Bounded compositions on scaling invariant Besov spaces

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For 0 < s < 1 < q < ∞, we characterize the homeomorphisms φ: ⁿ → ⁿ for which the composition operator f f ∘ φ is bounded on the homogeneous, scaling invariant Besov space Bˢₙ/ₛ,q(ⁿ), where the emphasis is on the case q=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 Fˢₙ/ₛ,q(ⁿ) with 0 < s < 1 and 0 < q ≤ ∞.