Mixed norm estimates for the Riesz transforms associated to Dunkl harmonic oscillators
aa r X i v : . [ m a t h . F A ] J u l MIXED NORM ESTIMATES FOR THE RIESZTRANSFORMS ASSOCIATED TO DUNKL HARMONICOSCILLATORS
PRADEEP BOGGARAPU AND S. THANGAVELU
Abstract.
In this paper we study weighted mixed norm estimates forRiesz transforms associated to Dunkl harmonic oscillators. The idea isto show that the required inequalities are equivalent to certain vectorvalued inequalities for operator defined in terms of Laguerre expansions.In certain cases the main result can be deduced from the correspondingresult for Hermite Riesz transforms. Introduction
Let G be a Coxeter group (finite reflection group) associated to a rootsystem R in R d , d ≥
2. We use the notation h ., . i for the standard innerproduct on R d . Let κ be a multiplicity function which is assumed to benon-negative and let h κ ( x ) = Y ν ∈ R + |h x, ν i| κ ( ν ) where R + is the set of all positive roots in R . Let T j , j = 1 , , . . . , d be thedifference-differential operators defined by T j f ( x ) = ∂f∂x j ( x ) + X ν ∈ R + κ ( ν ) ν j f ( x ) − f ( σ ν x ) h ν, x i . where σ ν is the reflection defined by ν . The Dunkl Laplacian ∆ κ is thendefined to be the operator ∆ κ = d X j =1 T j which can be explicitly calculated, see Theorem 4.4.9 in Dunkl-Xu [8]. TheDunkl harmonic oscillator is then defined by H d,κ = − ∆ κ + | x | which reduces to the Hermite operator H d = − ∆ + | x | when κ = 0. Mathematics Subject Classification.
Primary: 42C10, 47G40, 26A33. 43A90. Sec-ondary: 42B20, 42B35, 33C44.
Key words and phrases.
Reflection groups, Dunkl operators, Hermite and generalisedHermite functions, Riesz transforms, singular integrals, weighted inequalities.
Our aim in this paper is to study the L p mapping properties of Riesztransforms associated to the Dunkl harmonic oscillator. The spectral theoryof the operator H d,κ has been developed by R¨osler in [17]. The eigenfunctionsof H d,κ are called the generalised Hermite functions and denoted by Φ κµ , µ ∈ N d . It has been proved that they form an orthonormal basis for L ( R d , h κ dx ) . In analogy with the Riesz transforms associated to the Hermite operator,one can define the Riesz transforms R κj , R κ ∗ j , j = 1 , , . . . , d by R κj = (cid:16) T j + x j (cid:17) H − d,κ , R κ ∗ j = (cid:16) − T j + x j (cid:17) H − d,κ . Note that the operators R κj and R κ ∗ j are densely defined i.e., they are definedon the subspace V consisting of finite linear combinations of the generalisedHermite functions Φ κα . In the particular case of G = Z d treated in [15] theauthors have shown that the L norm of ( T j + x j )Φ κα behaves like (2 | α | + d + 2 γ ) / where γ = P ν ∈ R + κ ( ν ) . Since Φ κα are eigenfunctions of H d,κ witheigenvalues (2 | α | + d + 2 γ ) the operator H − d,κ defined by spectral theoremsatisfies H − d,κ Φ κα = (2 | α | + d + 2 γ ) − / Φ κα . From these two facts, it is clearthat the Riesz transforms defined on V satisfy the inequalities k R κj f k ≤ C k f k , k R κ ∗ j f k ≤ C k f k for all f ∈ V. Consequently, they extend to L as bounded linear operators.In [1] a very cute argument based on the fact that H d,κ = 12 d X j =1 (( T j + x j )( − T j + x j ) + ( − T j + x j )( T j + x j ))is used to show that the L boundedness on V holds for any reflection group G. We make use of these definitions and results in the sequel.If it can be shown that R κj and R κ ∗ j satisfy the inequalities k R κj f k p ≤ C k f k p , k R κ ∗ j f k p ≤ C k f k p for any 1 < p < ∞ whenever f ∈ V then by density arguments they can beextended to the whole of L p ( R d , h κ dx ), 1 < p < ∞ as bounded operators.This was proved in [15] by Nowak and Stempak in the particular case when G = Z d . For general Coxeter groups the boundedness properties of theRiesz transforms are proved by Amri in [1]. We refer to these two papers fordetails and further information on Riesz transforms associated to the Dunklharmonic oscillator. Weighted norm inequalities or mixed norm inequalitiesare not known for these Riesz transforms. In this paper our main goal is toestablish certain weighted mixed norm estimates for these operators.For α ≥ − , let A αp ( R + ) be the Muckenhoupt’s class of A p -weights on R + associated to the doubling measure dµ α ( t ) = t α +1 dt . Let dσ be the surfacemeasure on unit sphere S d − and let w be a positive function on R + . We denote by L p, ( R d , w ( r ) r d +2 γ − h κ ( ω ) dσ ( ω ) dr ) the space of all measurablefunctions f on R d for which Z ∞ (cid:16) Z S d − | f ( rω ) | h κ ( ω ) dσ ( ω ) (cid:17) p w ( r ) r d +2 γ − dr < ∞ . The p − th root of the above quantity is a norm with respect to which thespace becomes a Banach space. For 1 < p < ∞ the dual of the Ba-nach space L p, ( R d , w ( r ) r d +2 γ − h κ ( ω ) dσ ( ω ) dr ) is nothing but the space L p ′ , ( R d , w ( r ) r d +2 γ − h κ ( ω ) dσ ( ω ) dr ) where p ′ is the index conjugate to p. This follows from a general theorem proved in [5] since we can think of thespace L p, ( R d , w ( r ) r d +2 γ − h κ ( ω ) dσ ( ω ) dr ) as an L p space on R + of functionstaking values in the Hilbert space L ( S d − , h κ ( ω ) dσ ( ω )) taken with respectto the measure w ( r ) r d +2 γ − dr. Since L ( S d − , h κ ( ω ) dσ ( ω )) is a separableHilbert space, it can be identified with the sequence space l ( N ) and hence asimple independent proof also can be given for the fact about the dual. Wedenote by L p, G ( R d , w ( r ) r d +2 γ − h κ ( ω ) dσ ( ω ) dr ) the subspace of G -invariantfunctions in L p, ( R d , w ( r ) r d +2 γ − h κ ( ω ) dσ ( ω ) dr ) . Let V G stand for the set of all G -invariant functions in V. To see that V G is a nontrivial subspace of V we proceed as follows. Given a function f on R d we define the G-invariant function f by averaging over G. Thus f ( x ) = 1 | G | X g ∈ G f ( gx )where | G | stands for the cardinality of G . We claim that V G is preciselythe set of all f where f runs through V. Indeed, it is obvious that for any G -invariant f ∈ V we have f = f . On the other hand, if f ∈ V then f is G -invariant and f belongs to V . The latter can be easily seen as follows:Since f ∈ V , it is of the form f ( x ) = X α ∈ F c α Φ κα ( x ) , where F is a finite subset of N d . Since H d,κ is G -invariant and H d,κ Φ κα =(2 | α | + d + 2 γ )Φ κα (see section 2.2 below) it follows that H d,κ (Φ κα ) = (2 | α | + d + 2 γ )(Φ κα ) . Note that H d,κ is a self-adjoint operator with discrete spectrum. Moreover,each eigenspace is finite dimensional and { Φ κα : α ∈ N d } is an orthonormalbasis for L ( R d , h κ dx ) consisting of eigenfunctions of H d,κ . Consequently,(Φ κα ) which is an eigenfunction of H d,κ can be written as P | β | = | α | a β Φ κβ .This shows that f = P α ∈ F c α (Φ κα ) belongs to V . This proves our claim.Also note that the density of V in L p, ( R d , w ( r ) r d +2 γ − h κ ( ω ) dσ ( ω ) dr )implies the density of V G in L p, G ( R d , w ( r ) r d +2 γ − h κ ( ω ) dσ ( ω ) dr ) which is animmediate consequence of Minkowski’s inequality since the measures given PRADEEP BOGGARAPU AND S. THANGAVELU by h κ ( ω ) dσ ( ω ) and w ( r ) r d +2 γ − dr are G - invariant. In Subsection 2.4 wewill show that V is dense in L p, ( R d , w ( r ) r d +2 γ − h κ ( ω ) dσ ( ω ) dr ) for all w ∈ A d + γ − p ( R + ) , < p < ∞ . Thus, R κj and R κ ∗ j are well defined on the densesubspace V G . Theorem 1.1.
