Sobolev's inequality in Musielak–Orlicz–Morrey spaces of an integral form

Takao Ohno and Tetsu Shimomura

Mathematische Nachrichten, 2023, vol. 296, issue 9, 4152-4168

Abstract: In this paper, we are concerned with Sobolev's inequality for variable Riesz potentials Jα(·)σf$J_{\alpha (\cdot )}^{\sigma }f$ of functions f in Musielak–Orlicz–Morrey spaces of an integral form LΦ,ω(X)$\mathcal {L}^{\Phi , \omega }(X)$ over metric measure spaces. As an application and example, we give Sobolev's inequality for double‐phase functionals Φ(x,t)=tp(x)+a(x)tq(x)$\Phi (x,t) = t^{p(x)} + a(x) t^{q(x)}$, where p(·)$p(\cdot )$ and q(·)$q(\cdot )$ satisfy log‐Hölder conditions and a(·)$a(\cdot )$ is non‐negative, bounded and Hölder continuous of order θ∈(0,1]$\theta \in (0,1]$. Further, we obtain the result for Sobolev functions.

Date: 2023
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