Fractional Weighted Spherical Mean and Maximal Inequality for the Weighted Spherical Mean and Its Application to Singular Pde

Abstract

In this paper we establish a mean value property for the functions which is satisfied to Laplace-Bessel equation. Our results involve the generalized divergence theorem and the second Green's identities relating the bulk with the boundary of a region on which differential Bessel operators act. Also we design a fractional weighted mean operator, study its boundedness, obtain maximal inequality for the weighted spherical mean and get its boundedness. The connection between the boundedness of the spherical maximal operator and the properties of solutions of the Euler-Poisson-Darboux equation with Bessel operators is given as an application.