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[Math.CA] 1 Dec 2004 Inof Sion Oeof Role Eea.I at Vnthe Even Fact, in General RIESZ TRANSFORM AND RIESZ POTENTIALS FOR DUNKL TRANSFORM SUNDARAM THANGAVELU AND YUAN XU Abstract. Analogous of Riesz potentials and Riesz transforms are defined and studied for the Dunkl transform associated with a family of weighted functions that are invariant under a reflection group. The Lp boundedness of these operators is established in certain cases. 1. Introduction For a family of weighted functions, hκ, invariant under a finite reflection group, Dunkl transform is an extension of the Fourier transform that defines an isometry 2 Rd 2 of L ( ,hκ) onto itself. The basic properties of the Dunkl transforms have been studied by several authors, see [2, 4, 6, 7, 8, 11, 12] and the references therein. Giving the important role of Fourier transform in analysis, one naturally asks if it is possible to extend results established for the Fourier transform to the Dunkl transform. In analogous to the ordinary Fourier analysis, one can define a convolution op- erator and study various summabilities of the inverse Dunkl transforms. The con- volution is defined through a generalized translation operator, τy, which plays the role of f f( y) but is defined in the Dunkl transform side. The explicit expres- 7→ ·− sion of τyf is known only in some special cases and it is not a positive operator in p general. In fact, even the L boundedness of τy is not established in general. This is the main reason that only part of the results for the Fourier transforms has been extended to the Dunkl transform at the moment. Recently, in [11], the Lp theory for convolution operators was studied. In par- ticular, the Lp boundedness of the convolution operator is established in the case that the kernel is a suitable radial function. Furthermore, a maximal function is defined and shown to be of strong type (p,p) and weak type (1, 1). This provides arXiv:math/0412037v1 [math.CA] 1 Dec 2004 a handy tool for extending some results from the Fourier transform to the Dunkl transform. In the present paper we study the analogous of the Riesz potentials and the Riesz transforms for the Dunkl transform. We will study the boundedness of the Riesz potentials as well as the related Bessel potentials. The Riesz transforms are examples of singular integrals. A general theory of singular integral for the Dunkl transform appears to be out of reach at the moment. We will prove the Lp boundedness of the weighted Riesz transform only in a very special case of d = 1 Date: February 8, 2008. 1991 Mathematics Subject Classification. 42A38, 42B08, 42B15. Key words and phrases. Dunkl transforms, reflection invariance, Riesz transform, singular integrals. The work of YX was supported in part by the National Science Foundation under Grant DMS-0201669. 1 2 SUNDARAM THANGAVELU AND YUAN XU and G = Z2. Even in this simple case, however, the proof turns out to be rather nontrivial. The paper is organized as follows. In the next section we collect the background materials. In Section 3 we recall the definition of the ordinary Riesz transforms and Riesz potentials, and prove a weighted Lp boundedness for the Riesz potentials that will be used later in the paper. The weighted Riesz potentials and the Bessel potentials for the Dunkl transform will be studied in Section 4. The weighted Riesz transform is discussed in Section 5. Throughout this paper we use the convention that c denotes a generic constant, depending on d, p, κ or other fixed parameters, its value may change from line to line. 2. Preliminaries 2.1. Dunkl Transform. The Dunkl transform is associated to a weight function that is invariant under a reflection group. Let G be a finite reflection group on d R with a fixed positive root system R+, normalized so that v, v = 2 for all v R , where x, y denotes the usual Euclidean inner product.h i Let κ be a ∈ + h i nonnegative multiplicity function v κv defined on R+ with the property that κ = κ whenever σ is conjugate7→ to