Mixed norm estimates for the Riesz transforms associated to Dunkl harmonic oscillators

In this paper we study weighted mixed norm estimates for Riesz transforms associated to Dunkl harmonic oscillators. The idea is to show that the required inequalities are equivalent to certain vector valued inequalities for operator defined in terms of Laguerre expansions. In certain cases the main result can be deduced from the corresponding result for Hermite Riesz transforms.


Introduction
Let G be a Coxeter group (finite reflection group) associated to a root system R in R d , d ≥ 2. We use the notation ., . for the standard inner product on R d . Let κ be a multiplicity function which is assumed to be non-negative and let where R + is the set of all positive roots in R. Let T j , j = 1, 2, . . . , d be the difference-differential operators defined by where σ ν is the reflection defined by ν. The Dunkl Laplacian ∆ κ is then defined to be the operator which can be explicitly calculated, see Theorem 4.4.9 in Dunkl-Xu [7]. The Dunkl harmonic oscillator is then defined by which reduces to the Hermite operator H d = −∆ + |x| 2 when κ = 0.
Our aim in this paper is to study the L p mapping properties of Riesz transforms associated to the Dunkl harmonic oscillator. The spectral theory of the operator H d,κ has been developed by Rösler in [16]. The eigenfunctions of H d,κ are called the generalised Hermite functions and denoted by Φ κ µ , µ ∈ N d . It has been proved that they form an orthonormal basis for L 2 (R d , h 2 κ dx). In analogy with the Riesz transforms associated to the Hermite operator, one can define the Riesz transforms R κ j , R κ * j , j = 1, 2, . . . , d by Note that the operators R κ j and R κ * j are densely defined i.e., they are defined on the subspace V consisting of finite linear combinations of the generalised Hermite functions Φ κ α . In the particular case of G = Z d 2 treated in [14] the authors have shown that the L 2 norm of (T j + x j )Φ κ α behaves like (2|α| + d + 2γ) 1/2 where γ = ν∈R + κ(ν). Since Φ κ α are eigenfunctions of H d,κ with eigenvalues (2|α| + d + 2γ) the operator H d,κ Φ κ α = (2|α| + d + 2γ) −1/2 Φ κ α . From these two facts, it is clear that the Riesz transforms defined on V satisfy the inequalities R κ j f 2 ≤ C f 2 , R κ * j f 2 ≤ C f 2 for all f ∈ V. Consequently, they extend to L 2 as bounded linear operators. In [1] a very cute argument based on the fact that is used to show that the L 2 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 κ j and R κ * j satisfy the inequalities R κ j f p ≤ C f p , R κ * j f p ≤ C f p for any 1 < p < ∞ whenever f ∈ V then by density arguments they can be extended to the whole of L p (R d , h 2 κ dx), 1 < p < ∞ as bounded operators. This was proved in [14] by Nowak and Stempak in the particular case when G = Z d 2 . For general Coxeter groups the boundedness properties of the Riesz transforms are proved by Amri in [1]. We refer to these two papers for details and further information on Riesz transforms associated to the Dunkl harmonic oscillator. Weighted norm inequalities or mixed norm inequalities are not known for these Riesz transforms. In this paper our main goal is to establish certain weighted mixed norm estimates for these operators.
