Uniform decay of function norms
Zdeněk Mihula, Maximilián Pándy
TL;DR
The paper develops two new, operator-independent relations between quasi-Banach function spaces over infinite-measure spaces to address compactness issues, introducing the notions $X \overset{*}{\underset{\infty}{\hookrightarrow}} Y$ and $X \overset{*}{\underset{\mathrm{loc}}{\hookrightarrow}} Y$ and the extremal fundamental functions that govern these relations. It provides an abstract Lions-type compactness principle linking these relations and demonstrates how endpoint and Lorentz/Orlicz spaces fit into the framework. It offers concrete examples and a systematic analysis of embeddings in Lebesgue, Lorentz, and Orlicz spaces, and applies the principle to inhomogeneous Sobolev spaces on $\mathbb{R}^n$, illustrating compactness where mass can escape to infinity. The results yield practical criteria for compactness in infinite-measure settings and advance the understanding of Sobolev embeddings under radial symmetry and localization constraints.
Abstract
We introduce and study two new relations between function spaces over measure spaces of infinite measure, motivated by the question of establishing compactness. The first relation captures the uniform decay of function (quasi-)norms ``at infinity''. It appeared implicitly in the first author's recent work on the compactness of Sobolev embeddings of radially symmetric functions on $\mathbb{R}^n$. The second is a suitably localized version of the relation of almost-compact embeddings, which has been successfully used to study compactness in function spaces over measure spaces of finite measure, but becomes of no use in the case of infinite measure. Our framework is that of quasi-Banach function spaces, which need not be normable or rearrangement invariant. This level of generality leads us to introduce the notion of extremal fundamental functions associated with a (quasi-)Banach function space. We provide several concrete examples and establish an abstract compactness principle involving the new relations. Finally, we demonstrate a possible application of this principle to embeddings of inhomogeneous Sobolev spaces on $\mathbb{R}^n$.