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Fractional integrals and weakly singular integral equations of the first kind in then-dimensional ball

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Abstract

The purpose of the paper is to introduce and to investigate a new class of fractional integrals connected with balls in ℝn. A Riesz potentialI Ω α ρ over a ball Ω is represented by a composition of such integrals. Using this representation we obtain necessary and sufficient solvability conditions for the equationI Ω α ρ =f in the space Lpw) with a power weight w(x) and solve the equation in a closed form. The investigation is based on a special Fourier analysis adopted for operators commuting with rotations and dilations in ℝn.

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Author information

Correspondence to Boris Rubin.

Additional information

Supported in part by the National Council of Israel (grant no. 032-7251) and in part by the Edmund Landau Center for Research in Mathematical Analysis supported by the Minerva Foundation (Germany).

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Rubin, B. Fractional integrals and weakly singular integral equations of the first kind in then-dimensional ball. J. Anal. Math. 63, 55–102 (1994). https://doi.org/10.1007/BF03008419

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Keywords

  • Fractional Derivative
  • Singular Integral Equation
  • Solvability Condition
  • Inversion Formula
  • Fractional Integral