Bounded compositions on scaling invariant Besov spaces

Abstract

For $0<s<1<q<\infty$, we characterize the homeomorphisms $\phi :{\Re }^n\to {\Re }^n$ for which the composition operator $f\mapsto f\circ \phi$ is bounded on the homogeneous, scaling invariant Besov space ${\dot{B}}_{n/s,q}^s({\Re }^n)$, where the emphasis is on the case $q\ne n/s$, left open in the previous literature. We also establish an analogous result for Besov-type function spaces on a wide class of metric measure spaces as well, and make some new remarks considering the scaling invariant Triebel-Lizorkin spaces ${\dot{F}}_{n/s,q}^s({\Re }^n)$ with $0<s<1$ and $0<q\le \infty$.

20 pages; corrected typos, simplified assumptions for Proposition 3.5 and Theorem 3.6