Asymptotics and inversion of Riesz potentials through decomposition in radial and spherical parts
- 154 Downloads
It is known that radial symmetry is preserved by the Riesz potential operators and also by the hypersingular Riesz fractional derivatives typically used for inversion. In this paper, we collect properties, asymptotics, and estimates for the radial and spherical parts of Riesz potentials and for solutions to the Riesz potential equation of order one. Sharp estimates for spherical functions are provided in terms of seminorms, and a careful analysis of the radial part of a Riesz potential is carried out in elementary terms. As an application, we provide a two weight estimate for the inverse of the Riesz potential operator of order one acting on spherical functions.
KeywordsRiesz potentials Singular integrals Weighted spaces Radial functions Spherical symmetry
Mathematics Subject Classification47G40 45E99 26A33
- 1.De Nápoli, P.L., Drelichman, I.: Weighted convolution inequalities for radial functions. Ann. Mat. Pura Appl. (2013). doi: 10.1007/s10231-013-0370-6
- 7.Mazya, V.G.: Sobolev Spaces. Springer, Berlin (1985)Google Scholar
- 12.Rubin, B.: One-dimensional representation, inversion and certain properties of Riesz potentials of radial functions. Mat. Zametki 34(4), 521–533; English translation. Math. Notes 34(3–4), 751–757 (1983)Google Scholar