$L^p-L^q$ estimates for non-local heat and wave type equations on locally compact groups

TL;DR

This paper establishes $L^p-L^q$ estimates for heat and wave equations on locally compact groups, using integro-differential operators and invariant operators, with sharp asymptotic time behavior in some cases.

Contribution

It provides new $L^p-L^q$ estimates for non-local heat and wave equations on groups, extending classical results to a broader non-commutative setting.

Findings

Derived $L^p-L^q$ bounds for solutions on groups

Provided asymptotic time estimates, some of which are sharp

Extended analysis to integro-differential operators on groups

Abstract

We prove the $L^p-L^q$ $(1<p\leqslant 2\leqslant q<+\infty)$ norm estimates for the solutions of heat and wave type equations on a locally compact separable unimodular group $G$ by using an integro-differential operator in time and any positive left invariant operator (maybe unbounded) on $G$. We complement our studies by giving asymptotic time estimates for the solutions, which in some cases are sharp.