For α ≥ − 1 2 , let A α p (R + ) be the Muckenhoupt's class of A p -weights on R + associated to the doubling measure dµ α (t) = t 2α+1 dt. Let dσ be the surface measure on unit sphere S d−1 and let w be a positive function on R + . We denote by The p−th root of the above quantity is a norm with respect to which the space becomes a Banach space. For 1 < p < ∞ the dual of the Banach space L p,2 (R d , w(r)r d+2γ−1 h 2 κ (ω)dσ(ω)dr) is nothing but the space where p is the index conjugate to p. This follows from a general theorem proved in [4] since we can think of the space L p,2 (R d , w(r)r d+2γ−1 h 2 κ (ω)dσ(ω)dr) as an L p space on R + of functions taking values in the Hilbert space L 2 (S d−1 , h 2 κ (ω)dσ(ω)) taken with respect to the measure w(r)r d+2γ−1 dr. Since L 2 (S d−1 , h 2 κ (ω)dσ(ω)) is a separable Hilbert space, it can be identified with the sequence space l 2 (N) and hence a simple independent proof also can be given for the fact about the dual. We denote by L p,2 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 where |G| stands for the cardinality of G. We claim that V G is precisely the set of all f # where f runs through V. Indeed, it is obvious that for The latter can be easily seen as follows: Since f ∈ V , it is of the form This proves our claim.
Also note that the density of V in which is an immediate consequence of Minkowski's inequality since the measures given by h 2 κ (ω)dσ(ω) and w(r)r d+2γ−1 dr are G-invariant. In Subsection 2.4 we will show that V is dense in L p,2 (R d , w(r)r d+2γ−1 h 2 κ (ω)dσ(ω)dr) for all w ∈ A d 2 +γ−1 p (R + ), 1 < p < ∞. Thus, R κ j and R κ * j are well defined on the dense subspace V G . Theorem 1.1. Let d ≥ 2, 1 < p < ∞. Then for j = 1, 2, · · · , d the Riesz transforms R κ j and R κ * j initially defined on V G satisfy the estimates The proof of this theorem is based on the fact that on radial functions the Dunkl harmonic oscillator H d,κ coincides with the Hermite operator H d+2γ , when 2γ is an integer. More generally, using an analogue of Funk-Hecke formula for h-harmonics we can show that the mixed norm estimates for the Riesz transforms R κ j are equivalent to a vector valued inequality for a sequence of Laguerre Riesz transforms. When 2γ is an integer these inequalities can be deduced from the weighted norm inequalities satisfied by Hermite Riesz transforms. In the general case when 2γ is not an integer, we can appeal to a recent result of Ciaurri and Roncal [5].
The plan of the paper is as follows. In Section 2 we collect some facts from the spectral theory of Dunkl harmonic oscillators. Especially, we need an analogue of Mehler's formula for the generalised Hermite functions. We also collect some basic facts about h-harmonics which are analogues of spherical harmonics on S d−1 . The most important result is an analogue of Funk-Hecke formula for h-harmonics. In Section 3 we consider the vector R κ f = (R κ 1 , · · · , R κ d ) of Riesz transforms and show that mixed norm inequalities for can be reduced to vector valued inequalities for operators related to Laguerre expansions. In Section 4 we prove the required inequalities by considering the vector of Hermite Riesz transforms.
Though we have considered only the Riesz transforms in this paper, we can also treat multipliers (e.g. Bochner-Riesz means) for the Dunkl harmonic oscillator. Using the known results for the Hermite operator, we can prove an analogue of Theorem 1.1 for multipliers associated to Dunkl harmonic oscillator.

