RIESZ TRANSFORMS FOR THE DUNKL ORNSTEIN UHLENBECK OPERATOR. Dedicated to the memory of Professor Andrzej Hulanicki

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1 RIESZ TRANSFORMS FOR THE DUNKL ORNSTEIN UHLENBECK OPERATOR ADAM NOWAK, LUZ RONCAL, AND KRZYSZTOF STEMPAK Abstract. We propose a definition of Riesz transforms associated to the Ornstein Uhlenbec operator based on the Dunl Laplacian. In the case related to the group Z 2 it is proved that the Riesz transform is bounded on the corresponding L p spaces, 1 < p <. Dedicated to the memory of Professor Andrzej Hulanici 1. Introduction In the recent years Riesz transforms in the setting of orthogonal expansions related to general second order differential operators have been intensively studied. In particular, the first and third-named authors proposed a unified approach to the theory of this topic [12]. The investigation in the context of differential-difference operators was initiated very recently in [13], where Riesz transforms for the Dunl harmonic oscillator were defined and studied. The present paper is a continuation of the research from [13]. Now we consider the Ornstein Uhlenbec operator based on the Dunl Laplacian, and define and investigate related Riesz operators. Our results partially contribute to the Dunl theory, which has gained a considerable interest in various fields of mathematics as well as in theoretical physics during the last years. Given a finite reflection group G OR d ) and a G-invariant nonnegative multiplicity function : R [, ) on a root system R R d associated with the reflections of G, the Dunl differential-difference operators T j, j = 1,..., d, are defined by Tj fx) = j fx) + fx) fσ β x) β)β j, f C 1 R d ); β, x β R + here j is the jth partial derivative,, denotes the Euclidean inner product in R d, R + is a fixed positive subsystem of R, and σ β denotes the reflection in the hyperplane orthogonal to β. The Dunl operators T j, j = 1,..., d, form a commuting system this is an important feature, cf. [3]) of the first order differential-difference operators, and reduce 2 Mathematics Subject Classification: Primary 42C1, 42C2. Key words and phrases: Dunl operators, Dunl Laplacian, Ornstein-Uhlenbec operator, Riesz transforms, maximal operator, generalized Hermite polynomials. THIS PAPER HAS BEEN PUBLISHED IN: Colloq. Math ), Research of the first and third-named authors supported by MNiSW Grant N /4285. Research of the second-named author supported by grant MTM26-13-C3-3 of the DGI and by FPI grant of the University of La Rioja. 1

2 2 A. NOWAK, L. RONCAL, AND K. STEMPAK to j, j = 1,..., d, when. Moreover, Tj are homogeneous of degree 1 on P, the space of all polynomials in R d. In Dunl s theory the operator d ) = T 2 j plays the role of the Euclidean Laplacian in fact comes into play when ). It is homogeneous of degree 2 on P and symmetric in L 2 R d, w ), where w x) = β R + β, x 2β), if considered initially on Cc R d ). Note that w is G-invariant. For basic facts concerning Dunl s theory we refer the reader to the survey article by Rösler [15]. There, one can also find a discussion see [15, Section 3]) and extensive references concerning applications of Dunl s theory in mathematical physics. In this article we propose a definition of Riesz transforms associated to the operator L = + 2x, which is symmetric with respect to the measure 1.1) dµ x) = e x 2 w x) dx, and becomes the classic Ornstein Uhlenbec operator when. It occurs that L or rather its suitable self-adjoint extension L ) has a discrete spectrum and the corresponding eigenfunctions are the generalized Hermite polynomials defined and investigated by Rösler [14]. Then the formal definition Rj = δ j L ) 1/2, j = 1,..., d, with δ j = Tj being appropriate derivatives associated to L, rewritten properly in terms of the related expansions, delivers L 2 -bounded Riesz operators. In the one-dimensional case of a reflection group isomorphic to the group Z 2 we study L p mapping properties of the introduced Riesz transform in detail. As the main result Theorem 5.1) we prove that this operator is bounded on the corresponding L p spaces for 1 < p <. This can be regarded as a generalization of the one-dimensional L p results obtained by Mucenhoupt [8, 9] for the conjugate mappings related to classical Hermite and Laguerre expansions. We conjecture that an analogous result holds for arbitrary dimension d. In the Z d 2 group case we also consider an alternative Dunl Ornstein-Uhlenbec operator defined by means of the Dunl gradient rather than the Euclidean one. This variant of the operator seems to be more natural, at least from Riesz transforms theory point of view. In particular, suitably defined Riesz operators are L 2 -contractions, which is not the case of Rj. Finally, still in the Z d 2 group case, we obtain the wea type 1, 1) estimate for the maximal operator of the semigroup generated by the Dunl Ornstein-Uhlenbec operator. This extends the analogous results proved earlier by Sjögren [16] and Dinger [2] in the classical Hermite and Laguerre settings.

