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 Lp(Ωw) 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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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
- Fractional Derivative
- Singular Integral Equation
- Solvability Condition
- Inversion Formula
- Fractional Integral