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Isomorphisms of variable Hardy spaces associated with Schrödinger operators

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Let L:= −Δ + V be the Schrödinger operator on ℝn with n ≥ 3, where V is a non-negative potential satisfying Δ−1 (V) ∈ L∞(ℝn). Let w be an L-harmonic function, determined by V, satisfying that there exists a positive constant ö such that, for any x ∈ ℝn, 0 < δ ≤ w(x) ≤ 1. Assume that p(·): ℝn → (0, 1] is a variable exponent satisfying the globally log-Hölder continuous condition. In this article, the authors show that the mappings \(H_L^{p\left( \cdot \right)}\left( {{\mathbb{R}^n}} \right)f \mapsto wf \in {H^{p\left( \cdot \right)}}\left( {{\mathbb{R}^n}} \right)\) and \(H_L^{p\left( \cdot \right)}\left( {{\mathbb{R}^n}} \right)f \mapsto {\left( { - {\rm{\Delta }}} \right)^{1/2}}{L^{ - 1/2}}\left( f \right) \in {H^{p\left( \cdot \right)}}\left( {{\mathbb{R}^n}} \right)\) are isomorphisms between the variable Hardy spaces (ℝn), associated with L, and the variable Hardy spaces Hp(·)(ℝn).

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Acknowledgements

Junqiang ZHANG would like to thank Sibei YANG for a helpful discussion on the subject of this article.

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Correspondence to Dachun Yang.

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Junqiang Zhang was supported by the National Natural Science Foundation of China (11801555 and 11971058) and the Fundamental Research Funds for the Central Universities (2020YQLX02). Dachun Yang was supported by the National Natural Science Foundation of China (11971058, 11761131002 and 11671185).

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Zhang, J., Yang, D. Isomorphisms of variable Hardy spaces associated with Schrödinger operators. Acta Math Sci 41, 39–66 (2021). https://doi.org/10.1007/s10473-021-0103-7

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  • DOI: https://doi.org/10.1007/s10473-021-0103-7

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