Coxeter groups and Dunkl operators:
We assume that the reader is familiar with the notion of finite reflection groups associated to root systems. Given a root system R we define the reflection σ ν , ν ∈ R by Recall that ν, x 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 defined on R is called a multiplicity function if it is G invariant. We assume that our multiplicity function κ is non negative. The Dunkl operators T j defined by form a commuting family of operators. There exists a kernel E κ (x, ξ) which is a joint eigenfunction for all T j : This is the analogue of the exponential e x, ξ and Dunkl transform is defined in terms of E κ (ix, ξ). For all these facts we refer to Dunkl [6] and Dunkl-Xu [7]. The weight function associated to R and κ is defined by Recall that γ = ν∈R + κ(ν) and the multiplicity function κ(ν) is always assumed to be non-negative. We consider L p spaces defined with respect to the measure h 2 κ (x)dx. Note that h 2 κ (x) is homogeneous of degree 2γ.

Generalised Hermite functions:
In [16] Rösler has studied generalised Hermite polynomials associated to Coxeter groups. She has shown that there exists an orthonormal ba- They are also eigenfunctions of the Dunkl transform. For our purpose, the most important result is the generating function identity or the Mehler's formula for the generalised Hermite functions. For 0 < r < 1, one has see Theorem 3.12 in [16]. By taking r = e −2t , t > 0 we see that the kernel of the heat semigroup generated by −∆ κ + |x| 2 is given by We will make use of this kernel in the study of Riesz transforms.
Recall that the subspace V defined in the introduction is the algebraic span of the generalised Hermite functions Φ κ α , α ∈ N d . As every Φ κ α is a Schwartz function it follows that elements of V are also of Schwartz [21] the authors have shown that Bochner-Riesz means S δ R f , for large enough δ, converge to f in the norm as R → ∞ as long as 1 ≤ p < ∞.
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 We also need to know the density of V in certain weighted L p spaces. This will be addressed in Subsection 2.4 below.

h-harmonics and Funk-Hecke formula:
The best reference for this section is Chapter 5 of [7]. For the space ) there exists an orthonormal basis consisting of hharmonics. These are analogues of spherical harmonics and defined using ∆ κ in place ∆. A homogeneous polynomial P (x) is said to be a solid hharmonic if ∆ κ P (x) = 0. Restrictions of such solid harmonics to S d−1 are called spherical h-harmonics. The space In order to state the Funk-Hecke formula we need to recall the intertwining operator. It has been proved that there is an operator V κ satisfying The explicit form of V κ is not known, except in a couple of simple cases, but it is a useful operator. In particular, the Dunkl kernel is given by E κ (x, ξ) = V κ e ·, ξ (x). The operator V κ also intertwines h-harmonics (see Proposition 5.2.8 of [7]).
The classical Funk-Hecke formula for spherical harmonics states the following. For any continuous function f on [−1, 1] and a spherical harmonic Y m of degree m, one has the formula Here C λ m stand for ultraspherical polynomials of type λ and B(r, s) stands for the beta function. A similar formula is true for h-harmonics (see Theorem 5.3.4 in [7]); Let J δ (z) stand for Bessel function of type δ > −1 and define I δ (z) = e −i π 2 δ J δ (iz). If we take f (t) = e itz in the above we get (see page 204-205 in [3]). By taking f (t) = e t|x| |y| and making use of the above formula we get In view of this and (2.1) we have We will make use of this formula in calculating the action of e −tH d,κ on functions of the form g(r)Y h m (x ).

The density of V :
In this subsection we take up the issue of proving the density of V, defined in the introduction, in the spaces In order to do this we will make use of the Laguerre connection. For each δ ≥ − 1 2 we consider the Laguerre differential operator 2δ + 1 r d dr whose normalised eigenfunctions are given by where L δ k (r) are Laguerre polynomials of type δ. These functions form an orthonormal basis for L 2 (R + , dµ δ ), where dµ δ (r) = r 2δ+1 dr. The operator L δ generates the semigroup T δ t = e −tL δ whose kernel is given by The generating function identity ((1.1.47) in [20]) for Laguerre functions gives the explicit expression The Dunkl-Hermite semigroup e −tH d,κ generated by the operator H d,κ is an integral operator given by The relation between this semigroup and the Laguerre semigroups T δ t = e −tL δ is given by the following proposition. In what follows, Y h m,j , j = 1, 2, · · · , d(m), m = 0, 1, 2, · · · stands for the orthonormal basis for L 2 (S d−1 , h 2 κ (ω)dσ(ω)) described in Subsection 2.3.

