For a given quasiconformal mapping f:Rn→Rn,\documentclass[12pt]{minimal}
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\begin{document}$$f{\text {:}}\,{\mathbb {R}}^n\rightarrow {\mathbb {R}}^n,$$\end{document} the authors show that the boundedness or the unboundedness of the composition operator Cf\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf{C}_f$$\end{document} on Triebel–Lizorkin type spaces F˙p,qα,1/p-α/n(Rn)\documentclass[12pt]{minimal}
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\begin{document}$$\dot{\mathbf{F}}^{\alpha ,\,1/p-\alpha /n}_{p,\,q}({\mathbb {R}}^n)$$\end{document} or, more generally, Hajłasz–Triebel–Lizorkin type spaces M˙p,qα,1/p-α/n(Rn)\documentclass[12pt]{minimal}
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\begin{document}$$\dot{\mathbf{M}}^{\alpha ,\,1/p-\alpha /n}_{p,\,q}({\mathbb {R}}^n)$$\end{document} depends on the indexes α,p\documentclass[12pt]{minimal}
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\begin{document}$$\alpha ,\,p$$\end{document} and the degenerate sets of the Jacobian Jf,\documentclass[12pt]{minimal}
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\begin{document}$$J_f,$$\end{document} but not on the index q. Actually, the following dual relation is proved to be sharp to obtain the boundedness of Cf\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf{C}_f$$\end{document} on F˙p,qα,1/p-α/n(Rn)\documentclass[12pt]{minimal}
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\begin{document}$$\dot{\mathbf{F}}^{\alpha ,\,1/p-\alpha /n}_{p,\,q}({\mathbb {R}}^n)$$\end{document} and M˙p,qα,1/p-α/n(Rn):\documentclass[12pt]{minimal}
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\begin{document}$$\dot{\mathbf{M}}^{\alpha ,\,1/p-\alpha /n}_{p,\,q}({\mathbb {R}}^n){\text {:}}$$\end{document}αp<n-dim¯LEwhenEisbounded;n-dim¯GEwhenEisunbounded,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \alpha p<\left\{ \begin{array}{ll} n-\overline{\dim }_{L}E&{}\quad \mathrm{when}\,E\,\mathrm{is\,bounded};\\ n-\overline{\dim }_{G}E&{}\quad \mathrm{when}\, E \,\mathrm{is\,unbounded}, \end{array}\right. \end{aligned}$$\end{document}where α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document} is the regularity index, p the integration index and dim¯LE\documentclass[12pt]{minimal}
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\begin{document}$$\overline{\dim }_{L}E$$\end{document} (resp., dim¯GE\documentclass[12pt]{minimal}
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\begin{document}$$\overline{\dim }_{G}E$$\end{document}) the local (resp., global) self-similar Minkowski dimension of the degenerate set E of Jf.\documentclass[12pt]{minimal}
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\begin{document}$$J_f.$$\end{document} This is completely different from the results for Sobolev, BMO, Besov and Triebel–Lizorkin spaces, and extends the recent result for Q-spaces. Finally, the authors show that, if n-1<αp<n,\documentclass[12pt]{minimal}
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\begin{document}$$n-1<\alpha p<n,$$\end{document} then a homeomorphism for which the composition operator is bounded on M˙p,qα,1/p-α/n(Rn)\documentclass[12pt]{minimal}
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\begin{document}$$\dot{\mathbf{M}}^{\alpha ,\,1/p-\alpha /n}_{p,\,q} ({\mathbb {R}}^n)$$\end{document} must be quasiconformal.
