Hypoellipticity of Hankel convolution equations on Schwartz distribution spaces

Abstract

In this paper we give a sufficient condition in order that a Hankel convolution equation is hypoelliptic on Schwartz distributions.

Hypoelliptic convolution equations in $\cal{S}' (\Bbb R)$ for the Dunkl theory on $\Bbb R$

“…Analogous result is obtained by Trimèche [18] for the Dunkl convolution on R d . Similar characterizations are established for other convolutions, see for example [1], [11] and [19].…”

Hypoelliptic Jacobi–Dunkl Convolution of Distributions

“…Ehrempreis [5] and next L. Hörmander [6] have characterized the hypoelliptic convolution equations of distributions on R d by giving conditions on their classical Fourier transform on R d . The same results is proved in the case of the Hankel convolution equation in [1] and for the Jacobi convolution equation in [3]. But in [10] by simple proofs K. Trimèche has given a characterization of the hypoelliptic convolution of distributions on Chébli-Trimèche hypergroups, in terms of their generalized Fourier transform.…”

Hypoelliptic Dunkl Convolution Equations in the Space of Distributions on ${\Bbb R}^d$

“…This characterization was first given by L. Ehrenpreis [8] and next by L. Hörmander [11] in the case of the classical Fourier transform on R d . In [1,2] the authors have studied this characterization for the Hankel, Jacobi, and Chébli-Trimèche transforms. We remark that their proof of the existence of a parametrix is complicated.…”