Trace Operators in Besov and Triebel–Lizorkin Spaces

Abstract

We determine the trace of Besov spaces ${\mathfrak{B}}_{p,q}^s({\mathbb{R}}^n)$ and Triebel–Lizorkin spaces ${\mathfrak{F}}_{p,q}^s({\mathbb{R}}_n)$ – characterized via atomic decompositions – on hyperplanes ${\mathbb{R}}^m$, $n>m\in \mathbb{N}$, for parameters $0<p,q\le \infty$ and $s>\frac{n-m}{p}$. The limiting case $s=\frac{n-m}{p}$ is investigated as well. Our results remain valid considering the classical spaces ${\mathbf{B}}_{p,q}^s$, ${\mathbf{F}}_{p,q}^s$ – defined via differences. Finally, we include some density assertions, which imply that the trace does not exist when $s<\frac{n-m}{p}$.