Dini Lipschitz functions for the Dunkl transform in the Space $$\mathrm {L}^{2}(\mathbb {R}^{d},w_{k}(x)dx)$$ L 2 ( R d , w k ( x ) d x )

Mohamed El Hamma,Radouan Daher

doi:10.1007/s12215-015-0195-9

Mohamed El Hamma, Radouan Daher

Open Access

https://doi.org/10.1007/s12215-015-0195-9

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Abstract

Using a generalized spherical mean operator, we obtain an analog of Theorem 5.2 in Younis (J Math Sci 9(2),301–312 1986) for the Dunkl transform for functions satisfying the $d$-Dunkl Dini Lipschitz condition in the space $\mathrm {L}^{2}(\mathbb {R}^{d},w_{k}(x)dx)$, where $w_{k}$ is a weight function invariant under the action of an associated reflection group.

Highlights

Introduction and preliminariesYounis Theorem 5.2 [13] characterized the set of functions in L2(R) satisfying the Cauchy Lipschitz condition by means of an asymptotic estimate growth of the norm of their Fourier transforms, namely, we have the followingTheorem 1.1 [13] Let f ∈ L2(R)
R −2α2β as h −→ 0, 0 < α < 1, β ≥ 0, as r −→ +∞, where F stands for the Fourier transform of f
The Dunkl transform shares several properties with its counterpart in the classical case, we mention here in particular that Parseval Theorem holds in Lk2 = Lk2(Rd ) = Lk2(Rd, wk (x)d x), when both f and f are in Lk1(Rd ), we have the inversion formula f (x) = f (ξ )Ek (i x, ξ )wk (ξ )dξ, x ∈ Rd

Summary

Introduction and preliminaries

Younis Theorem 5.2 [13] characterized the set of functions in L2(R) satisfying the Cauchy Lipschitz condition by means of an asymptotic estimate growth of the norm of their Fourier transforms, namely, we have the following. The theory of Dunkl operators provides generalizations of various multivariable analytic structures, among others we cite the exponential function, the Fourier transform and the translation operator. The Dunkl transform shares several properties with its counterpart in the classical case, we mention here in particular that Parseval Theorem holds in Lk2 = Lk2(Rd ) = Lk2(Rd , wk (x)d x), when both f and f are in Lk1(Rd ), we have the inversion formula f (x) = f (ξ )Ek (i x, ξ )wk (ξ )dξ, x ∈ Rd. Sd−1 where τx Dunkl translation operator (see [11,12]), η is the normalized surface measure on the unit sphere Sd−1 in Rd and set dηk (y) = wk (x)dη(y), ηk is a W -invariant measure on Sd−1 and dk = ηk (Sd−1).

Dini Lipschitz condition

New results on Dini Lipschitz class