$L^p$-$L^q$ Multipliers on commutative hypergroups

TL;DR

This paper proves H"ormander's $L^p$-$L^q$ boundedness of Fourier multipliers on commutative hypergroups, extending classical harmonic analysis results to a broader algebraic setting and applying them to PDEs.

Contribution

It establishes Paley and Hausdorff-Young-Paley inequalities for commutative hypergroups and demonstrates $L^p$-$L^q$ boundedness of spectral multipliers for generalized radial Laplacians.

Findings

Proved H"ormander's $L^p$-$L^q$ boundedness for hypergroup Fourier multipliers.

Derived embedding theorems and heat kernel asymptotics for generalized radial Laplacians.

Applied results to the well-posedness of nonlinear PDEs.

Abstract

The main purpose of this paper is to prove H\"ormander's $L^p$-$L^q$ boundedness of Fourier multipliers on commutative hypergroups. We carry out this objective by establishing 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 Ch\'{e}bli-Trim\`{e}che 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. Finally, we present applications of the obtained results to study the well-posedness of nonlinear partial differential equations.