Work

Quantitative Analysis of Optimal Sobolev-Lorentz Embeddings with $$\alpha $$-Homogeneous Weights

Year: 2025

Type: article

Abstract: Optimal weighted Sobolev-Lorentz embeddings with homogeneous weights in open convex cones are established, with the exact value of the optimal constant. These embeddings are non-compact, and this paper investigates the structure of their non-compactness quantitatively. Opposite to the previous results in this direction, the non-compactness in this case does not occur uniformly over all sub-domains of the underlying domain, and the problem is not translation invariant, and so these properties cannot be exploited here. Nevertheless, by developing a new approach based on a delicate interplay between the size of suitable extremal functions and the size of their supports, the exact values of the (ball) measure of non-compactness and of all injective strict s-numbers (in particular, of the Bernstein numbers) are obtained. Moreover, it is also shown that the embedding is not strictly singular.

Source: Journal of Geometric Analysis

DOI: 10.1007/s12220-025-01946-0

Authors: Petr Gurka, Jan Lang, Zdeněk Mihula

Institutions: Czech University of Life Sciences Prague, Charles University, College of Polytechnics Jihlava, Czech Technical University in Prague, The Ohio State University

Language: English

Topic: Advanced Harmonic Analysis Research

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,290

Funders: Ministerstvo Školství, Mládeže a Tělovýchovy, Grantová Agentura České Republiky, Chinese Academy of Agricultural Sciences