Trudinger‐type inequalities for variable Riesz potentials of functions in Musielak–Orlicz–Morrey spaces over metric measure spaces

We study Trudinger‐type inequalities for variable Riesz potentials Jα(·),τf$J_{\alpha (\cdot ), \tau }f$ of functions in Musielak–Orlicz–Morrey spaces over bounded metric measure spaces. As a good example, we obtain Trudinger‐type inequalities for double‐phase functionals Φ(x,t)=tp(x)+a(x)tq(x)$\Phi (x,t) = t^{p(x)} + a(x) t^{q(x)}$. As an application, we introduce Musielak–Orlicz–Morrey–Hajłasz–Sobolev spaces and give Trudinger‐type inequalities for Sobolev functions.