3 RIESZ TRANSFORMS FOR THE DUNKL ORNSTEIN UHLENBECK OPERATOR 3 The paper is organized as follows. In Section 2 we define, in an appropriate L 2 space, Riesz transforms in the context of the Dunl Ornstein-Uhlenbec operator based on the general Dunl Laplacian. Section 3 introduces the particular Dunl setting related to the group Z d 2. In Section 4 we establish the abovementioned wea type 1, 1) estimate for the heat semigroup maximal operator in the Z d 2 group case Theorem 4.1). Section 5 is devoted to the Z d 2 Riesz-Dunl transforms, and the main result of the paper is stated Theorem 5.1). In Section 6 we gather several facts from the theory of classical Laguerre expansions needed in the proof of the main result. In particular, we establish L p -boundedness, 1 < p <, of the left and right shift operators in the Laguerre setting Theorem 6.3); this result is new and of independent interest. The proof of Theorem 5.1 is given at the end of Section 6. Eventually, in Section 7 we discuss Riesz operators related to the already mentioned variant of the Dunl Ornstein-Uhlenbec operator. Throughout the paper we use a fairly standard notation. Given a multi-index n N d, we write n = n n d ; x denotes the Euclidean norm of x R d, and e j the jth coordinate vector in R d. For a nonnegative weight function w on R d, by L p R d, w), 1 p <, we denote the usual Lebesgue spaces related to the measure dwx) = wx)dx we will often abuse slightly the notation and use the same symbol w for the measure induced by a density w). Similarly, when w is a nonnegative weight function on R d + =, ) d, we write L p R d +, w) for the relevant Lebesgue spaces. 2. The general setting Similarly to numerous framewors discussed in the literature, see [12], it is reasonable to define, at least formally, the Riesz transforms R 1,..., R d, associated with L as 2.1) R j = δ j L ) 1/2 Π ; here L is a suitable self-adjoint extension in L 2 R d, µ ) of L, Π is a projection annihilating the eigenspace of L corresponding to the eigenvalue, and δ j s are appropriately defined first order differential-difference operators. In the present setting we define the jth partial derivative δ j related to L by δ j = T j. A short calculation shows that the formal adjoint of δ j in L 2 R d, µ ) is To be precise, this means that δ j = T j + 2x j. 2.2) δ j f, g = f, δ j g, f, g C 1 c R d ), where, is the canonical inner product in L 2 R d, µ ). One of the facts which motivate the definition 2.1) is that, as a direct computation shows, L + d + 2γ) = 1 2 d δ j δ j + δ j δ j ), γ = β R + β).

4 4 A. NOWAK, L. RONCAL, AND K. STEMPAK In the setting of general Dunl s theory Rösler [14] constructed systems of naturally associated multivariable generalized Hermite polynomials H n, so that {H n : n N d } is a complete orthogonal system in L 2 R d, µ ), cf. [14, Corollary 3.5 i)]. Noteworthy, for the construction leads to suitably normalized) classical Hermite polynomials. Moreover, H n are eigenfunctions of L, L H n = 2 n H n. From now on we will always consider the generalized Hermite polynomials normalized by dividing by their L 2 R d, µ ) norms. For the sae of clarity, polynomials of the normalized system will be denoted by H n. The operator defined on the domain DomL ) = L f = n N d 2 n f, H n H n, {f L 2 R d, µ ) : n Nd 2 n f, H n 2 < }, is a self-adjoint extension of L considered on Cc R d ) as the natural domain the inclusion Cc R d ) DomL ) may be easily verified). The spectrum of L is the discrete set {2m : m N}, and the spectral decomposition of L is L f = 2m Pmf, f DomL ), where the spectral projections are m= Pmf = f, Hn Hn. n =m Then, letting Π be the orthogonal projection onto the orthogonal complement of the subspace spanned by the constant function H,...,), we have L 1/2 Π f = 2m) 1/2 Pmf, m=1 and this superposition is clearly a bounded operator on L 2 R d, µ ). We now furnish the rigorous definition of Rj on L 2 R d, µ ). Let E be the dense subspace of L 2 R d, µ ) spanned by {Hn : n N d }. Note that E precisely consists of all polynomials in R d. Moreover, E is stable under the action of L 1/2, Π, δ j, δj, and 2.2) is valid also for f E. Then for f E we may define strictly the Riesz transforms by 2.1), and these are bounded operators on E. Indeed, letting R j = δj L 1/2 Π we see that for f E R j f 2 L 2 R d, µ ) R j f 2 L 2 R d, µ ) + R j f 2 L 2 R d, µ ) = δj δ j L 1/2 Π f, L 1/2 Π f + δ j δj L 1/2 Π f, L 1/2 d ) i=1 ) δ i δ i + δ i δi 1/2 L Π f, L 1/2 Π f Π f

