## Abstract

The purpose of the paper is to introduce and to investigate a new class of fractional integrals connected with balls in ℝ^{n}. A Riesz potential*I*
_{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 equation*I*
_{Ω}
^{α}
ρ =*f* in the space L_{p}(Ω*w*) 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 ℝ^{n}.

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## 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 the*n*-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