Bounded compositions on scaling invariant Besov spaces

TL;DR

This paper characterizes the boundedness of composition operators on certain scaling invariant Besov spaces and extends the results to metric measure spaces and Triebel-Lizorkin spaces, addressing previously open cases.

Contribution

It provides a complete characterization of homeomorphisms inducing bounded composition operators on homogeneous Besov spaces for the case q ≠ n/s, filling a gap in the literature.

Findings

Characterization of bounded composition operators on Besov spaces for q ≠ n/s

Extension of results to Besov-type spaces on metric measure spaces

Remarks on scaling invariant Triebel-Lizorkin spaces

Abstract

For $0 < s < 1 < q < \infty$, we characterize the homeomorphisms $\varphi : \real^n \to \real^n$ for which the composition operator $f \mapsto f \circ \varphi$ is bounded on the homogeneous, scaling invariant Besov space $\dot{B}^s_{n/s,q}(\real^n)$, where the emphasis is on the case $q\not=n/s$, 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 $\dot{F}^s_{n/s,q}(\real^n)$ with $0 < s < 1$ and $0 < q \leq \infty$.