5 RIESZ TRANSFORMS FOR THE DUNKL ORNSTEIN UHLENBECK OPERATOR 5 = 2 L + d + 2γ)L 1/2 Π f, L 1/2 Π f = 2 Π f 2 L 2 R d, µ ) 2 + d + 2γ) f 2 L 2 R d, µ ). + 2d + 2γ) L 1/2 Π f 2 L 2 R d, µ ) It follows that the unique extension of Rj to the whole L 2 R d, µ ) is given by Rj f = 2 n ) 1/2 f, Hn δ j Hn, n > the series being convergent in L 2 R d, µ ) and its sum being independent of the order of summation. 3. Preliminaries to the Z d 2 group case Consider the finite reflection group generated by σ j, j = 1,..., d, σ j x 1,..., x j,..., x d ) = x 1,..., x j,..., x d ), and isomorphic to Z d 2 = {, 1} d. The reflection σ j is in the hyperplane orthogonal to e j, the jth coordinate vector in R d. Thus R = {± 2e j : j = 1,..., d}, R + = { 2e j : j = 1,..., d}, and for a nonnegative multiplicity function : R [, ) which is Z d 2-invariant only values of on R + are essential. Hence we may thin = α 1 + 1/2,..., α d + 1/2), α j 1/2. We write α j +1/2 in place of seemingly more appropriate α j since, for the sae of clarity, it is convenient for us to stic to the notation used in the Laguerre polynomial setting. In what follows the symbols T α j, δ j, α, µ α, L α, and so on, denote the objects introduced in Section 2 and related to the present particular setting. Thus the Dunl differentialdifference operators T α j, j = 1,..., d, are now given by T α j fx) = j fx) + α j + 1/2) fx) fσ jx) x j, f C 1 R d ), and the explicit form of the Dunl Laplacian is α fx) = d 2 f x 2 j x) + 2α j + 1 x j Note that α, when restricted to the even subspace f x) α j + 1/2) fx) fσ jx) x j x 2 j ). 3.1) { f C 1 R d ) : j = 1,..., d, fx) = fσ j x) }, coincides with the multi-dimensional Bessel differential operator d 2 j + 2α j+1 x j j ), and consequently L α = α + 2x reduces to the Laguerre-type operator 3.2) + 2x d 2α j + 1 x j. x j

6 6 A. NOWAK, L. RONCAL, AND K. STEMPAK The corresponding measure µ α has a product structure of the form dµ α x) = d x j 2αj+1 e x2 j dxj = 2 α d/2 e x 2 β R + β, x 2β) dx, x R d ; for simplicity we neglect the constant factor in comparison with 1.1). In dimension one, for the reflection group Z 2 see [14, Example 3.3 2)]) and the multiplicity parameter α + 1/2, α 1/2, one obtains as the corresponding normalized) generalized Hermite polynomials ) 1/2 H2nx) α = 1) n n! L α nx 2 ), Γn + α + 1) H2n+1x) α = 1) n n! Γn + α + 2) ) 1/2 xl α+1 n x 2 ), where n N and L α n denotes the Laguerre polynomial of degree n and order α, see [6, p. 76]. Note that these Hn α are, up to multiplicative constants, the genuine generalized Hermite polynomials Hn α+1/2 on R, as defined and studied by Chihara [1]. For α = 1/2 the Hn α become the classical normalized) Hermite polynomials, see [6, p. 81]. In the multidimensional setting, corresponding to the group Z d 2, the generalized Hermite polynomials are obtained by taing tensor products of the one-dimensional Hn. α Thus for a multi-index α [ 1/2, ) d, H α nx) = H α 1 n 1 x 1 )... H α d n d x d ), x R d, n N d. The system {H α n : n N d } is an orthonormal basis in L 2 R d, µ α ) consisting of eigenfunctions of L α ; recall that L α H α n = 2 n H α n. 4. Z d 2 heat semigroup maximal operator Let {Tt α } t> be the heat-diffusion semigroup generated by L α, Tt α f = e 2mt Pmf, α f L 2 R d, µ α ). m= Then the integral representation of Tt α is Tt α fx) = Gt α x, y)fy) dµ α y), x R d, R d where the heat ernel is expressed in terms of the Hn, α Gt α x, y) = e 2mt Hnx)H α ny). α m= n =m The oscillating series defining Gt α x, y) can be summed and we get 4.1) Gt α x, y) = x, y), G α,ε t ε {,1} d

