Convolution inequalities for Besov and Triebel–Lizorkin spaces, and applications to convolution semigroups
Volume 262 / 2022
Studia Mathematica 262 (2022), 93-119
MSC: Primary 46E35; Secondary 60J76, 60G51, 35K25.
DOI: 10.4064/sm210127-23-3
Published online: 12 July 2021
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.