Let d ≥ , < p < ∞ . Then for j = 1 , , · · · , d the Riesztransforms R κj and R κ ∗ j initially defined on V G satisfy the estimates Z ∞ (cid:16) Z S d − | R κj f ( rω ) | h κ ( ω ) dσ ( ω ) (cid:17) p w ( r ) r d +2 γ − dr ≤ C j ( w, p, κ ) Z ∞ (cid:16) Z S d − | f ( rω ) | h κ ( ω ) dσ ( ω ) (cid:17) p w ( r ) r d +2 γ − dr for all f ∈ V G , w ∈ A d + γ − p ( R + ) . Consequently R κj and R κ ∗ j can be extendedas a bounded linear operators from L p, G ( R d , w ( r ) r d +2 γ − h κ ( ω ) dσ ( ω ) dr ) into L p, ( R d , w ( r ) r d +2 γ − h κ ( ω ) dσ ( ω ) dr ) . The proof of this theorem is based on the fact that on radial functions theDunkl harmonic oscillator H d,κ coincides with the Hermite operator H d +2 γ ,when 2 γ is an integer. More generally, using an analogue of Funk-Heckeformula for h-harmonics we can show that the mixed norm estimates forthe Riesz transforms R κj are equivalent to a vector valued inequality for asequence of Laguerre Riesz transforms. When 2 γ is an integer these in-equalities can be deduced from the weighted norm inequalities satisfied byHermite Riesz transforms. In the general case when 2 γ is not an integer, wecan appeal to a recent result of Ciaurri and Roncal [6].The plan of the paper is as follows. In Section 2 we collect some facts fromthe spectral theory of Dunkl harmonic oscillators. Especially, we need ananalogue of Mehler’s formula for the generalised Hermite functions. Wealso collect some basic facts about h-harmonics which are analogues ofspherical harmonics on S d − . The most important result is an analogueof Funk-Hecke formula for h-harmonics. In Section 3 we consider the vec-tor R κ f = ( R κ , · · · , R κd ) of Riesz transforms and show that mixed norminequalities for |R κ f | = (cid:16) P dj =1 | R κj f | (cid:17) can be reduced to vector valuedinequalities for operators related to Laguerre expansions. In Section 4 weprove the required inequalities by considering the vector of Hermite Riesztransforms.Though we have considered only the Riesz transforms in this paper, wecan also treat multipliers (e.g. Bochner-Riesz means) for the Dunkl har-monic oscillator. Using the known results for the Hermite operator, wecan prove an analogue of Theorem 1.1 for multipliers associated to Dunklharmonic oscillator. Preliminaries
Coxeter groups and Dunkl operators:
We assume that the readeris familiar with the notion of finite reflection groups associated to root sys-tems. Given a root system R we define the reflection σ ν , ν ∈ R by σ ν x = x − h ν, x i| ν | ν. Recall that h ν, x i is the inner product on R d . These reflections σ ν , ν ∈ R generate a finite group which is called a Coxeter group. A function κ definedon R is called a multiplicity function if it is G invariant. We assume thatour multiplicity function κ is non negative. The Dunkl operators T j definedby T j f ( x ) = ∂∂x j f ( x ) + X ν ∈ R + κ ( ν ) ν j f ( x ) − f ( σ ν x ) h ν, x i . form a commuting family of operators. There exists a kernel E κ ( x, ξ ) whichis a joint eigenfunction for all T j : T j E κ ( x, ξ ) = ξ j E κ ( x, ξ ) . This is the analogue of the exponential e h x, ξ i and Dunkl transform is definedin terms of E κ ( ix, ξ ). For all these facts we refer to Dunkl [7] and Dunkl-Xu[8]. The weight function associated to R and κ is defined by h κ ( x ) = Y ν ∈ R + |h x, ν i| κ ( ν ) . Recall that γ = P ν ∈ R + κ ( ν ) and the multiplicity function κ ( ν ) is alwaysassumed to be non-negative. We consider L p spaces defined with respect tothe measure h κ ( x ) dx . Note that h κ ( x ) is homogeneous of degree 2 γ .2.2. Generalised Hermite functions:
In [17] R¨osler has studied gener-alised Hermite polynomials associated to Coxeter groups. She has shownthat there exists an orthonormal basis Φ κα , α ∈ N d for L ( R d , h κ ( x ) dx ) con-sisting of functions for which Φ κα ( x ) e | x | are polynomials. Moreover, theyare eigenfunctions of the Dunkl harmonic oscillator: (cid:16) − ∆ κ + | x | (cid:17) Φ κα = (2 | α | + d + 2 γ )Φ κα . They are also eigenfunctions of the Dunkl transform. For our purpose, themost important result is the generating function identity or the Mehler’sformula for the generalised Hermite functions. For 0 < r <
1, one has X α ∈ N d Φ κα ( x )Φ κα ( y ) r | α | = c d (1 − r ) − d − γ e − (cid:16) r − r (cid:17) ( | x | + | y | ) E κ (cid:16) rx − r , y (cid:17) PRADEEP BOGGARAPU AND S. THANGAVELU see Theorem 3.12 in [17]. By taking r = e − t , t > − ∆ κ + | x | is given by(2.1) K t ( x, y ) = c d,γ (sinh 2 t ) − d − γ e − (coth 2 t )( | x | + | y | ) E κ (cid:16) x sinh 2 t , y (cid:17) . We will make use of this kernel in the study of Riesz transforms.Recall that the subspace V defined in the introduction is the algebraicspan of the generalised Hermite functions Φ κα , α ∈ N d . As every Φ κα is aSchwartz function it follows that elements of V are also of Schwartz class.It is known that V is dense in L p ( R d , h κ ( x ) dx ), 1 ≤ p < ∞ . Indeed, in [22]the authors have shown that Bochner-Riesz means S δR f , for large enough δ ,converge to f in the norm as R → ∞ as long as 1 ≤ p < ∞ . Since S δR f ∈ V for any f ∈ L p ( R d , h κ ( x ) dx ) it follows that V is dense in L p ( R d , h κ ( x ) dx ).The same thing can be proved using the fact that the heat semigroup, e − tH d,κ generated by H d,κ is strongly continuous in each of L p ( R d , h κ ( x ) dx ), 1 ≤ p < ∞ . We also need to know the