7 RIESZ TRANSFORMS FOR THE DUNKL ORNSTEIN UHLENBECK OPERATOR 7 where the component ernels are given by G α,ε t x, y) = e 2t α 1 e 4t ) exp 1 d e 4t 1 x 2 + y 2)) d x i y i ) ε I xi y i ) α i i +ε i sinh 2t x i y i ) α i+ε i, with I ν being the modified Bessel function of the first ind and order ν, see [6, Chapter 5]. This formula can be deduced, for instance, from a relation with the setting considered in [13, Section 3] and the facts invoed there. Indeed, it is easy to see that Gt α x, y) = e 2t α +d) e x 2 + y 2 )/2 G α t x, y), with G α t x, y) defined in [13, Section 3]. Then the decomposition G α t x, y) = ε {,1} G α,ε d t x, y), together with the explicit form of G α,ε t x, y), shows 4.1). Consider the maximal operator T α f = sup t> Tt α f. By Stein s general maximal theorem [18, p. 73], T α is bounded on L p R d, µ α ) for 1 < p. The case p = 1 is more subtle. The following theorem is a consequence of Dinger s result [2] in the classical Laguerre setting. In fact it generalizes analogous multi-dimensional results for classical Hermite [16] and Laguerre [2] settings, which in one dimension were obtained originally by Mucenhoupt [7]. i=1 Theorem 4.1. Let α [ 1/2, ) d. Then T α satisfies the wea type 1, 1) inequality µ α { x R d : T α fx) > λ } C λ f L 1 R d, µ α), λ >. Proof. Denote ε o =,..., ). By Soni s inequality [17], we see that I ν+1 z) < I ν z), z >, ν 1/2, < G α t x, y) 2 d G α,εo t x, y), t >, x, y R d. Since both G α,εo t x, y) and the density of the measure µ α are even with respect to each coordinate, it follows that 2 d T α fx) sup t> G α,ε o t R d δ { 1,1} d sup t> x, y) fy) dµ α y) R d + G α,εo t x1,..., x d ), y ) f δ y) dµ α y) δ { 1,1} d T α,εo f δ x 1,..., x d ) ), where f δ x) = fδ 1 x 1,..., δ d x d ). Thus it suffices to show the wea type 1, 1) for the maximal operator T α,εo in R d +. But T α,εo is, up to a constant factor and the change of variable R d + x x 2 R d +, the Laguerre maximal operator considered by Dinger [2]. The relevant wea type 1, 1) estimate is stated in [2, Theorem 1], see also the accompanying comments explaining validity of the result for any type multi-index. An important consequence of Theorem 4.1 is the almost everywhere convergence T α t f f as t +, for f L 1 R d, µ α ).

