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arXiv · 2307.03127

Quantitative analysis of optimal Sobolev-Lorentz embeddings with $\alpha$-homogeneous weights

Petr Gurka·Jan Lang·Zdeněk Mihula
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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.

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Petr Gurka, Jan Lang, Zdeněk Mihula. 2023-07-06. Quantitative analysis of optimal Sobolev-Lorentz embeddings with $\alpha$-homogeneous weights. https://doi.org/10.1007/s12220-025-01946-0

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