One-sided Fractional Integrals and Riesz Potentials on a Spherical Cap

Abstract

We introduce fractional integrals on the $n$-dimensional spherical cap, study their boundednes in weighted $L^p$ spaces and obtain explicit inversion formulas. The results are applied to the inversion problem for Riesz potentials on a spherical cap. In the case $n=2$, this problem arises in electrostatics, elasticity, and some other applied areas. The main tools are stereographic projection, hypersingular integrals of the Marchaud type, Poisson integrals, and spherical harmonic decompositions.