8 8 A. NOWAK, L. RONCAL, AND K. STEMPAK 5. Z d 2 Riesz transforms Recall that our choice of appropriate derivatives δ j is motivated by the decomposition L α + 2 α + 2d) = 1 2 d ) δj δ j + δ j δj. First we shall see how δ j s act on H α n. It is sufficient to consider the one-dimensional situation and then distinguish between the even and odd cases. Recall that δ j = T α j ; in the one-dimensional case we simply write δ in place of δ 1. For n N and α 1/2, combining the fact that H α 2n is an even function with the identity 5.1) see [6, )], one easily obtains d dy Lα ny) = L α+1 n 1y), δh α 2n = 4nH α 2n 1; here, and also later on, we use the convention that H α m L α m if m <. Similarly, combining the fact that H α 2n+1 is an odd function with 5.1) and the identities 5.2) yl α+2 n 1y) + α + 1)L α+1 n y) = yl α+1 n y) + n + 1)L α n+1y) = n + α + 1)L α ny), which in turn can be deduced from 5.1), [6, )] and [6, )], one gets δh α 2n+1 = 4n + 4α + 4H α 2n. Summarizing, in d dimensions, for n N d and α [ 1/2, ) d we have where m j n, α) = δ j H α n = m j n, α)h α n e j, { 2nj, if n j is even, 2nj + 4α j + 2, if n j is odd; by the convention, H n ej if n j =. Note that for each j the system {δ j Hn α : n j 1} is orthogonal in L 2 R d, µ α ). The rigorous definition of the Riesz transforms on L 2 R d, µ α ) is provided by the orthogonal series 5.3) Rj α f = m j n, α) f, Hn α α Hn e α 2 n j, n > from which the L 2 -boundedness can easily be seen directly. Notice however, that Rj α is not a contraction on L 2 R d, µ α ) if α j > 1/2. Our main result, Theorem 5.1 below, is an extension of Mucenhoupt s L p results [8, 9] for the conjugate mappings related to classical Hermite and Laguerre expansions. Theorem 5.1. Let d = 1 and assume that α 1/2. Then for each 1 < p < the Riesz transform R α 1, defined on L 2 R, µ α ) by 5.3), extends to a bounded operator on L p R, µ α ).

9 RIESZ TRANSFORMS FOR THE DUNKL ORNSTEIN UHLENBECK OPERATOR 9 We conjecture that an analogous result holds for arbitrary dimension d and α [ 1/ 2, ) d, but proving this seems to be a rather difficult tas. In contrast with the maximal operator, it is not possible to deduce the result in a straightforward manner from the nown results [11] in the Laguerre setting. Nor the technique of square functions used in [11] seems to be suitable enough in the present context. The proof of Theorem 5.1 is partially based on nown results in the classical Laguerre setting. To show the L p estimate we split a function into its even and odd parts. Then the Riesz transform of the even part can be identified with the Riesz-Laguerre transform for which the relevant bound is nown. Treatment of the odd part is less straightforward. The Riesz operator coincides, up to a shift and multiplier operators, with the adjoint of the Riesz-Laguerre transform. Thus to get the desired estimate we need to invoe a suitable multiplier theorem and to establish L p -boundedness of a shift operator in the Laguerre setting. The next section gathers the abovementioned auxiliary results. The proof of the main theorem is furnished at the end of Section Laguerre setting results and proof of Theorem 5.1 The one-dimensional setting discussed below is equivalent to the classical Laguerre polynomial setting, from which it emerges by the change of variable x x 2 on R +. Thus all relevant definitions and results can be directly translated from the original to squared Laguerre setting. In what follows we always assume that α 1/2. The restriction of µ α to R + will be denoted by the same symbol. Consider the operator 3.2) in dimension one, L α = d2 2α + 1 2x2 d dx2 x dx, which is positive and symmetric in L 2 R +, µ α ). eigenfunctions of L α, L α L α nx 2 ) = 4nL α nx 2 ), The polynomials L α nx 2 ), n N, are and the set {L α nx 2 ) : n N} forms an orthogonal basis in L 2 R +, µ α ). In the sequel it is convenient to normalize this system in L 2 R +, µ α ) and consider the polynomials ) 1/2 ϕ α 2n! nx) = L α Γn + α + 1) nx 2 ). The definition of the Riesz transform in the squared Laguerre setting is inherited from the classical Laguerre setting see [9] or [11]), and hence is induced by the mapping R α ϕ : ϕ α n ψ α n 1, n N, where ψ 1 α and {ψn α : n N} is another orthonormal basis of L 2 R +, µ α ) consisting of the polynomials ) 1/2 ψnx) α 2n! = xl α+1 n x 2 ). Γn + α + 2)