density of V in certain weighted L p spaces.This will be addressed in subsection 2.4 below.2.3. h-harmonics and Funk-Hecke formula: The best reference for thissection is Chapter 5 of [8]. For the space L ( S d − , h κ ( ω ) dσ ( ω )) there ex-ists an orthonormal basis consisting of h-harmonics. These are analogues ofspherical harmonics and defined using ∆ κ in place ∆. A homogeneous poly-nomial P ( x ) is said to be a solid h-harmonic if ∆ κ P ( x ) = 0. Restrictions ofsuch solid harmonics to S d − are called spherical h-harmonics. The space L ( S d − , h κ dσ ) is the orthogonal direct sum of the finite dimensional spaces H dm consisting of h-harmonics of degree m . We can choose an orthonor-mal basis Y hm,j , j = 1 , , . . . , d ( m ), d ( m ) = dim ( H hm ) so that the collection { Y hm,j : j = 1 , , . . . , d ( m ) , m = 0 , , , . . . } is an orthonormal basis for L ( S d − , h κ dσ ).In order to state the Funk-Hecke formula we need to recall the inter-twining operator. It has been proved that there is an operator V satisfying T j V = V ∂∂x j . The explicit form of V is not known, except in a couple of sim-ple cases, but it is a useful operator. In particular, the Dunkl kernel is givenby E κ ( x, ξ ) = V e h· , ξ i ( x ). The operator V also intertwines h-harmonics (seeProposition 5.2.8 of [8]).The classical Funk-Hecke formula for spherical harmonics states the fol-lowing. For any continuous function f on [ − ,
1] and a spherical harmonic Y m of degree m , one has the formula Z S d − f ( h x ′ , y ′ i ) Y m ( y ′ ) dσ ( y ′ ) = λ m ( f ) Y m ( x ′ ) where λ m ( f ) is a constant defined by λ m ( f ) = B ( d − , ) C d − m (1) Z − f ( t ) C d − m ( t )(1 − t ) d − dt. Here C λm stand for ultraspherical polynomials of type λ and B ( r, s ) stands forthe beta function. A similar formula is true for h-harmonics (see Theorem5.3.4 in [8]); Z S d − V f ( x ′ , · )( y ′ ) Y hm ( y ′ ) h κ ( y ′ ) dσ ( y ′ ) = λ m ( f ) Y hm ( x ′ )where λ m ( f ) = B ( d − + γ, ) C d − γm (1) Z − f ( t ) C d − γm ( t )(1 − t ) d − + γ dt. Let J δ ( z ) stand for Bessel function of type δ > − I δ ( z ) = e − i π δ J δ ( iz ). If we take f ( t ) = e itz in the above we get B ( d − + γ, ) C d − γm (1) Z − e itz C d − γm ( t )(1 − t ) d − + γ dt = c d,γ J d + γ + m − ( z ) z d + γ − (see page 204-205 in [3]). By taking f ( t ) = e t | x | | y | and making use of theabove formula we get Z S d − E κ ( x, y ) Y hm ( y ′ ) h κ ( y ′ ) dσ ( y ′ ) = c d,γ I d + γ + m − ( | x | | y | )( | x | | y | ) d + γ − Y hm ( x ′ ) . In view of this and (2.1) we have Z S d − K t ( rx ′ , sy ′ ) Y hm ( y ′ ) h κ ( y ′ ) dσ ( y ′ )(2.2) = c d,γ (sinh 2 t ) − e − (coth 2 t )( r + s ) I d + γ + m − ( rs sinh 2 t )( rs ) d + γ − Y hm ( x ′ ) . We will make use of this formula in calculating the action of e − tH d,κ onfunctions of the form g ( r ) Y hm ( x ′ ) . The density of V : In this subsection we take up the issue of prov-ing the density of V, defined in the introduction, in the weighted mixednorm spaces L p, ( R d , w ( r ) r d +2 γ − h κ ( ω ) dσ ( ω ) dr ) for 1 < p < ∞ and w ∈ A d/ γ − p ( R + ) . In order to do this we will make use of the Laguerre connec-tion. For each δ ≥ − we consider the Laguerre differential operator L δ = − d dr + r − δ + 1 r ddr whose normalised eigenfunctions are given by ψ δk ( r ) = (cid:16) k + 1)Γ( k + δ + 1) (cid:17) L δk ( r ) e − r PRADEEP BOGGARAPU AND S. THANGAVELU where L δk ( r ) are Laguerre polynomials of type δ . These functions form anorthonormal basis for L ( R + , dµ δ ), where dµ δ ( r ) = r δ +1 dr . The operator L δ generates the semigroup T δt = e − tL δ whose kernel is given by K δt ( r, s ) = ∞ X k =0 e − (4 k +2 δ +2) t ψ δk ( r ) ψ δk ( s ) . (2.3)The generating function identity ((1.1.47) in [21]) for Laguerre functionsgives the explicit expression K δt ( r, s ) = (sinh 2 t ) − e − (coth 2 t )( r + s ) ( rs ) − δ I δ (cid:16) rs sinh 2 t (cid:17) (2.4)where I δ ( z ) = e − i π δ J δ ( iz ) is the modified Bessel function.The Dunkl-Hermite semigroup e − tH d,κ generated by the operator H d,κ isan integral operator given by e − tH d,κ f ( x ) = Z R d f ( y ) K t ( x, y ) h κ ( y ) dy where K t ( x, y ) is the kernel defined in (2.1). The relation between thissemigroup and the Laguerre semigroups T δt = e − tL δ is given by the followingproposition. In what follows, Y hm,j , j = 1 , , · · · , d ( m ), m = 0 , , , · · · standsfor the orthonormal basis for L ( S d − , h κ ( ω ) dσ ( ω )) described in subsection2.3. Proposition 2.1.
For any Schwartz class function f on R d let e f m,j ( r ) = r − m Z S d − f ( rω ) Y hm,j ( ω ) h κ ( ω ) dσ ( ω ) . Then we have the relation Z S d − e − tH d,κ f ( rω ) Y hm,j ( ω ) h κ ( ω ) dσ ( ω ) = c d,γ r m (cid:16) T d/ m + γ − t e f m,j (cid:17) ( r ) . The proof of this proposition is immediate from the expressions (2.1) and(2.4) for the kernels of e − tH d,κ and T d/ m + γ − t and the Funk-Hecke formula.We make use of the following Lemma in order to prove that V is a densesubspace of L p, ( R d , w ( r ) r d +2 γ − h κ ( ω ) dσ ( ω ) dr ) . That V is a subspace fol-lows immediately from the lemma as every member of V being a finite linearcombination of Φ κα is a Schwartz class function. Lemma 2.2.
Let ≤ p < ∞ and f be a Schwartz class function on R d .Then Z ∞ (cid:16) Z S d − | f ( rω ) | h κ ( ω ) dσ ( ω ) (cid:17) p w ( r ) r d +2 γ − dr < ∞ whenever w ∈ A d/ γ − p ( R + ) . Proof.
