Nemytzkij operators on Besov and Triebel-Lizorkin spaces with power weights: necessary conditions

Let G:R -> R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G:\mathbb {R\rightarrow R}$$\end{document} be a continuous function. We investigate necessary conditions on G such that {G(f):f is an element of Ap,qs(Rn,|<middle dot>|alpha)}subset of Ap,qs(Rn,|<middle dot>|alpha)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \{G(f):f\in A_{p,q}<^>{s}(\mathbb {R}<^>{n},|\cdot |<^>{\alpha })\}\subset A_{p,q}<^>{s}(\mathbb {R}<^>{n},|\cdot |<^>{\alpha }) \end{aligned}$$\end{document}holds. Here Ap,qs(Rn,|<middle dot>|alpha)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{p,q}<^>{s}(\mathbb {R}<^>{n},|\cdot |<^>{\alpha })$$\end{document} stands for either the Besov space Bp,qs(Rn,|<middle dot>|alpha)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{p,q}<^>{s}(\mathbb {R}<^>{n},|\cdot |<^>{\alpha })$$\end{document} or the Triebel-Lizorkin space Fp,qs(Rn,|<middle dot>|alpha)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_{p,q}<^>{s}(\mathbb {R}<^>{n},|\cdot |<^>{\alpha })$$\end{document}. These spaces unify and generalize many classical function spaces such as Sobolev spaces with power weights.