Proposition 2.1. For any Schwartz class function
Then we have the relation 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/2+m+γ−1 t and the Funk-Hecke formula.
We make use of the following lemma in order to prove that V is a dense subspace of L p,2 (R d , w(r)r d+2γ−1 h 2 κ (ω)dσ(ω)dr). That V is a subspace follows immediately from the lemma as every member of V being a finite linear combination of Φ κ α is a Schwartz class function.
Proof. First we observe that if f is a Schwartz class function on R d , then Let δ = d/2 + γ − 1 and the above integral can be written as The first integral on the left hand side of the above is finite as f 0 is continuous and w is locally integrable. And the second integral can be written as The A p -weight condition on w implies R 0 w(r)r 2δ+1 dr ≤ CR 2p(δ+1) for R > 0, see page 252, Eqn.16 in [12]. Choose a positive integer N such that N > 2(δ + 1). Finally we see that The second inequality in the above is due to the facts that (1 + r) N f 0 (r) is bounded on R + and 1 + r j ≥ 2 j−1 . This proves the lemma.
We are now in a position to prove the density of V in the mixed norm space y) is a Schwartz function, it follows from Lemma 2.2 that e −tH d,κ g is well defined. Moreover, by Mehler's formula Consequently, e −tH d,κ g = 0 for all t > 0 in view of (2.5). In view of Proposition 2.1 it follows that for any m = 0, 1, 2, . . ., j = 1, 2, · · · , d(m) Hence we only need to conclude that the above implies g m,j = 0 for all m and j which leads to a contradiction.
But this follows from the theory of Laguerre semigroups. Indeed, what we have is Here w ∈ A d/2+γ−1 p (R + ) and f m,j ∈ L p (R + , w(s)dµ d/2+γ−1 (s)). Once again, the above can be rewritten as (R + ) we conclude that g m,j = 0 for all m and j.
Finally, we briefly indicate how the strong continuity of T t d/2+m+γ−1 can be proved. It is almost trivial to prove that the kernel of this semigroup satisfies the estimates stated in Proposition 3.4 of Ciaurri-Roncal [5]. Actually, we need not care about the uniformity in m. These estimates in turn can be used to prove that T t d/2+m+γ−1 f is dominated by the maximal function M d/2+γ−1 f adapted to the space of homogeneous type (R + , dµ d/2+γ−1 ). As this maximal function is known to be bounded on , see e.g. Duoandikoetxea [8], we This completes the proof.

Preliminaries on Riesz transforms:
As in the case of Hermite operator which corresponds to the case κ = 0, we define the Riesz transforms R κ j , j = 1, 2, . . . , d associated to the Dunkl harmonic oscillator H d,κ by d,κ is defined by spectral theorem. More precisely, where Φ κ α are the generalised Hermite functions and (f, We can also define R κ * j by as in the Hermite case. It is easy to see that R κ j are bounded on L 2 (R d , h 2 κ (x)dx), see Proposition 2.1 in [1] . In the same paper, Amri has proved that R κ j are singular integral operators whose kernels satisfy a modified Calderón-Zygmund condition and hence by a theorem of Amri and Sifi [2] they are all bounded on L p (R d , h 2 κ (x)dx), 1 < p < ∞.
In the case of Hermite operator, the Riesz transforms satisfy weighted norm estimates. More precisely, if w ∈ A p (R d ), then R 0 j are bounded on L p (R d , wdx), 1 < p < ∞. This has been proved by Stempak and Torrea [19] and it follows from the fact that the kernels of R 0 j satisfy standard Calderón-Zygmund conditions. In the present situation we do not have weighted inequalities for the Riesz transforms R κ j . Later we will show that the weighted inequalities for R 0 j can be used to prove mixed norm inequalities for the Hermite Riesz transforms which will then be used to prove similar results for R κ j .
Assume that 2γ is an integer. Then the action of H d,κ on radial functions coincides with that of H d+2γ on radial functions. More generally, let f ( where on the right hand side g is considered as a radial function on R d+2γ+2m . It is also possible to write e −tH d+2γ+2m 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, we plan to expand the given function f on R d in terms of h-harmonics. In order to find out the action of Riesz transforms on individual terms which are of the form g(|x|)Y h m (ω) we need formulas for the action of T j on such terms. More generally we let ∇ κ = (T 1 , T 2 , · · · , T d ) stand for the Dunkl gradient which is the sum of the gradient ∇ = ∂ ∂x 1 , · · · , ∂ ∂x d and E κ where Let ∇ 0 be the spherical part of ∇. Then the Dunkl gradient is written as For ξ ∈ R d , let T ξ stand for the Dunkl derivative given by If one of f and g is G-invariant then remains true. Moreover, we also know that In view of this we get We will make use of these properties in the following calculation.
We begin with some simple observations. When f is a radial function we have and consequently where (∇ 0 ) j stand for the j th component of ∇ 0 . To see this, consider in view of Euler's formula. On the other hand Comparing this with the earlier expression we get the assertion.
We will now evaluate the same integral using the expression Hence the proposition will be proved if we show that whenever n = m. In view of (1) of the proposition it suffices to show that But this is obvious, since the space H h n is invariant under the action of the orthogonal group. This completes the proof of (4).
If Y m,j and Y m,k are h-harmonics of the same degree which are orthogonal to each other, then we cannot claim that 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
where λ d (m, γ) = m(m + 2λ κ ), with λ κ = γ + d−2 2 . Proof. Proceeding as in the proof of Proposition 3.1 and noting that ∇ κ 0 Y m,j , ω = 0 in view of (1) and the G-invariance we get whenever Y m,j is orthogonal to Y m,k . When they are not orthogonal, the constant λ d (m, γ) is given by Simplifying we obtain the expression for λ d (m, γ).

