Bounded compositions on scaling invariant Besov spaces

For $0 < s < 1 < q < \infty$, we characterize the homeomorphisms $φ: \real^n \to \real^n$ for which the composition operator $f \mapsto f \circ φ$ 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 $0 < q \leq \infty$.

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