Convolution inequalities for Besov and Triebel--Lizorkin spaces, and applications to convolution semigroups

Abstract

We establish convolution inequalities for Besov spaces $B_{p,q}^s$ and Triebel--Lizorkin spaces $F_{p,q}^s$. As an application, we study the mapping properties of convolution semigroups, considered as operators on the function spaces $A_{p,q}^s$, $A\in \{B,F\}$. Our results apply to a wide class of convolution semigroups including the Gauß--Weierstraß semigroup, stable semigroups and heat kernels for higher-order powers of the Laplacian $(-\Delta {)}^m$, and we can derive various caloric smoothing estimates.

Accepted for publication in Studia Mathematica