First we observe that if f is a Schwartz class function on R d , thenthe function f ( r ) := (cid:16) R S d − | f ( rω ) | h κ ( ω ) dσω (cid:17) is a continuous functionon R + and for every positive integer N there exists C N > f ( r ) ≤ C N (1 + r ) − N for all r ∈ R + . To prove the lemma, it is enough toprove that Z ∞ ( f ( r )) p w ( r ) r d +2 γ − dr < ∞ . Let δ = d/ γ − Z ∞ ( f ( r )) p w ( r ) r δ +1 dr = (cid:18)Z + Z ∞ (cid:19) (( f ( r )) p w ( r ) r δ +1 dr ) . The first integral on the left hand side of the above is finite as f is continuousand w is locally integrable. And the second integral can be written as ∞ X j =1 Z j − ≤ r< j ( f ( r )) p w ( r ) r δ +1 dr which can be bounded by ∞ X j =1 ( f ( r j )) p Z j w ( r ) r δ +1 dr where r j ∈ [2 j − , j ] are the points at which f attains maximum on [2 j − , j ].Such r j ’s exist in the closed interval [2 j − , j ], since f is continuous. The A p -weight condition on w implies R R w ( r ) r δ +1 dr ≤ CR p ( δ +1) for R >
0, seepage 252, Eqn.16 in [13]. Choose a positive integer N such that N > δ +1).Finally we see that Z ∞ ( f ( r )) p w ( r ) r δ +1 dr ≤ ∞ X j =1 ((1 + r j ) N f ( r j )) p (1 + r j ) − Np pj ( δ +1) ≤ C ∞ X j =1 − jNp pj ( δ +1) ≤ C ∞ X j =1 − jp ( N − δ +1)) < ∞ . The second inequality in the above is due to the facts that (1 + r ) N f ( r ) isbounded on R + and 1 + r j ≥ j − . This proves the lemma. (cid:3) We are now in a position to prove the density of V in the mixed norm space L p, ( R d , w ( r ) r d +2 γ − h κ ( ω ) dσ ( ω ) dr ) for 1 < p < ∞ , w ∈ A d/ γ − p ( R + ).If V is not dense in L p, ( R d , w ( r ) r d +2 γ − h κ ( ω ) dσ ( ω ) dr ), by duality thereexists a nontrivial function f ∈ L p ′ , ( R d , w ( r ) r d +2 γ − h κ ( ω ) dσ ( ω ) dr ) (where p + p ′ = 1) such that(2.5) Z R d f ( y )Φ κα ( y ) w ( | y | ) h κ ( y ) dy = 0for all α ∈ N d . Since w ∈ A d/ γ − p ( R + ) if and only if w − p ′ ∈ A d/ γ − p ′ ( R + )it follows that the function g defined by g ( y ) = f ( y ) w ( | y | ) belongs to L p ′ , ( R d , w − p ′ ( r ) r d +2 γ − h κ ( ω ) dσ ( ω ) dr ). Since the heat kernel K t ( x, y ) isa Schwartz function, it follows from Lemma 2.2 that e − tH d,κ g is well de-fined. Moreover, by Mehler’s formula e − tH d,κ g ( x ) = X α ∈ N d e − (2 | α | + d +2 γ ) t (cid:18)Z R d g ( y )Φ κα ( y ) h κ ( y ) dy (cid:19) Φ κα ( x ) . Consequently, e − tH d,κ g = 0 for all t > m = 0 , , , . . . , j = 1 , , · · · , d ( m )( T d/ m + γ − t e g m,j )( r ) = 0 . Hence we only need to conclude that the above implies e g m,j = 0 for all m and j which leads to a contradiction.But this follows from the theory of Laguerre semigroups. Indeed, whatwe have is Z ∞ ( rs ) m K d/ m + γ − t ( r, s ) f m,j ( s ) w ( s ) s d +2 γ − ds = 0 . Here w ∈ A d/ γ − p ( R + ) and f m,j ∈ L p ′ ( R + , w ( s ) dµ d/ γ − ( s )). Once again,the above can be rewritten as Z ∞ ( rs ) m K d/ m + γ − t ( r, s ) g m,j ( s ) s d +2 γ − ds = 0for all t > . Note that the function g m,j ( s ) = f m,j ( s ) w ( s ) belongs to L p ′ ( R + , w − p ′ ( s ) dµ d/ γ − ( s )) with w − p ′ ∈ A d/ γ − p ′ ( R + ). Invoking thefact that the modified Laguerre semigroup e T td/ m + γ − defined by e T td/ m + γ − h ( r ) = Z ∞ ( rs ) m K d/ m + γ − t ( r, s ) h ( s ) s d +2 γ − ds is strongly continuous on L p ′ ( R + , u ( s ) s d +2 γ − ds ) for any u ∈ A d/ γ − p ′ ( R + )we conclude that g m,j = 0 for all m and j .Finally, we briefly indicate how the strong continuity of e T td/ m + γ − canbe proved. It is almost trivial to prove that the kernel of this semigroupsatisfies the estimates stated in Proposition 3.4 of Ciaurri-Roncal [6]. Ac-tually, we need not care about the uniformity in m . These estimates inturn can be used to prove that e T td/ m + γ − f is dominated by the maximalfunction M d/ γ − f adapted to the homogeneous space ( R + , dµ d/ γ − ). As this maximal function is known to be bounded on L p ( R + , wdµ d/ γ − ), w ∈ A d/ γ − p ( R + ), see e.g. Duoandikoetxea [9], we conclude that e T td/ m + γ − is strongly continuous on L p ( R + , wdµ d/ γ − ), w ∈ A d/ γ − p ( R + ), 1 < p < ∞ . This completes the proof.3. Riesz transforms for the Dunkl harmonic Oscillator κ = 0, we define the Riesz transforms R κj , j = 1 , , . . . , d associated to theDunkl harmonic oscillator H d,κ by R κj f = ( T j + x j ) H − d,κ f where H − d,κ is defined by spectral theorem. More precisely, H − d,κ f = X α (2 | α | + 2 γ + d ) − ( f, Φ κα )Φ κα where Φ κα are the generalised Hermite functions and ( f, Φ κα ) = R R d f ( x )Φ κα ( x ) h κ ( x ) dx . We can also define R κ ∗ j by as in the Hermite case. It is easy tosee that R κj are bounded on L ( R d , h κ ( x ) dx ), see Proposition 2.1 in [1] . Inthe same paper, Amri has proved that R κj are singular integral operatorswhose kernels satisfy a modified Calderon-Zygmund condition and hence bya theorem of Amri and Sifi [2] they are all bounded on L p ( R d , h κ ( x ) dx ),1 < p < ∞ .In the case of Hermite operator, the Riesz transforms satisfy weightednorm estimates. More precisely, if w ∈ A p ( R d ), then R j are bounded on L p ( R d , wdx ), 1 < p < ∞ . This has been proved by Stempak and Torrea [20]and it follows from the fact that the kernels of R j satisfy standard Calderon-Zygmund conditions. In the present situation we do not have weighted in-equalities for the Riesz transforms R κj . Later we will show that the weightedinequalities for R j can be used to prove mixed norm inequalities for the Her-mite Riesz transforms which will then be used to prove similar results for R κj .Assume that 2 γ is an integer. Then the action of H d,κ on radial functionscoincides with that of H d +2 γ on radial functions. More generally, let f ( x ) = g ( r ) Y h ( ω ), r = | x | , ω ∈ S d − where Y h is h-harmonic of degree m . ThenMehler’s formula for the generalised Hermite functions along with Funk-Hecke formula yields the result e − tH d,κ f ( x ) = ce − tH d +2 γ +2 m g ( | x | ) Y h ( ω )where on the right hand side g is considered as a radial function on R d +2 γ +2 m .It is also possible to write e − tH d +2 γ +2 m g ( | x | ) in terms of Laguerre semigroup.We will make use of these observations in the proof of our main result. More on h-harmonics:
As indicated in the previous subsection, weplan to expand the given function f on R d in terms of h-harmonics. In orderto find out the action of Riesz transforms on individual terms which are ofthe form g ( | x | ) Y hm ( ω ) we need formulas for the action of T j on such terms.More generally we let ∇ κ = ( T , T , · · · , T d ) stand for the Dunkl gradientwhich is the sum of the gradient ∇ = (cid:16) ∂∂x , · · · , ∂∂x d (cid:17) and E κ where E κ f ( x ) = X ν ∈ R + κ ( ν ) f ( x ) − f ( σ ν x ) h x, ν i ν. Let ∇ be the spherical part of ∇ . Then the Dunkl gradient is written as ∇ κ = ω ∂∂r + 1 r ∇ κ with ∇ κ = ∇ + E κ standing for the spherical part of the Dunkl gradient,where E κ f ( ω ) = P ν ∈ R + κ ( ν ) f ( ω ) − f ( σ ν ω ) h ω,ν i ν for functions f defined on S d − .For ξ ∈ R d , let T ξ stand for the Dunkl derivative given by T ξ f = ∂ ξ f + X ν ∈ R + κ ( ν ) h ν, ξ i f ( x ) − f ( σ ν x ) h x, ν i . If one of f and g is G -invariant then T ξ ( f g ) = f T ξ g + ( T ξ f ) g remains true. Moreover, we also know that Z R d T ξ f ( x ) g ( x ) h κ ( x ) dx = − Z R d f ( x ) T ξ g ( x ) h κ ( x ) dx. In view of this we get Z R d h∇ κ f ( x ) , ∇ κ g ( x ) i h κ ( x ) dx = − Z R d ∆ κ f ( x ) g ( x ) h κ ( x ) dx We will make use of these properties in the following calculation.We begin with some simple observations. When f is a radial function wehave ∇ κ ( f g ) = f ∇ κ g + g ∇ κ f and consequently ∇ κ ( f g )( rω ) = f ( r ) ∇ κ g ( rω ) + g ( rω ) ∂f∂r ω. (3.1)Let Y m be a homogeneous polynomial of degree m on R d . Then d X j =1 ( ∇ ) j ( ω j Y m ( ω )) = ( d − Y m ( ω )(3.2) where ( ∇ ) j stand for the j th component of ∇ . To see this, consider d X j =1 ∂∂x j ( x j Y m ( x )) = dY m ( x ) + d X j =1 x j ∂∂x j Y m ( x ) = ( m + d ) Y m ( x )in view of Euler’s formula. On the other hand d X j =1 ∂∂x j ( x j Y m ( x )) = d X j =1 ∂∂x j ( r m +1 ω j Y m ( ω )) . Since ∂∂x j = ω j ∂∂r + r ( ∇ ) j it follows that d X j =1 ∂∂x j ( x j Y m ( x )) = ( m + 1) Y m ( x ) + d X j =1 r m ( ∇ ) j ( ω j Y m ( ω )) . Comparing this with the earlier expression we get the assertion.
