Sobolev’s Inequality for Musielak–orlicz–morrey Spaces Over Metric Measure Spaces

Abstract

Our aim in this paper is to establish a generalization of Sobolev’s inequality for Riesz potentials $J_{\unicode[STIX]{x1D6FC}(\cdot )}^{\unicode[STIX]{x1D70E}}f$ of functions $f$ in Musielak–Orlicz–Morrey spaces $L^{\unicode[STIX]{x1D6F7},\unicode[STIX]{x1D705}}(X)$ . As a corollary we obtain Sobolev’s inequality for double phase functionals with variable exponents.

Riesz spaces with generalized Orlicz growth

“…In this article, we further extend and improve their result to the generalized Orlicz case and answer a question regarding the Riesz-Medvedev variation by Appell, Banaś and Merentes [2]. Generalized Orlicz spaces, also known as Musielak-Orlicz spaces, have been studied with renewed intensity recently [14,23,26,28] as have related PDE [4,5,6,9,10,17,21,29]. A contributing factor is that they cover both the variable exponent case ϕ(x, t) := t p(x) [11] and the double phase case ϕ(x, t) := t p + a(x)t q [3], as well as their many variants: perturbed variable exponent, Orlicz variable exponent, degenerate double phase, Orlicz double phase, variable exponent double phase, triple phase and double variable exponent.…”

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

“…Let $$\alpha (\cdot )$$ be a measurable function on $$X$$ such that $$$$\begin{equation*} 0<\alpha ^- :=\inf _{x \in X} \alpha (x)\le \sup _{x \in X} \alpha (x) =: \alpha ^+<\infty . \end{equation*}$$$$As in [11] by Hajłasz and Koskela and [13, 31, 32], we consider a Riesz‐type potential which is better suited to the metric measure case. We define the Riesz potential $$J_{\alpha (\cdot ), \tau }f$$ of order $$\alpha (\cdot )$$ for $$\tau \ge 1$$ and a locally integrable function $$f$$ on $$X$$ by …”

Extension in generalized Orlicz spaces

“…Generalized Orlicz spaces, also known as Musielak-Orlicz spaces, and related differential equations have been studied with increasing intensity recently, see, e.g., the references [1,2,3,4,6,7,9,14,15,16] published since 2018. This year, we published the monograph [10] in which we present a new framework for the basics of these spaces.…”