Vanishing integrability for generalized Riesz potentials in weighted Morrey–Orlicz spaces and differentiability

Our aim in this paper is to establish integrabilities for generalized Riesz potentials R_{\alpha,m;x_0}f(x) = \int_{\R^n}R_{\alpha,m;x_0}(x,y) f(y) dy of functions f in weighted Morrey–Orlicz spaces, where x_{0}\in \R^{n} , R_{\alpha}(x) = |x|^{\alpha -n} , and R_{\alpha,m;x_0}(x,y) = R_{\alpha}(x-y) - \sum_{|\ell| \le m-1}\frac{(x-x_{0})^{\ell}}{\ell !}(D^{\ell}R_{\alpha})(x_{0}-y), which was first introduced by Hayman for the integral representation of subharmonic functions and then by the first author for the integral representation of Sobolev functions. As an application, we discuss integral mean differentiability of Riesz potentials with exceptional sets whose size are evaluated by (\alpha,p) -capacity.