Proposition 3.1.
Let Y n and Y m be h-harmonic polynomials of degree n and m respectively. Then we have the following identities. Let ρ κ Y n ( ω ) = P ν ∈ R + κ ( ν ) Y n ( σ ν ω ) . (1) h∇ κ Y n ( ω ) , ω i = γY n ( ω ) − ρ κ Y n ( ω )(2) h∇ κ Y n ( x ) , ω i = r n − (( n + γ ) Y n ( ω ) − ρ κ Y n ( ω ))(3) P dj =1 ( ∇ κ ) j ( ω j Y n ( ω )) = ( d + γ − Y n ( ω ) + ρ κ Y n ( ω )(4) R S d − h∇ κ Y n ( ω ) , ∇ κ Y m ( ω ) i h κ ( ω ) dσ ( ω ) = 0 if n = m. Proof. (1) follows from the definition of ∇ κ = ∇ + E κ and the fact that h∇ Y n ( ω ) , ω i = 0 for any homogeneous polynomial, see Lemma 2.2 in [16].(2) follows from (1) since ∇ κ Y n ( x ) = nr n − Y n ( ω ) ω + r n − ∇ κ Y n ( ω ) . (3.3)To prove (3) use the definition of ∇ κ = ∇ + E κ ; d X j =1 ( ∇ κ ) j ( ω j Y n ( ω )) = d X j =1 ( ∇ ) j ( ω j Y n ( ω )) + d X j =1 ( E κ ) j ( ω j Y n ( ω ))and d X j =1 ( E κ ) j ( ω j Y n ( ω )) = d X j =1 X ν ∈ R + κ ( ν ) ω j Y n ( ω ) − ( σ ν ω ) j Y n ( σ ν ω ) h ν, ω i ν j = γY n ( ω ) − X ν ∈ R + κ ( ν ) Y n ( σ ν ω ) h σ ν ω, ν ih ν, ω i Since h σ ν ω, ν i = h ω, σ ν ν i = −h ω, ν i we get (3) in view of (3.2) and thedefinition of ρ κ . Finally, in order to prove (4) we evaluate the integral Z R d h∇ κ f ( x ) , ∇ κ g ( x ) i h κ ( x ) dx where f ( x ) = e − | x | Y n ( x ) and g ( x ) = e − | x | Y m ( x ) in two different ways.As we have already observed, the above integral is equal to − Z R d ∆ κ f ( x ) g ( x ) h κ ( x ) dx. The Dunkl Laplacian decomposes as (see Dunkl-Xu [8] )∆ κ = ∂ ∂r + 2 λ κ + 1 r ∂∂r + 1 r ∆ κ, = p ( ∂ r ) + 1 r ∆ κ, where λ κ = γ + d − and p ( ∂ r ) = ∂ ∂r + λ κ +1 r ∂∂r . Thus∆ κ f ( x ) = p ( ∂ r )( r n e − r ) Y n ( ω ) + r n − e − r ∆ κ, Y n ( ω ) . Since h-harmonics are eigenfunctions of the spherical part ∆ κ, we have∆ κ, Y n ( ω ) = − n ( n + λ κ ) Y n ( ω )and consequently,∆ κ f ( x ) = p ( ∂ r )( r n e − r ) Y n ( ω ) − n ( n + λ κ ) r n − e − r Y n ( ω ) . Clearly, integrating the above against g ( x ) = e − r Y m ( ω ) produces 0 when-ever m = n .We will now evaluate the same integral using the expression ∇ κ = ω ∂∂r + r ( ∇ + E κ ). Note that ∇ κ f ( x ) = ( nr n − − r n +1 ) e − r Y n ( ω ) ω + r n − e − r ∇ κ Y n ( ω )with a similar expression for ∇ κ g ( x ). Thus h∇ κ f ( x ) , ∇ κ g ( x ) i involves termsof the form Y n ( ω ) Y m ( ω ), Y n ( ω ) h ω, ∇ κ Y m ( ω ) i , Y m ( ω ) h ω, ∇ κ Y n ( ω ) i and h∇ κ Y n ( ω ) , ∇ κ Y m ( ω ) i . Hence the proposition will be proved if we show that Z S d − Y n ( ω ) h ω, ∇ κ Y m ( ω ) i h κ ( ω ) dσ ( ω ) = 0whenever n = m . In view of (1) of the proposition it suffices to show that Z S d − Y n ( ω )( X ν ∈ R + κ ( ν ) Y m ( σ ν ω )) h κ ( ω ) dσ ( ω ) = 0 . But this is obvious, since the space H hn is invariant under the action of theorthogonal group. This completes the proof of (4). (cid:3) If Y m,j and Y m,k are h-harmonics of the same degree which are orthogonalto each other, then we cannot claim that Z S d − h∇ κ Y m,j ( ω ) , ∇ κ Y m,k ( ω ) i h κ ( ω ) dσ ( ω ) = 0 . This is clear from the above proof. However, if we assume that Y m,j and Y m,k are both G -invariant, then the orthogonally relation holds. Proposition 3.2.