The vector of Riesz transforms:
In this subsection we consider the vector of Riesz transforms Rf = (R κ 1 f, · · · , R κ d f ) and show that for G-invariant functions, the mixed norm estimates for Rf, Rf If we let F = (−∆ κ + |x| 2 ) − 1 2 f , then F is also G-invariant and hence This expansion is justified since the operator (−∆ κ +|x| 2 ) − 1 2 is bounded on This can be easily seen as follows: Since the kernel K t (x, y) of the semigroup e −tH d,κ We are now ready to prove the following.
and consequently As Y h m,j 's are G-invariant we can make use of Proposition 3.2. Also, note that ω, ∇ κ 0 Y h m,j (ω) = 0. Therefore, integrating out Rf (rω), Rf (rω) over S d−1 and making use of the orthogonality relations we get the proposition.

The Laguerre connection and a proof of Theorem 1.1:
In view of the above proposition, Theorem 1.1 will be proved once we show that . We now show that the above inequalities for A i , i = 1, 2 can be interpreted as certain vector valued inequalities for Laguerre Riesz transforms.
For each δ ≥ − 1 2 the Laguerre differential operator L δ has been introduced in Subsection 2.4. The Laguerre functions ψ δ k are eigenfunctions of L δ and the semigroup generated by L δ is denoted by e −tL δ or T δ t . Using spectral theory we can define L − 1 2 δ which is also given by the integral The operators R δ = ∂ ∂r + r L − 1 2 δ are called Laguerre Riesz transforms and they have been studied in [13]. It is known that they are bounded on L p (R + , dµ δ ), 1 < p < ∞. Recently Ciaurri and Roncal [5] have proved the following vector inequality.
We only need to prove the above inequality when the right hand side is finite. If (f m ) is a sequence with this property then each function f m belongs to L 2 (R + , w(r)dµ δ ) which will then imply thatf m ∈ L 2 (R + , w(r) dµ δ+m ) so that R δ+mf m are well defined. A similar remark applies to L − 1 2 δ+mf m which appears in the next theorem. Actually it is enough to prove the inequality when the sequence (f m ) is finite with a constant C independent of the number of terms in the sequence. In the same paper [5] they have also proved the following inequality.
We claim that the required inequalities for A 1 and A 2 can be deduced from the above two theorems. Recall that F m,j is defined as which can be expressed in terms of the semigroup e −tH d,κ as follows: Use Proposition 2.1 stated in the Subsection 2.4 to conclude that From these expressions for F m,j and ( ∂ ∂r + r)F m,j (r) it is clear that the weighted inequalities for A 1 and A 2 follow from Theorem 3.4 and 3.5.
In the next section we give a simple proof Theorem 3.4 and 3.5 when 2γ is an integer.