σ in G; then v κ is a G-invariant u v u v 7→ v function. The weight function hκ is defined by (2.1) h (x)= x, v κv , x Rd. κ |h i| ∈ v∈YR+ This is a positive homogeneous function of degree γκ := v∈R+ κv, and it is invari- ant under the reflection group G. P To define the Dunkl transform we will also need the intertwining operator Vκ. Let denote Dunkl’s differential-difference operators defined by [1] Dj f(x) f(xσ ) f(x)= ∂ f(x)+ k − v v,ε , 1 j d, Dj j v x, v h j i ≤ ≤ vX∈R+ h i d where ε1,...,εd are the standard unit vectors of R and σv denote the reflection 2 with respect to the hyperplane perpendicular to v, xσv := x 2( x, v / v )v, Rd d d − hd i k k x . The operators j , 1 j d, map n to n−1, where n denotes the space∈ of homogeneous polynomialsD ≤ ≤ of degree nPin d variables,P andP they mutually commute; that is, i j = j i, 1 i, j d. The intertwining operator Vκ is a linear operator determinedD D D uniquelyD ≤ by ≤ V , V 1=1, V = V ∂ , 1 i d. κPn ⊂Pn κ Di κ κ i ≤ ≤ Zd The explicit formula of Vκ is not known in general. For the group G = 2, hκ(x)= d x κi , it is an integral transform i=1 | i| Q d 2 κi−1 (2.2) Vκf(x)= bκ f(x1t1,...,xdtd) (1 + ti)(1 ti ) dt. d − [−1,1] i=1 Z Y It is known that V is a positive operator [6]; that is, p 0 implies V p 0. κ ≥ κ ≥ RIESZ TRANSFORM AND RIESZ POTENTIALS FOR DUNKL TRANSFORM 3 (x) ihx,yi d Let E(x,iy) = Vκ e , x, y R , where the superscript means that V is ∈ κ applied to the x variable. For f L1(Rd,h2 ), the Dunkl transform is defined by ∈ κ 2 (2.3) f(y)= ch f(x)E(x, iy)hκ(x)dx Rd − Z −1 2 −kxk2/2 where ch is the constantb defined by ch = Rd hκ(x)e dx. If κ = 0 then V = id and the Dunkl transform coincides with the usual Fourier transform. If κ R d = 1 and G = Z2, then the Dunkl transform is related closely to the Hankel transform on the real line. Some of the properties of the Dunkl transform is collected below ([2, 4]). 1 Rd 2 Rd Proposition 2.1. (1) For f L ( ,hκ), f is in C0( ). ∈ 1 Rd 2 (2) When both f and f are in L ( ,hκ) we have the inversion formula b 2 fb(x)= E(ix,y)f(y)hκ(y)dy. Rd Z 2 Rd 2 (3) The Dunkl transform extends to an isometryb of L ( ,hκ). 2.2. Generalized translation operator. Let y Rd be given. The generalized translation operator f τ f is defined on L2(Rd,h∈2 ) by the equation 7→ y κ (2.4) τ f(x)= E(y, ix)f(x), x Rd. y − ∈ d It plays the role of the ordinary translation τyf = f( y) of R , since the Fourier d −ihx,ybi ·− transform of τy is given by τyf(x)= e f(x). The generalized translation operator has been studied in [6, 7, 11, 12]. The 2 definition gives τyf as an Ldfunction. Let usb define (Rd)= f L1(Rd; h2 ) : f L1(Rd; h2 ) . Aκ { ∈ κ ∈ κ } d Then (2.4) holds pointwise. Note that κ(R ) is contained in the intersection of 1 Rd 2 ∞ A b2 Rd 2 L ( ; hκ) and L and hence is a subspace of L ( ; hκ). The operator τy satisfies the following properties: Proposition 2.2. Assume that f (Rd) and g L1(Rd; h2 ) is bounded. Then ∈ Aκ ∈ κ 2 2 (1) τyf(ξ)g(ξ)hκ(ξ)dξ = f(ξ)τ−yg(ξ)hκ(ξ)dξ. Rd Rd Z Z (2) τ f(x)= τ f( y). y −x − A formula of τyf is known, at the moment, only in two cases. One is in the case of G = Z and h (x)= x κ on R ([5]) 2 κ | | 1 1 2 2 x y (2.5) τyf(x)= f x + y 2xyt 1+ − Φκ(t)dt 2 2 2 −1 − x + y 2xyt Z 1p − 1 2 2 p x y + f x + y 2xyt 1 − Φκ(t)dt, 2 −1 − − − x2 + y2 2xyt Z − 2pκ−1 where Φκ(t)= bκ(1 + t)(1 t ) , from which also followsp a formula of τyf in the Zd − case of G = 2. The explicit formula implies the boundendess of τyf. Let κ,p p Rd 2 k·k denote the norm of L ( ,hκ). Proposition 2.3. Let G = Zd. For f Lp(Rd,h2 ), 1 p , 2 ∈ κ ≤ ≤ ∞ τ f c f . k y kκ,p ≤ k kκ,p 4 SUNDARAM THANGAVELU AND YUAN XU Another case where a formula for τyf is known is when f are radial functions, f(x)= f ( x ), and G being any reflection group ([7]), 0 k k (2.6) τ f(x)= V f x 2 + y 2 2 x y x′, (y′), y κ 0 k k k k − k k k kh ·i from which it follows thathτ f(px) 0 for all y Rd if f(x)= fi( x ) 0. y ≥ ∈ 0 k k ≥ Several essential properties of τyf is established for f being radial functions.
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