Let Y m,j and Y m,k be h-harmonics of degree m which are G -invariant. Then Z S d − h∇ κ Y m,j ( ω ) , ∇ κ Y m,k ( ω ) i h κ ( ω ) dσ ( ω )(3.4) = λ d ( m, γ ) Z S d − Y m,j ( ω ) Y m,k ( ω ) h κ ( ω ) dσ ( ω ) where λ d ( m, γ ) = m ( m + λ κ ) , with λ κ = γ + d − .Proof. Proceeding as in the proof of Proposition 3.1 and noting that h∇ κ Y m,j , ω i = 0 in view of (1) and the G -invariance we get Z S d − h∇ κ Y m,j ( ω ) , ∇ κ Y m,k ( ω ) i h κ ( ω ) dσ ( ω ) = 0whenever Y m,j is orthogonal to Y m,k . When they are not orthogonal, theconstant λ d ( m, γ ) is given by λ d ( m, γ ) = A d ( m, γ ) − B d ( m, γ ) − C d ( m, γ ) D d ( m, γ )where A d ( m, γ ) = m ( m + λ κ ) Z ∞ e − r r d +2 γ +2 m − dr,B d ( m, γ ) = Z ∞ p ( ∂ r )( r m e − r ) e − r r d +2 γ + m − dr,C d ( m, γ ) = Z ∞ ( mr m − − r m +1 ) e − r r d +2 γ − dr and D d ( m, γ ) = Z ∞ e − r r d +2 γ +2 m − dr. Simplifying we obtain the expression for λ d ( m, γ ). (cid:3) The vector of Riesz transforms:
In this subsection we considerthe vector of Riesz transforms R f = ( R κ f, · · · , R κd f ) and show that for G -invariant functions, the mixed norm estimates for hR f, R f i can be reducedto certain vector valued inequalities.Let L G ( S d − , h κ ( ω ) dσ ) stand for the subspace of L ( S d − , h κ ( ω ) dσ ) con-sisting of G -invariant functions. Each space H hm can be decomposed into thesubspace ( H hm ) G consisting of G -invariant h-harmonics in H hm and its orthog-onal complement. We choose an orthonormal basis Y hm,j ; j = 1 , , . . . , d ( m ), d ( m ) ≤ d ( m ) for ( H hm ) G and then augment it with an orthonormal basis Y hm,j , d ( m ) < j ≤ d ( m ) for the orthogonal complement. Thus, we get anorthonormal basis { Y hm,j : 1 ≤ j ≤ d ( m ) , m ∈ N } for L ( S d − , h κ ( ω ) dσ ) such that for each m , Y hm,j , ≤ j ≤ d ( m ) are G -invariant. It is easyto see that { Y hm,j : 1 ≤ j ≤ d ( m ) , m ∈ N } is an orthonormal basisfor L G ( S d − , h κ ( ω ) dσ ). Indeed, if f is G -invariant and orthogonal to all Y hm,j , ≤ j ≤ d ( m ) , m ∈ N then for any Y hm,k , k > d ( m ) we have γ Z S d − f ( ω ) Y hm,k ( ω ) h κ ( ω ) dσ ( ω )= X ν ∈ R + κ ( ν ) Z S d − f ( σ ν ω ) Y hm,k ( ω ) h κ ( ω ) dσ ( ω )= Z S d − f ( ω ) (cid:16) X ν ∈ R + κ ( ν ) Y hm,k ( σ ν ω ) (cid:17) h κ ( ω ) dσ ( ω ) . As P ν ∈ R + κ ( ν ) Y hm,k ( σ ν ω ) is G -invariant, it can be written as P d ( m ) j =1 c k,j Y hm,j and consequently f is orthogonal to Y hm,k . Let L G ( R d , h κ ( x ) dx ) stand forthe subspace of L ( R d , h κ ( x ) dx ) consisting of G -invariant functions. Thuswe note that if f ∈ L p, G ( R d , r d +2 γ − h κ ( ω ) dσ ( ω ) dr ) ∩ L G ( R d , h κ ( x ) dx ) thenwe have the expansion f ( rω ) = ∞ X m =0 d ( m ) X j =1 f m,j ( r ) Y hm,j ( ω )where f m,j ( r ) = R S d − f ( rω ) Y hm,j ( ω ) h κ ( ω ) dσ ( ω ) . Note that V G is a subspaceof L p, G ( R d , r d +2 γ − h κ ( ω ) dσ ( ω ) dr ) ∩ L G ( R d , h κ ( x ) dx ) . If we let F = ( − ∆ κ + | x | ) − f , then F is also G -invariant and hence F ( rω ) = ∞ X m =0 d ( m ) X j =1 F m,j ( r ) Y hm,j ( ω ) . This expansion is justified since the operator ( − ∆ κ + | x | ) − is bounded on L ( R d , h κ ( x ) dx ) and it takes G -invariant functions into G -invariant func-tions. We remark that V G is also invariant under ( − ∆ κ + | x | ) − . Thiscan be easily seen as follows: Since the kernel K t ( x, y ) of the semigroup e − tH d,κ satisfies K t ( gx, gy ) = K t ( x, y ), g ∈ G , e − tH d,κ preserves G -invariantfunctions. Consequently, ( − ∆ κ + | x | ) − f is G -invariant whenever f is.We are now ready to prove the following. Proposition 3.3.
Let d ≥ and < p < ∞ . For functions f in the space L p, G ( R d , r d +2 γ − h κ ( ω ) dσ ( ω ) dr ) ∩ L G ( R d , h κ ( x ) dx ) , we have Z S d − hR f ( rω ) , R f ( rω ) i h κ ( ω ) dσ ( ω ) = A ( r ) + A ( r ) where A ( r ) = ∞ X m =0 d ( m ) X j =1 (cid:12)(cid:12)(cid:12)(cid:16) ∂∂r + r (cid:17) F m,j ( r ) (cid:12)(cid:12)(cid:12) and A ( r ) = ∞ X m =0 d ( m ) X j =1 λ d ( m, γ ) r | F m,j ( r ) | . Proof. As R f = ( ∇ κ + x )( − ∆ κ + | x | ) − f we see that R f ( rω ) = ( ω ∂∂r + rω + 1 r ∇ κ ) F ( rω ) . Now ( ω ∂∂r + rω + 1 r ∇ κ )( F m,j ( r ) Y hm,j )( ω )= ( ∂∂r + r ) F m,j ( r ) Y hm,j ( ω ) ω + 1 r F m,j ( r ) ∇ κ Y hm,j ( ω ) , and consequently R f ( rω ) = ∞ X m =0 d ( m ) X j =1 (cid:16) ∂∂r + r (cid:17) F m,j ( r ) Y hm,j ( ω ) ω + 1 r F m,j ( r ) ∇ κ Y hm,j ( ω ) . As Y hm,j ’s are G -invariant we can make use of Proposition 3.2 . Also, notethat h ω, ∇ κ Y hm,j ( ω ) i = 0. Therefore, integrating out hR f ( rω ) , R f ( rω ) i over S d − and making use of the orthogonality relations we get the proposition. (cid:3) The Laguerre connection and a proof of Theorem 1.1:
In viewof the above proposition, Theorem 1.1 will be proved once we show that Z ∞ A i ( r ) p w ( r ) r d +2 γ − dr ≤ C Z ∞ (cid:16) ∞ X m =0 d ( m ) X j =1 | f m,j ( r ) | (cid:17) p w ( r ) r d +2 γ − dr for i = 1 , w ∈ A n + γ − p ( R + ). Actually, we get Z ∞ (cid:16) Z S d − hR f ( rω ) , R f ( rω ) i h κ ( ω ) dσ ( ω ) (cid:17) p w ( r ) r d +2 γ − dr ≤ c Z ∞ (cid:16) Z S d − | f ( rω ) | h κ ( ω ) dσ ( ω ) (cid:17) p w ( r ) r d +2 γ − dr for all G -invariant functions f in L p, ( R d , w ( r ) r d +2 γ − h κ ( ω ) dσ ( ω ) dr ). Wenow show that the above inequalities for A i , i = 1 , For each δ ≥ − the Laguerre differential operator L δ has been introducedin subsection 2.4. The Laguerre functions ψ δk are eigenfunctions of L δ andthe semigroup generated by L δ is denoted by e − tL δ or T δt . Using spectraltheory we can define L − δ which is also given by the integral L − δ = 1 √ π Z ∞ e − tL δ t − dt. The operators R δ = (cid:16) ∂∂r + r (cid:17) L − δ are called Laguerre Riesz transformsand they have been studied in [14]. It is known that they are bounded on L p ( R + , dµ δ ), 1 < p < ∞ . Recently Ciaurri and Roncal [6] have proved thefollowing vector inequality. Theorem 3.4.
Let δ ≥ − and < p < ∞ . Then Z ∞ (cid:16) ∞ X m =0 r m | R δ + m ˜ f m ( r ) | (cid:17) p w ( r ) dµ δ ( r ) ≤ C Z ∞ (cid:16) ∞ X m =0 | f m ( r ) | (cid:17) p w ( r ) dµ δ ( r ) for all w ∈ A δp ( R + ) . Here ˜ f m ( r ) = r − m f m ( r ) . We only need to prove the above inequality when the right hand side isfinite. If ( f m ) is a sequence with this property then each function f m belongsto L p ( R + , w ( r ) dµ δ ) which will then imply that ˜ f m ∈ L p ( R + , w ( r ) dµ δ + m )so that R δ + m ˜ f m are well defined. A similar remark applies to L − δ + m ˜ f m which appears in the next theorem. Actually it is enough to prove theinequality when the sequence ( f m ) is finite with a constant C independentof the number of terms in the sequence. In the same paper [6] they havealso proved the following inequality. Theorem 3.5.
