ ω ) calculus for ay ω (, π ) We deduce the followig Corollary 36 Let < p < ad Y = Y (Ω ) be a UMD Baach lattice Let ψ (, π ) For f L p (R d, h κ; Y ), defie the maximal fuctio M ψ (f)(x) = sup z Σψ exp( za)f(x, ) Y The M ψ (f) κ,p C ψ f 7