On Trudinger-type inequalities in Musielak–Orlicz–Morrey spaces of an integral form over metric measure spaces

Abstract

We establish Trudinger-type inequalities for variable Riesz potentials $J_{\alpha (\cdot ),\tau }f$ of functions $f$ in Musielak–Orlicz–Morrey spaces of an integral form over metric measure spaces $X$. As an application and example, we give Trudinger’s inequality for double-phase functionals with variable exponents. Finally, we prove the result for Sobolev functions satisfying a Poincaré inequality in $X$.