Let δ ≥ − and < p < ∞ . Then Z ∞ (cid:16) ∞ X m =0 m r m − | L − δ + m ˜ f m ( r ) | (cid:17) p w ( r ) dµ δ ( r ) ≤ C Z ∞ (cid:16) ∞ X m =0 | f m ( r ) | (cid:17) p w ( r ) dµ δ ( r ) for all w ∈ A δp ( R + ) . Here ˜ f m ( r ) = r − m f m ( r ) . We claim that the required inequalities for A and A can be deducedfrom the above two theorems. Recall that F m,j is defined as F m,j ( r ) = Z S d − ( − ∆ κ + | x | ) − f ( rω ) Y hm,j ( ω ) h κ ( ω ) dσ ( ω ) which can be expressed in terms of the semigroup e − tH d,κ as follows: F m,j ( r ) = 1 √ π Z S d − (cid:16) Z ∞ e − tH d,κ f ( rω ) t − dt (cid:17) Y hm,j ( ω ) h κ ( ω ) dσ ( ω )= 1 √ π Z ∞ (cid:16) Z S d − e − tH d,κ f ( rω ) Y hm,j ( ω ) h κ ( ω ) dσ ( ω ) (cid:17) t − dt. Use Proposition 2.1 stated at the end of subsection 2.4 to conclude that F m,j ( r ) = c d,γ r m L − d + γ + m − ˜ f m,j ( r ) . Consequently,( ∂∂r + r ) F m,j ( r ) = c d,γ r m R d + γ + m − ˜ f m,j ( r ) + c d,γ mr F m,j ( r ) . From these expressions for F m,j and ( ∂∂r + r ) F m,j ( r ) it is clear that theweighted inequalities for A and A follow from Theorem 3.4 and 3.5 .In the next section we give a simple proof Theorem 3.4 and 3.5 when 2 γ is an integer.4. Riesz transforms for the Hermite operator
Hermite operator in spherical coordinates:
The Hermite oper-ator H = − ∆ + | x | admits a family of eigenfunctions viz., the Hermitefunctions Φ α , α ∈ N d which forms an orthonormal basis for L ( R d ). On theother hand there is another family of orthonormal basis given by˜ ϕ m,j,l ( x ) = (cid:16) j + 1)Γ( m − j + d ) (cid:17) L d − m − jj ( | x | ) Y m − j,l ( x ) e − | x | where m ≥ j = 0 , , . . . , [ m ], l = 1 , , . . . , d ( m − j ), Y m − j,l ( x ) aresolid spherical harmonics and L δk are Laguerre polynomials of type δ . TheHermite operator in spherical coordinates takes the form H = − ∂ ∂r − d − r ∂∂r + r − r ∆ where ∆ is the spherical Laplacian on S d − . It can be shown that H = A ∗ A + d , where A = (cid:16) ∂∂r + r (cid:17) ω + 1 r ∇ where ∇ is the spherical part of the gradient and A ∗ = − (cid:16) ∂∂r − r (cid:17) ω − r ( div ) where ( div ) is the spherical part of the divergence. It is therefore naturalto look at the vector valued Riesz transform AH − f . The natural space suitable for studying this is the mixed norm space L p, ( R d , w ( r ) r d − drdσ ( ω ))consisting of functions for which Z ∞ (cid:16) Z S d − | f ( rω ) | dσ ( ω ) (cid:17) p w ( r ) r d − dr < ∞ . In [6] Ciaurri and Roncal have proved the following theorem.
Theorem 4.1.
Let d ≥ , < p < ∞ and w ∈ A d − p ( R + ) . Then kh AH − f, AH − f i k L p, ( R d ,wr d − drdσ ( ω )) ≤ C k f k L p, ( R d ,wr d − drdσ ( ω )) (4.1) for all f in the algebraic span of Hermite functions with a constant C independent of f. Consequently, the above inequality remains valid for all f ∈ L p, ( R d , wr d − drdσ ( ω )) . For the Hermite operator we also have the standard Riesz transforms R j = A j H − studied by several authors in the literature, see [21] and [20].It is well known that R j are Calderon-Zygmund singular integral operatorsand hence satisfy the weighted norm inequalities (cid:16) Z R d | R j f ( x ) | p w ( x ) dx (cid:17) p ≤ C (cid:16) Z R d | f ( x ) | p w ( x ) dx (cid:17) p for every w ∈ A p ( R d ), 1 < p < ∞ . This has been proved by Stempakand Torrea in [20]. We will give an easy proof of the above theorem ofCiaurri and Roncal based on the connection between AH − and the vector Rf = ( R f, · · · , R d f ). Theorem 4.2.
Let d ≥ and < p < ∞ . Then the inequality (4.1) for AH − stated in the previous theorem holds for all finite linear combinationof Hermite functions if and only if k (cid:16) d X j =1 | R j f ( x ) | (cid:17) k L p, ( R d ,wr d − drdσ ( ω )) ≤ C k f k L p, ( R d ,wr d − drdσ ( ω )) . (4.2) for all such functions. The proof of this theorem is easy. We have already observed in the pre-vious section that the mixed norm estimates for the Riesz transforms R j f ,(which corresponds to κ = 0 of Theorem 1.1) is equivalent to the weightednorm inequalities for A ( r ) and A ( r ) appearing in Proposition 3.3 . Ourclaim is substantiated by comparing this with the proof of Theorem 2.1 in[6]. The terms they call O ( f ) and O ( f ) are precisely our terms A ( r ) and A ( r ) respectively.4.2. A simple proof of Theorem 4.1:
We now give a simple proof ofmixed norm estimates (4.2) for the (standard) Riesz transforms associatedto the Hermite operator, which implies Theorem 4.1. When 2 γ is an integerit also implies the weighted norm inequalities for A ( r ) and A ( r ) and hencewe get another proof of Theorem 1.1 without using the result of Ciaurri and Roncal [6].We will be following an idea of Rubio de Francia. This method describedbriefly in [18] is based on an extension of a theorem of Marcinkiewicsz andZygmund as expounded in Herz and Riviere [12]. Indeed, we make use ofthe following lemma which can be found in [12]
Lemma 4.3.
