Boundedness of maximal operators and Sobolev inequalities on Musielak-Orlicz spaces over unbounded metric measure spaces

We prove the boundedness of the Hardy–Littlewood maximal operator $M_{\lambda },\lambda \ge 1$, on Musielak-Orlicz spaces $L^{\Phi }\left(X\right)$ over unbounded metric measure spaces as an extension of earlier results, where λ is its modification rate. As an application of the boundedness of $M_{\lambda }$, we establish a generalization of Sobolev inequalities for the variable Riesz potentials $I_{\alpha (\cdot ),\tau }f,\tau \ge 1$, on $L^{\Phi }\left(X\right)$ over unbounded metric measure spaces, where τ is its modification rate. As a corollary, we show the boundedness of $M_{\lambda }$ and Sobolev inequalities for $I_{\alpha (\cdot ),\tau }f$ for double phase functionals with variable exponents. Our results are new even for the doubling metric measure setting in that the underlying spaces need not be bounded.