10 1 A. NOWAK, L. RONCAL, AND K. STEMPAK By Plancherel s theorem, R α ϕ extends uniquely to a contraction on L 2 R +, µ α ), which we denote by the same symbol. Notice that ϕ α n and ψ α n coincide, up to constant factors independent of n and α, with the generalized Hermite polynomials H α 2n and H α 2n+1, respectively. In view of Mucenhoupt s result [9, Theorem 3 b)], see also [11, Theorem 13], we have the following Theorem 6.1. Let α 1/2 and 1 < p <. Then R α ϕf L p R +, µ α) C f L p R +, µ α), with a constant C independent of f L 2 L p R +, µ α ). It is immediate that the adjoint operator R α ϕ), taen in L 2 R +, µ α ), is determined by the mapping R α ψ : ψ α n ϕ α n+1, n N, whose unique) extension to L 2 R +, µ α ) still denoted by the same symbol) is precisely the adjoint of R α ϕ. Consequently, by Theorem 6.1 and duality we see that for 1 < p < 6.1) R α ψf L p R +, µ α) C f L p R +, µ α), with a constant C independent of f L 2 L p R +, µ α ). The next ingredient that will be needed in the proof of Theorem 5.1 is the multiplier theorem below. It is a direct translation to the squared Laguerre setting of [5, Theorem 3.4], after specifying it to one dimension and taing β = 1. Theorem 6.2. Let 1 < p < and α 1/2. Assume that h is a function analytic in a neighborhood of the origin. Let {ξn)} n N be a sequence of real numbers such that ξn) = hn 1 ) for n n. Then the multiplier operator given by satisfies M ξ : ϕ α n ξn)ϕ α n M ξ f L p R +, µ α) C f L p R +, µ α), with a constant C independent of f L 2 L p R +, µ α ). Finally, we establish L p -boundedness of the left and right shift operators related to the system {ϕ α n}. Changing the variable leads to the analogous result for the system of normalized) Laguerre polynomials. This may be regarded as an extension of the result stated in [4, Proposition 3.3 a)]. Theorem 6.3. Let α 1/2 and 1 < p <. Then the shift operators given by satisfy S L : ϕ α n ϕ α n 1, S L f L p R +, µ α) C f L p R +, µ α), S R : ϕ α n ϕ α n+1, with a constant C independent of f L 2 L p R +, µ α ). S R f L p R +, µ α) C f L p R +, µ α),

11 RIESZ TRANSFORMS FOR THE DUNKL ORNSTEIN UHLENBECK OPERATOR 11 Proof. First, observe that by duality it suffices to prove the statement only for S R, the adjoint of S L in L 2 R +, µ α ). Then the estimate we need to justify is 6.2) a n ϕ α n+1x) p dµ α x) C a n ϕ α nx) p dµ α x). n= Next notice that by means of Theorem 6.2 the tas of showing 6.2) can be reduced to proving the estimate n ) n + α + 1 b nl α n+1x 2 ) p dµ α x) C b n L α nx 2 ) p dµ α x). n= Indeed, to get 6.2) let ξn) = n= n= n+α+1 n+1 and apply first 6.3) with b n = ξn)a n and then use Theorem 6.2 for the multiplier ξn). It remains to prove 6.3). We invoe the formula, see 5.2), n + 1 n + α + 1 Lα n+1y) = L α y ny) n + α + 1 Lα+1 n y) and use it to estimate the left-hand side in 6.3). We get n + 1 n + α + 1 b nl α n+1x 2 ) p dµ α x) n= 2 p b n L α nx 2 ) p dµ α x) + 2 p x 2 n + α + 1 b nl α+1 n x 2 ) p dµ α x). n= To deal with the last integral we apply the identity see Koshlyaov s formula [6, p. 94]) x 2 n + α + 1 Lα+1 n x 2 ) = 2 x 2α This produces x 2 n + α + 1 b nl α+1 n x 2 ) p dµ α x) = n= n= x 2 x 2α L α ny 2 ) y 2α+1 dy. x ) b n L α ny 2 ) y 2α+1 dy p dµ α x). Now the desired estimate is a consequence of weighted Hardy s inequality x 6.4) gy) dy p x 2α1 p)+1 e x2 dx C gx) p x 2α+1)1 p) e x2 dx. But it is nown, see for instance [1, Theorem 1], that the sufficient and necessary) condition for 6.4) to hold is ) 1/p r ) 6.5) sup x α1 p) e x dx x α e x 1 1/p p 1 dx < r> r this condition is, by the change of variable x 2 x, equivalent to [1, 1.3)] with suitably chosen weights U, V ). Thus the proof will be finished once we verify 6.5). The decays at + of the integrated expressions in 6.5) are essentially determined by the exponentials. So neglecting the power factors at the price of adding a positive constant to both exponents, we see that when r n=

