Trudinger-type inequalities for double-phase functionals with variable exponents over metric measure spaces

We prove Trudinger-type inequalities for variable Riesz potentials $I_{\alpha(\cdot),\tau}f$ of functions in Musielak—Orlicz—Morrey spaces $L^{\Phi,\kappa,\theta}(X)$ over bounded non-doubling metric measure spaces $X$, under conditions on $\Phi$ which are weaker than those considered in the previous paper by Hurri-Syrjänen and the authors in 2023. We also discuss the case when $\Phi$ is the double phase functional with variable exponents.