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

Gómez Cobos, Santiago; Restrepo, Joel E.; Ruzhansky, Michael

Comptes Rendus. Mathématique, Tome 362 (2024) no. G11, pp. 1331-1336

We prove the $L^p-L^q$ $(1<p⩽2⩽q<+\infty )$ 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.

On montre les estimations de norme $L^p-L^q$ $(1<p⩽2⩽q<+\infty )$ pour les solutions des équations dites «  de type chaleur  » et «  de type onde  » définies sur un groupe localement compact, séparable et unimodulaire $G$ en utilisant un opérateur intégro-différentiel sur le temps et un opérateur positif invariant á gauche quelconque sur $G.$ De plus, on donne des estimations de temps asymptotiques pour ces solutions, qui deviennent des estimations optimales dans quelques cas.

Classification : 43A15, 43A85, 45K05
Keywords: Locally compact groups, heat type equations, wave type equations, asymptotic estimates, non-local operators
Mots-clés : Groupes localement compacts, équations de type chaleur, équations de type onde, estimations asymptotiques, opérateurs non locaux

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