Abstract
We study \(\ell ^r\)-valued extensions of linear operators defined on Lebesgue spaces with variable exponent. Under some natural (and usual) conditions on the exponents, we characterize \(1\le r\le \infty \) such that every bounded linear operator \(T:L^{q(\cdot )}(\Omega _2, \mu )\rightarrow L^{p(\cdot )}(\Omega _1, \nu )\) has a bounded \(\ell ^r\)-valued extension. We consider both non-atomic measures and measures with atoms and show the differences that can arise. We present some applications of our results to weighted norm inequalities of linear operators and vector-valued extensions of fractional operators with rough kernel.
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Acknowledgements
We want to thank our friend Sheldy Ombrosi for useful conversations and comments regarding this work. We also thank the anonymous referees for many suggestions and comments that helped to improve the manuscript. Finally, we gratefully acknowledge the support provided by the scientific system of Argentina. Furthermore, we wish to express our concern regarding the ongoing defunding that universities and scientific agencies are currently facing. This work was partially supported by CONICET PIP 11220200102366CO, ANPCyT PICT 2018-04104, and UNCOMA PIN I 04/B251. The first author has a doctoral fellowship from CONICET.
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Communicated by Manuel Maestre.
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Bonich, M., Carando, D. & Mazzitelli, M. Marcinkiewicz–Zygmund inequalities in variable Lebesgue spaces. Banach J. Math. Anal. 18, 36 (2024). https://doi.org/10.1007/s43037-024-00344-y
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DOI: https://doi.org/10.1007/s43037-024-00344-y