Hermite operator in spherical coordinates:
The Hermite operator H = −∆ + |x| 2 admits a family of eigenfunctions viz., the Hermite functions Φ α , α ∈ N d which forms an orthonormal basis for L 2 (R d ). On the other hand there is another family of orthonormal basis given bỹ for all such functions.
The proof of this theorem is easy. We have already observed in the previous 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 weighted norm inequalities for A 1 (r) and A 2 (r) appearing in Proposition 3.3. Our claim is substantiated by comparing this with the proof of Theorem 2.1 in [5]. The terms they call O 1 (f ) and O 2 (f ) are precisely our terms A 1 (r) and A 2 (r) respectively.

A simple proof of Theorem 4.1:
We now give a simple proof of mixed norm estimates (4.2) for the (standard) Riesz transforms associated to the Hermite operator, which implies Theorem 4.1. When 2γ is an integer it also implies the weighted norm inequalities for A 1 (r) and A 2 (r) and hence we get another proof of Theorem 1.1 without using the result of Ciaurri and Roncal [5].
We will be following an idea of Rubio de Francia. This method described briefly in [17] is based on an extension of a theorem of Marcinkiewicsz and Zygmund as expounded in Herz and Riviere [11]. Indeed, we make use of the following lemma which can be found in [11] The idea of Rubio de Francia is as follows (we are indebted to Gustavo Garrigos for bringing this to our attention). Suppose T : dx) is a bounded linear operator. Then by the lemma of Herz and and as H − 1 2 commutes with ρ(k) it follows that This leads us to 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 lemma of Herz and Riviere, R j extends as a bounded operator R j on When w is radial, it can be easily checked that we obtain which translates into the required inequality (4.2) by (4.7) and taking u to be coordinate vectors.

Higher order Riesz transforms:
In this section we show that Theorem 1.1 remains true for higher order Riesz transforms associated to the Hermite operator H d . As explained in Sanjay-Thangavelu [18], operators of the form where P is a solid bigraded harmonic of total degree (m + n) and G(P ) is the Weyl correspondence of P , are natural analogues of higher order Riesz transforms. When P (z) = |α|=m,|β|=n c α,β z αzβ is a solid harmonic, Geller [10] has shown that G(P ) = |α|=m,|β|=n c α,β A α A * β where A = (A 1 , · · · , A d ), A * = (A * 1 , · · · , 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 2 ) have been studied in [18]. There, by using a transference result of Mauceri it has been shown that G(P )H −( m+n 2 ) are all bounded on L p (R d ), 1 < p < ∞.
However, we can also directly prove the boundedness of G(P )H −( m+n 2 ) . In fact, ∞ 0 G(P )e −tH t m+n 2 −1 dt and hence the kernel K P (x, y) of R P := G(P )H −( m+n 2 ) is given by where K t is the kernel of e −tH which is explicitly known. Though it is tedious, it is not difficult to show that K P is a Calderón-Zygmund kernel (see  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 we can prove for all f ∈ L p,2 (R d , w(r)r d−1 drdσ(ω)).
The proof is similar to that of Theorem 4.1. Consider H m,n the space of all bigraded 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 and we have an irreducible unitary representation, denoted by R(σ) supported by H m,n . We choose an orthonormal basis Y j , j = 1, 2, . . . , 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 This operator has a very nice transformation property. Let ρ(σ)f (z) = f (σ −1 z) stand for the action of U (d) on functions on C d . This lemma has been essentially proved in [18], see the proof of Theorem 1.4. Once we have the above Lemma we can easily prove Theorem 4.4. Indeed, from the lemma we have With the same notation as in the proof of Theorem 4.1, the above reads as where we keep ζ ∈ S 2d−1 fixed. Then by similar calculations, using the Lemma of Herz-Riviere we can obtain the desired inequality for T (·, ζ) and hence for any R P j f . This completes the proof of Theorem 4.4.