Boundedness Results for the Fractional Wolff Potential in Morrey Spaces over Homogeneous Groups

Abstract

Let G be a homogeneous Lie group of dimension ϰ. In this article, we investigate the mapping properties of the Wolff-type potential Wϑ,2 on G, associated with a homogeneous quasi-norm, and show that it coincides, up to an explicit dimensional constant, with the Riesz potential of doubled order 2ϑ on G. Using this identity together with Hedberg’s trick and a dyadic decomposition of the convolution kernel, we establish single-weight and two-weight boundedness inequalities for Wϑ,2 in the global Morrey spaces Mpμ(G). In contrast to the linear Riesz and Bessel–Riesz potentials, the Wolff potential belongs to a broader family of operators associated with quasilinear elliptic equations of p-Laplace type. We believe that these boundedness properties and the associated inequalities play an important role in the further advancement of nonlinear potential theory.