– Multipliers on Commutative Hypergroups

Abstract

The main purpose of this paper is to prove Hörmander’s $L^p$ – $L^q$ boundedness of Fourier multipliers on commutative hypergroups. We carry out this objective by establishing the Paley inequality and Hausdorff–Young–Paley inequality for commutative hypergroups. We show the $L^p$ – $L^q$ boundedness of the spectral multipliers for the generalised radial Laplacian by examining our results on Chébli–Trimèche hypergroups. As a consequence, we obtain embedding theorems and time asymptotics for the $L^p$ – $L^q$ norms of the heat kernel for generalised radial Laplacian.