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

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Abstract

Let \(G:\mathbb {R\rightarrow R}\) be a continuous function. We investigate necessary conditions on G such that

$$\begin{aligned} \{G(f):f\in A_{p,q}^{s}(\mathbb {R}^{n},|\cdot |^{\alpha })\}\subset A_{p,q}^{s}(\mathbb {R}^{n},|\cdot |^{\alpha }) \end{aligned}$$

holds. Here \(A_{p,q}^{s}(\mathbb {R}^{n},|\cdot |^{\alpha })\) stands for either the Besov space \(B_{p,q}^{s}(\mathbb {R}^{n},|\cdot |^{\alpha })\) or the Triebel-Lizorkin space \(F_{p,q}^{s}(\mathbb {R}^{n},|\cdot |^{\alpha })\). These spaces unify and generalize many classical function spaces such as Sobolev spaces with power weights.

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Acknowledgements

We thank the referee for carefully reading the manuscript and providing valuable suggestions. This work is funded by the General Direction of Higher Education and Training under Grant No. C00L03UN280120220004 and by The General Directorate of Scientific Research and Technological Development, Algeria.

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  1. Laboratory of Functional Analysis and Geometry of Spaces, Department of Mathematics, Faculty of Mathematics and Informatics, M’sila University, PO Box 166, Ichebelia, 28000, M’sila, Algeria

    Douadi Drihem

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Drihem, D. Nemytzkij operators on Besov and Triebel-Lizorkin spaces with power weights: necessary conditions. Boll Unione Mat Ital (2024). https://doi.org/10.1007/s40574-024-00413-y

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