Riesz potential in generalized Morrey spaces on the Heisenberg group

Abstract

We consider the Riesz potential operator , on the Heisenberg group $$ {{\mathbb{H}}_n} $$ in generalized Morrey spaces $$ {M_{{p,\varphi }}}\left( {{{\mathbb{H}}_n}} \right) $$ and find conditions for the boundedness of as an operator from $$ {M_{{p,\varphi 1}}}\left( {{{\mathbb{H}}_n}} \right) $$ to $$ {M_{{p,\varphi 2}}}\left( {{{\mathbb{H}}_n}} \right) $$ , 1 < p < ∞, and from $$ {M_{{1,\varphi 1}}}\left( {{{\mathbb{H}}_n}} \right) $$ to a weak Morrey space $$ W{M_{{1,\varphi 2}}}\left( {{{\mathbb{H}}_n}} \right) $$ . The boundedness conditions are formulated it terms of Zygmund type integral inequalities. Based on the properties of the fundamental solution of the sub-Laplacian on $$ {{\mathbb{H}}_n} $$ , we prove two Sobolev–Stein embedding theorems for generalized Morrey and Besov–Morrey spaces. Bibliography: 40 titles.