Titchmarsh ’ s Theorem for the Dunkl transform in the space L 2 ( R d , wk ( x ) dx )

Using a generalized spherical mean operator, we obtain a generalization of Titchmarsh’s theorem for the Dunkl transform for functions satisfying the (ψ,α, β)-Dunkl Lipschitz condition in L(R, wk(x)dx).


Intoduction and preliminaries
In [10], E. C. Titchmarsh characterized the set of functions in L 2 (R) satisfying the Cauchy Lipschitz condition for the Fourier transform, namely we have Theorem 1.1.Let α ∈ (0, 1) and assume that f ∈ L 2 (R).Then the following are equivalents where F stands for the Fourier transform of f .
The main aim of this paper is to establish a generalization of Theorem 1.1 in the Dunkl transform setting by means of the generalized spherical mean operator.
In this paper we consider the Dunkl operators T j , j = 1, 2, ..., d, which are the differential-difference operators introduced by C.F. Dunkl in [3].These operators are very important in pure mathematics and in physics.
We consider R d with the Euclidean scalar product h., .iand |x| = p hx, xi.For α ∈ R d \{0}, let σ α be the reflection in the hyperplane A function k : R −→ C on a root system R is called a multiplicity function, if it is invariant under the action of the associated reflection group W .
If one regards k as a function on the corresponding reflections, this means that k is constant on the conjugacy classes of reflections in W .
We consider the weight function where w k is W -invariant and homogeneous of degree 2γ where We let η be the normalized surface measure on the unit sphere S d−1 in R d and set Introduced by C.F. Dunkl in [3] the Dunkl operators T j , 1 ≤ j ≤ d, on R d associated with the reflection group W and the multiplicity function k are the first-order differential-difference operators given by where α j = hα, e j i; (e 1 , ...., e d ) being the canonical basis of R d and This kernel has unique holomorphic extension to M. Rösler has proved in [8] the following integral representation for the Dunkl kernel where µ x is a probability measure on R d with support in the closed ball B(0, |x|) of center 0 and raduis |x|. where In particulier where the constant c k is given by According to [4,6,9] we have the following results: 1.When both f and b f are in L 1 k (R d ), we have the inversion formula . (Plancherel's theorem) The Dunkl transform on S(R d ), the space of Schwartz functions, extends uniquely to an isometric isomorphism on K. Trimèche has introduced [11] the Dunkl translation operators and The generalized spherical mean operator for f ∈ L 2 k (R d ) is defined by From [7], we have For p ≥ − 1 2 , we introduce the normalized Bessel functuion j p defined by where Γ is the gamma-function.
Lemma 1.3.[1] The following inequalities are fulfilled Lemma 1.4.The following inequality is true with |x| ≥ 1, where c > 0 is a certain constant.
here I is the unit operator.
From formula (1.1) and proposition 1.5, we have By Parseval's identity, we obtain The lemma is proved

Main Results
In this section we give the main result of this paper.We need first to define (ψ, α, β)-Dunkl Lipschitz class.