On the interpolation constants for variable Lebesgue spaces |
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Authors: | Oleksiy Karlovych Eugene Shargorodsky |
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Institution: | 1. Centro de Matemática e Aplicações, Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Quinta da Torre, Caparica, Portugal;2. Department of Mathematics, King's College London, Strand, London, UK |
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Abstract: | For and variable exponents and with values in 1, ∞], let the variable exponents be defined by The Riesz–Thorin–type interpolation theorem for variable Lebesgue spaces says that if a linear operator T acts boundedly from the variable Lebesgue space to the variable Lebesgue space for , then where C is an interpolation constant independent of T. We consider two different modulars and generating variable Lebesgue spaces and give upper estimates for the corresponding interpolation constants Cmax and Csum, which imply that and , as well as, lead to sufficient conditions for and . We also construct an example showing that, in many cases, our upper estimates are sharp and the interpolation constant is greater than one, even if one requires that , are Lipschitz continuous and bounded away from one and infinity (in this case, ). |
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Keywords: | Calderón product complex method of interpolation interpolation constant Riesz–Thorin interpolation theorem variable Lebesgue space |
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