Generalized fractional integral operators on Musielak–Orlicz spaces over unbounded metric measure spaces
DOI:
https://doi.org/10.2140/mscand.2026.132.289Keywords:
maximal functions, fractional integral operators, Musielak–Orlicz spaces, Sobolev's inequality, metric measure spaces, nondoubling measure, multiphase functionalAbstract
We establish the boundedness of generalized fractional integral operators $I_{\rho,\tau}$, $\tau\ge1$, on Musielak–Orlicz spaces $L^{\Phi}(X)$ over unbounded nondoubling metric measure spaces $X$, where $\rho$ depends on $x \in X$. As an application, we give Sobolev's inequality in the framework of multiphase functionals with variable exponents $$ \Phi(x,t) = t^{p(x)} + a(x) t^{q(x)}+ b(x) t^{s(x)}, \quad x \in X, \, t \ge 0, $$ where $p(\,\cdot\,)$, $q(\,\cdot\,)$ and $s(\,\cdot\,)$ are log-Hölder continuous, $p(x)<q(x) \le s(x)$ for $x\in X$ and $a(\,\cdot\,)$ and $b(\,\cdot\,)$ are nonnegative, bounded and Hölder continuous.
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