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

The purpose of the paper is to introduce and to investigate a new class of fractional integrals connected with balls in ℝⁿ. 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(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.