Quantitative analysis of optimal Sobolev-Lorentz embeddings with $α$-homogeneous weights
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913764556668928 |
|---|---|
| author | Gurka, Petr Lang, Jan Mihula, Zdeněk |
| author_facet | Gurka, Petr Lang, Jan Mihula, Zdeněk |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_03127 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Quantitative analysis of optimal Sobolev-Lorentz embeddings with $α$-homogeneous weights Gurka, Petr Lang, Jan Mihula, Zdeněk Functional Analysis 46E35, 47B06 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. |
| title | Quantitative analysis of optimal Sobolev-Lorentz embeddings with $α$-homogeneous weights |
| topic | Functional Analysis 46E35, 47B06 |
| url | https://arxiv.org/abs/2307.03127 |