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

TL;DR

This paper develops convolution inequalities for Besov and Triebel--Lizorkin spaces and applies these results to analyze the smoothing properties of various convolution semigroups, including heat kernels and stable processes.

Contribution

It introduces new convolution inequalities for Besov and Triebel--Lizorkin spaces and explores their implications for the mapping properties of convolution semigroups.

Findings

Established convolution inequalities for $B_{p,q}^s$ and $F_{p,q}^s$ spaces.

Analyzed the smoothing effects of Gaussian, stable, and higher-order heat semigroups.

Derived caloric smoothing estimates for various 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 Gau{\ss}--Weierstra{\ss} semigroup, stable semigroups and heat kernels for higher-order powers of the Laplacian $(-\Delta)^m$, and we can derive various caloric smoothing estimates.