Hypoelliptic Jacobi--Dunkl convolution of distributions (Q2462241)

The hypoelliptic convolution equations of distributions were characterized by means of their Fourier transform by \textit{L.\,Ehrenpreis} [Am.\ J.\ Math.\ 82, 522--588 (1960; Zbl 0098.08401)] and later by \textit{L.\,Hörmander} [Math.\ Scand.\ 9, 178--181 (1961; Zbl 0102.11003)]. The second author [\textit{K.\,Trimèche}, Glob.\ J.\ Pure Appl.\ Math.\ 1, No.\,3, 251--271 (2005; Zbl 1106.46304)] gave characterizations of hypoelliptic convolution of distributions on Chébli--Trimèche hypergroups in terms of their generalized Fourier transform. In the paper under review, the hypoelliptic convolution equations of distributions are characterized by means of their Jacobi--Dunkl transform. To this end, a previous study of the harmonic analysis associated to the Jacobi--Dunkl operator is accomplished.