Generalized fractional integral operators on Musielak–Orlicz–Morrey spaces of an integral form over metric measure spaces

Takao Ohno and Tetsu Shimomura

Mathematische Nachrichten, 2025, vol. 298, issue 12, 3729-3756

Abstract: In this paper, we discuss the boundedness of generalized fractional integral operators Iρ,τ$I_{\rho,\tau }$ on Musielak–Orlicz–Morrey spaces of an integral form LΦ,ω,θ1(X)$\mathcal {L}^{\Phi,\omega, \theta _1}(X)$ over bounded non‐doubling metric measure spaces X$X$, where both ρ$\rho$ and ω$\omega$ depend on x∈X$x \in X$. As an application, we give Sobolev‐type inequalities for multiphase functions Φ(x,t)=tp(x)+a(x)tq(x)+b(x)ts(x),x∈X,t≥0,$$\begin{equation*} \hspace*{4pc}\Phi (x,t) = t^{p(x)} + a(x) t^{q(x)}+ b(x) t^{s(x)}, \ x \in X, \ t \ge 0, \end{equation*}$$where p(·)$p(\cdot)$, q(·)$q(\cdot)$, and s(·)$s(\cdot)$ are log‐Hölder continuous, p(x)

Date: 2025
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