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Singular operators and Fourier multipliers in weighted Lebesgue spaces with variable index

  • To the 100th Anniversary of Birthday of Solomon Grigor’Evich Mikhlin
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Abstract

Mikhlin’s ideas and results related to the theory of spaces L p(·)ρ with nonstandard growth are developed. These spaces are called Lebesgue spaces with variable index; they are used in mechanics, the theory of differential equations, and variational problems. The boundedness of Fourier multipliers and singular operators on the spaces L p(·)ρ are considered. All theorems are derived from an extrapolation theorem due to Rubio de Francia. The considerations essentially use theorems on the boundedness of operators and maximal Hardy-Littlewood functions on Lebesgue spaces with constant index.

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Authors and Affiliations

  1. Razmadze Mathematical Institute, Academy of Sciences of Georgia, ul. M. Aleksidze 1, Tbilisi, 0193, Georgia

    V. M. Kokilashvili

  2. University of Algarve, Campus de Gambelas, Faro, 8000-810, Portugal

    S. G. Samko

Authors
  1. V. M. Kokilashvili
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  2. S. G. Samko
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Additional information

Dedicated to the memory of Solomon Grigor’evich Mikhlin

Original Russian Text © V.M. Kokilashvili, S.G. Samko, 2008, published in Vestnik Sankt-Peterburgskogo Universiteta. Seriya 1. Matematika, Mekhanika, Astronomiya, 2008, No. 2, pp. 56–68.

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Kokilashvili, V.M., Samko, S.G. Singular operators and Fourier multipliers in weighted Lebesgue spaces with variable index. Vestnik St.Petersb. Univ.Math. 41, 134–144 (2008). https://doi.org/10.3103/S1063454108020076

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  • DOI: https://doi.org/10.3103/S1063454108020076

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