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In this chapter, we start by deriving a variable exponent Sobolev theorem for the fractional integral of variable-order \(\alpha (x)\) over a bounded open set \(\varOmega \) in a quasi-metric measure space \((X, d, \mu )\) with quasi-distance d and measure \(\mu \) , where the dimension \(n(x)\) comes from the growth condition on the measure \(\mu \) . Based on this theorem, applications to spherical potential operators, fractional integrals on Carleson curves, and anisotropic potentials are also obtained.

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Fractional Integrals of Variable-Order Lebesgue and Morrey Spaces

  • 章节

  • 发表日期：

    2026年

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  • Vakhtang Kokilashvili,
  • Alexander Meskhi,
  • Humberto Rafeiro,
  • Stefan Samko

摘要

In this chapter, we start by deriving a variable exponent Sobolev theorem for the fractional integral of variable-order \(\alpha (x)\) over a bounded open set \(\varOmega \) in a quasi-metric measure space \((X, d, \mu )\) with quasi-distance d and measure \(\mu \) , where the dimension \(n(x)\) comes from the growth condition on the measure \(\mu \) . Based on this theorem, applications to spherical potential operators, fractional integrals on Carleson curves, and anisotropic potentials are also obtained.