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Riesz-Kolmogorov theorem in variable exponent Lebesgue spaces and its applications to Riemann-Liouville fractional differential equations

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Abstract

In this paper, we obtain the necessary and sufficient condition of the pre-compact sets in the variable exponent Lebesgue spaces, which is also called the Riesz-Kolmogorov theorem. The main novelty appearing in this approach is the constructive approximation which does not rely on the boundedness of the Hardy-Littlewood maximal operator in the considered spaces such that we do not need the log-Hölder continuous conditions on the variable exponent. As applications, we establish the boundedness of Riemann-Liouville integral operators and prove the compactness of truncated Riemann-Liouville integral operators in the variable exponent Lebesgue spaces. Moreover, applying the Riesz-Kolmogorov theorem established in this paper, we obtain the existence and the uniqueness of solutions to a Cauchy type problem for fractional differential equations in variable exponent Lebesgue spaces.

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Acknowledgements

Baohua Dong was supported by the Startup Foundation for Introducing Talent of Nanjing University of Information Science and Technology (Grant No. 2017r098). Zunwei Fu was supported by National Natural Science Foundation of China (Grant Nos. 11671185 and 11771195) and National Science Foundation of Shandong Province (Grant No. ZR2017MA041). Jingshi Xu was supported by National Natural Science Foundation of China (Grant No. 11761026).

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Authors and Affiliations

  1. Department of Mathematics, Nanjing University of Information Science and Technology, Nanjing, 210044, China

    Baohua Dong

  2. Department of Mathematics, Linyi University, Linyi, 276005, China

    Zunwei Fu

  3. Department of Computer Science, The University of Suwon, Hwaseong, 445-743, Korea

    Zunwei Fu

  4. Department of Mathematics, Hainan Normal University, Haikou, 571158, China

    Jingshi Xu

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  1. Baohua Dong
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  2. Zunwei Fu
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  3. Jingshi Xu
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Correspondence to Zunwei Fu.

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Dong, B., Fu, Z. & Xu, J. Riesz-Kolmogorov theorem in variable exponent Lebesgue spaces and its applications to Riemann-Liouville fractional differential equations. Sci. China Math. 61, 1807–1824 (2018). https://doi.org/10.1007/s11425-017-9274-0

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  • DOI: https://doi.org/10.1007/s11425-017-9274-0

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