Let ( G, µ ) and ( H, ν ) be arbitrary measure spaces and T : L p ( G ) → L p ( G ) a bounded linear operator. Then if p ≤ q ≤ or p ≥ q ≥ ,there exists a bounded linear operator ˜ T : L p ( G ; L q ( H )) → L p ( G ; L q ( H )) with k ˜ T k ≤ k T k such that for g ∈ L p ( G ; L q ( H )) of the form g ( x, ξ ) = f ( ξ ) u ( x ) where f ∈ L p ( G ) and u ∈ L q ( H ) we have ( ˜ T g )( ξ, x ) = ( T f )( ξ ) u ( x ) . The idea of Rubio de Francia is as follows (we are indebted to GustavoGarrigos for bringing this to our attention). Suppose T : L p ( R d , dx ) → L p ( R d , dx ) is a bounded linear operator. Then by the lemma of Herz andRiviere, it has an extension ˜ T to H valued functions on R d where H isthe Hilbert space L ( K ), K = SO ( d ). Moreover, the extension satisfies( ˜ T ˜ f )( x, k ) = T g ( x ) h ( k ) if ˜ f ( x, k ) = g ( x ) h ( k ), x ∈ R d , k ∈ SO ( d ). Given f ∈ L p ( R d , dx ) consider ˜ f ( x, k ) = f ( kx ). Then R R d ( R K | ˜ f ( x, k ) | dk ) p dx canbe calculated as follows. If x = rω , ω ∈ S d − , ˜ f ( x, k ) = f ( rkω ) and hence Z K | ˜ f ( x, k ) | dk = Z K ω (cid:16) Z K/K ω | f ( rkω ) | dµ (cid:17) dν (4.3)where K ω = { k ∈ K : kω = ω } is the isotropy subgroup of K , dν is theHaar measure on K ω and dµ is the K ω invariant measure on K/K ω whichcan be identified with S d − . Hence Z K | ˜ f ( x, k ) | dk = c Z S d − | f ( rω ) | dσ ( ω ) . (4.4)Therefore, Z R d (cid:16) Z K | ˜ f ( x, k ) | dk (cid:17) p dx = c ′ Z ∞ (cid:16) Z S d − | f ( rω ) | dσ ( ω ) (cid:17) p r d − dr. (4.5)Let us define ρ ( k ) f ( x ) = f ( kx ) so that ˜ f ( x, k ) = ρ ( k ) f ( x ). If T commuteswith rotation i.e. T ρ ( k ) = ρ ( k ) T then˜ T ˜ f ( x, k ) = T ( ρ ( k ) f )( x ) = ρ ( k )( T f )( x ) = ( T f )( kx ) . The boundedness of ˜ T on L p ( R d , H ) gives Z R d (cid:16) Z K | T f ( kx ) | dk (cid:17) p dx ≤ C Z R d (cid:16) Z K | f ( kx ) | dk (cid:17) p dx (4.6)which translates into the mixed norm estimate for T . Given a unit vector u ∈ S d − let us consider the operator T u f = P dj =1 u j R j f ( x ) where R j = A j H − are the Hermite Riesz transforms. This operator T u is not rotation invariant but has a nice transformation property underthe action of SO ( d ). Indeed, T u f ( x ) = ( x · u + u · ∇ ) H − f ( x )and as H − commutes with ρ ( k ) it follows that T u ρ ( k ) f = ρ ( k ) T ku f or T k − u ρ ( k ) f = ρ ( k ) T u f. This leads us to T u f ( kx ) = d X j =1 ( k − u ) j R j ( ρ ( k ) f )( x ) . We make use of this in proving the mixed norm estimate (4.2) .The operator R j are singular integral operators and hence bounded on L p ( R d , wdx ) for any weight function w ∈ A p ( R d ), 1 < p < ∞ . By the lemmaof Herz and Riviere, R j extends as a bounded operator e R j on L p ( R d , H ; wdx )where H = L ( S d − ) and e R j ( ρ ( k ) f )( x ) = R j ( ρ ( k ) f )( x ). When w is radial,it can be easily checked that k ρ ( k ) f ( x ) k pL p ( R d , H ; wdx ) = Z R d (cid:16) Z K | ρ ( k ) f ( x ) | dk (cid:17) p w ( x ) dx = c Z ∞ (cid:16) Z S d − | f ( rω ) | dσ ( ω ) (cid:17) p w ( r ) r d − dr. (4.7)Moreover, by the result of Duoandikoetxea et al (Theorem 3.2 in [10]), aradial weight w belongs to A p ( R d ) if and only if w ( r ) ∈ A d − p ( R + ). Fromthe identity T u f ( kx ) = d X j =1 ( k − u ) j R j ( ρ ( k ) f )( x )= d X j =1 ( k − u ) j e R j ( ρ ( k ) f )( x )we obtain k T u f ( kx ) k L p ( R d , H ; wdx ) ≤ C d X j =1 k e R j ( ρ ( k ) f )( x ) k L p ( R d , H ; wdx ) ≤ C d X j =1 k ρ ( k ) f ( x ) k L p ( R d , H ; wdx ) which translates into the required inequality (4.2) by (4.7) and taking u tobe coordinate vectors. Higher order Riesz transforms:
In this section we show that The-orem 1.1 remains true for higher order Riesz transforms associated to theHermite operator H d . As explained in Sanjay-Thangavelu [19], operatorsof the form R P f = G ( P ) H − ( m + n ) where P is a solid bigraded harmonic oftotal degree ( m + n ) and G ( P ) is the Weyl correspondence of P , are naturalanalogues of higher order Riesz transforms. When P ( z ) = X | α | = m, | β | = n c α,β z α ¯ z β is a solid harmonic, Geller [11] has shown that G ( P ) = X | α | = m, | β | = n c α,β A α A ∗ β where A = ( A , · · · , A d ), A ∗ = ( A ∗ , · · · , A ∗ d ). In particular when P ( z ) = z α (resp. ¯ z α ), G ( P ) = A α (resp. A ∗ α ). The Riesz transforms G ( P ) H − ( m + n ) have been studied in [19]. There, by using a transference result of Mauceri ithas been shown that G ( P ) H − ( m + n ) are all bounded on L p ( R d ), 1 < p < ∞ .However, we can also directly prove the boundedness of R P = G ( P ) H − ( m + n ) .In fact, G ( P ) H − ( m + n ) = 1Γ( m + n ) Z ∞ G ( P ) e − tH t m + n − dt and hence the kernel K P ( x, y ) of R P is given by K P ( x, y ) = 1Γ( m + n ) Z ∞ G ( P ) K t ( x, y ) t m + n − dt where K t is the kernel of e − tH which is explicitly known. Though it istedious, it is not difficult to show that K P is a Calderon-Zygmund kernel(see Stempak-Torrea [20] for the case m + n = 1). Hence the Riesz transforms R P are bounded on L p ( R d , wdx ), 1 < p < ∞ , w ∈ A p ( R d ). Using this wecan prove Theorem 4.4.
Let P be a solid harmonic of bidegree ( m, n ) , < p < ∞ and w ∈ A d − p ( R + ) . Then there exists C > such that Z ∞ (cid:16) Z S d − | R P f ( rω ) | dσ ( ω ) (cid:17) p w ( r ) r d − dr ≤ C Z ∞ (cid:16) Z S d − | f ( rω ) | dσ ( ω ) (cid:17) p w ( r ) r d − dr for all f ∈ L p, ( R d , w ( r ) r d − drdσ ( ω )) . The proof is similar to that of Theorem 4.1. Consider H m,n the space of allbigraded spherical harmonics of bidegree ( m, n ). If Y ∈ H m,n then P ( z ) = | z | m + n Y ( z ′ ), z = | z | z ′ is a solid harmonic. The group U ( d ) acts on H m,n andwe have an irreducible unitary representation, denoted by R ( σ ) supported by H m,n . We choose an orthonormal basis Y j , j = 1 , , . . . , d ( m, n ) and let P j stand for the corresponding solid harmonics. Consider the operator T which takes L p ( C d ) into L p ( C d , H m,n ) given by the prescription T f ( z, ζ ) = d ( m,n ) X j =1 R P j f ( z ) Y j ( ζ ) . This operator has a very nice transformation property. Let ρ ( σ ) f ( z ) = f ( σ − z ) stand for the action of U ( d ) on functions on C d . Lemma 4.5.
For any σ ∈ U ( d ) we have T f ( z, σ − ζ ) = d ( m,n ) X j =1 ρ ( σ ) R P j ρ ( σ − ) f ( z ) Y j ( ζ ) . This lemma has been essentially proved in [19], see the proof of Theorem1.4. Once we have the above Lemma we can easily prove Theorem 4.4.Indeed, from the lemma we have
T f ( σz, ζ ) = d ( m,n ) X j =1 R P j ρ ( σ − ) f ( z ) Y j ( σζ ) . With the same notation as in the proof of Theorem 4.1, the above reads as ρ ( σ − ) T f ( z, ζ ) = d ( m,n ) X j =1 ˜ R P j ˜ f ( z, σ − ) Y j ( σζ ) . where we keep ζ ∈ S d − fixed. Then by similar calculations, using theLemma of Herz-Riviere we can obtain the desired inequality for T ( · , ζ ) andhence for any R P j f . This completes the proof of Theorem 4.4. Acknowledgments
We are very thankful to G. Garrigos for pointing out the method of Rubiode Francia. We are also very thankful to the referee whose persistent demandfor details has greatly improved the exposition. The first author is thankfulto CSIR, India, for the financial support. The work of the second authoris supported by J. C. Bose Fellowship from the Department of Science andTechnology (DST) and also by a grant from UGC via DSA-SAP.
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