12 12 A. NOWAK, L. RONCAL, AND K. STEMPAK is large, say r 1, the whole expression under the supremum is dominated by a constant. On the other hand, for x close to + the exponential factors can be neglected. Then taing into account small r and integrating the power factors shows that the expression under the supremum is controlled by a positive power of r. The conclusion follows. Remar 6.4. The Laguerre setting results of this section hold in fact for a wider range α > 1 of the type parameter. This remar concerns in particular Theorem 6.3, and the proof given above is valid also for α 1, 1/2). We are now in a position to prove Theorem 5.1. Given f L 2 L p R, µ α ), decompose it into its even and odd parts, f = f e + f o. To prove the theorem it is sufficient to show the L p estimates 6.6) R α 1 f e L p R, µ α) C f e L p R, µ α), R α 1 f o L p R, µ α) C f o L p R, µ α). Since the generalized Hermite polynomial Hn α is even if n is even and is odd for n odd, expansions of f e and f o are given only by even and odd Hn s, α respectively. Moreover, in view of 5.3), R1 α f e is odd and R1 α f o is even. Due to these symmetries we consider the operators Re α and Ro α on L 2 R +, µ α ) emerging naturally from restrictions of R1 α to the subspaces of L 2 R, µ α ) of even and odd functions, respectively. Clearly, 6.6) will follow once we show suitable L p estimates for Re α and Ro α. Observe that by 5.3) we have Re α : ϕ α n ψn 1, α Ro α : ψn α n + α + 1 n + 1/2 ϕα n. Thus Re α coincides with Rϕ, α and the corresponding L p estimate is provided by Theorem 6.1. On the other hand, Ro α is related to the mapping Rψ α by means of shift and multiplier operators, Ro α = M ξ S L Rψ, α n + α + 1 ξn) = n + 1/2. Consequently, the relevant L p estimate follows by 6.1) and Theorems 6.2 and 6.3. The proof of Theorem 5.1 is complete. 7. Alternative Z d 2 Dunl Ornstein-Uhlenbec operator In this section we consider the Laplacian L α = α + 2x α, a variant of the Dunl Ornstein-Uhlenbec operator based on the Dunl gradient α = ) T1 α,..., Td α. This variant seems to be more natural than L α for defining Riesz transforms, at least in the Z d 2 group case. It occurs that Riesz transforms naturally associated with L α are contractions in L 2 R d, µ α ), which is not the case of Rj α related to L α. Moreover, the context

13 RIESZ TRANSFORMS FOR THE DUNKL ORNSTEIN UHLENBECK OPERATOR 13 of L α is better related to the classical Laguerre setting, as will be seen below. Similarly as L α, when restricted to the even subspace 3.1) L α coincides with the Laguerre-type operator 3.2), and for α = 1/2,..., 1/2) it reduces to the classic Ornstein-Uhlenbec operator. Below we eep the notation introduced in previous sections. It is straightforward to chec that L α admits the decomposition d L α = δj δ j. In fact the decomposition + 2x = d δ j δ j, = T 1,..., T d ), δ j = T j, also holds in the general setting from Section 2. It follows that L α is formally symmetric and nonnegative in L 2 R d, µ α ). Thus it is reasonable, see [12], to define formally the Riesz transforms associated with L α by R α j = δ j L α ) 1/2 Π, j = 1,..., d. The multi-dimensional generalized Hermite polynomials are eigenfunctions of L α, L α Hn α = 2 n + ) d [ 4α j + 2) Hn α = mj n, α) ] ) 2 Hn. α {j:n j odd} Let L α be the self-adjoint extension of L α whose spectral decomposition is given by the H α n. Then the rigorous definition of R α j f for f being a generalized Hermite) polynomial is R j α = δ L 1/2 j α Π. Rewriting this in terms of the corresponding expansions leads to the orthogonal series 7.1) Rα j f = m j n, α) d f, Hn α α H α n > [m jn, α)] 2 n e j, which provides a definition of the Riesz operators on L 2 R d, µ α ). Clearly, by Plancherel s theorem the mapping f R 1 α f R d αf 2 is a contraction on L 2 R d, µ α ), and even an isometry on the orthogonal complement of the constant function H α,...,). We now state an analogue of Theorem 5.1 in the context of L α. Theorem 7.1. Let d = 1 and α 1/2. Then for each 1 < p < the Riesz transform R α 1, defined on L 2 R, µ α ) by 7.1), extends to a bounded operator on L p R, µ α ). Proof. We proceed as in the proof of Theorem 5.1 and arrive at the operators R α e and R α o acting on L 2 R +, µ α ). The conclusion will follow once we show suitable L p estimates for these two operators. Notice that by 7.1) we have R α e : ϕ α n ψ α n 1, Rα o : ψ α n ϕ α n. Thus R α e = R α ϕ and R α o = S L R α ψ. Now the relevant Lp estimates are consequences of 6.1) and Theorems 6.1 and 6.3.

14 14 A. NOWAK, L. RONCAL, AND K. STEMPAK Acnowledgment. This research was started during the sojourn in Wroc law of the second-named author, who wants to than Politechnia Wroc lawsa for the support and hospitality. References [1] T.S. Chihara, An introduction to orthogonal polynomials, Gordon and Breach, New Yor, [2] U. Dinger, Wea type 1, 1) estimates of the maximal function for Laguerre semigroup in finite dimensions, Rev. Mat. Iberoamericana ), [3] C.F. Dunl, Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc ), [4] G. Gasper, W. Trebels, Applications of weighted Laguerre transplantation theorems, Methods Appl. Anal ), [5] P. Graczy, J.J. Loeb, I. López, A. Nowa, W. Urbina, Higher order Riesz transforms, fractional derivatives, and Sobolev spaces for Laguerre expansions, J. Math. Pures Appl ), [6] N.N. Lebedev, Special functions and their applications, Revised Edition, Dover Publications, Inc., New Yor, [7] B. Mucenhoupt, Poisson integrals for Hermite and Laguerre expansions, Trans. Amer. Math. Soc ), [8] B. Mucenhoupt, Hermite conjugate expansions, Trans. Amer. Math. Soc ), [9] B. Mucenhoupt, Conjugate functions for Laguerre expansions, Trans. Amer. Math. Soc ), [1] B. Mucenhoupt, Hardy s inequality with weights, Studia Math ), [11] A. Nowa, On Riesz transforms for Laguerre expansions, J. Funct. Anal ), [12] A. Nowa, K. Stempa, L 2 -theory of Riesz transforms for orthogonal expansions, J. Fourier Anal. Appl ), [13] A. Nowa, K. Stempa, Riesz transforms for the Dunl harmonic oscillator, Math. Z ), [14] M. Rösler, Generalized Hermite polynomials and the heat equation for Dunl operators, Commun. Math. Phys ), [15] M. Rösler, Dunl operators: theory and applications, Orthogonal polynomials and special functions Leuven, 22), , Lecture Notes in Math. 1817, Springer, Berlin, 23. [16] P. Sjögren, On the maximal function for the Mehler ernel, Harmonic analysis Cortona, 1982), 73 82, Lecture Notes in Math. 992, Springer, Berlin, [17] R.P. Soni, On an inequality for modified Bessel functions, J. Math. and Phys ), [18] E.M. Stein, Topics in Harmonic Analysis Related to the Littlewood-Paley Theory, Annals of Math. Studies, Vol. 63, Princeton Univ. Press, Princeton, NJ, 197. Adam Nowa, Instytut Matematyi i Informatyi, Politechnia Wroc lawsa, Wyb. Wyspiańsiego 27, 5 37 Wroc law, Poland address: Adam.Nowa@pwr.wroc.pl Luz Roncal, Departamento de Matemáticas y Computación, Universidad de La Rioja, Edificio J.L. Vives, Calle Luis de Ulloa s/n, 264 Logroño, Spain address: luz.roncal@unirioja.es

15 RIESZ TRANSFORMS FOR THE DUNKL ORNSTEIN UHLENBECK OPERATOR 15 Krzysztof Stempa, Instytut Matematyi i Informatyi, Politechnia Wroc lawsa, Wyb. Wyspiańsiego 27, 5 37 Wroc law, Poland, and Katedra Matematyi i Zastosowań Informatyi, Politechnia Opolsa, Mio lajczya 5, Opole, Poland address: Krzysztof.Stempa@pwr.wroc.pl