Non-strict singularity of optimal Sobolev embeddings
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866914274123710464 |
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| author | Lang, Jan Mihula, Zdeněk |
| author_facet | Lang, Jan Mihula, Zdeněk |
| contents | We investigate the operator-theoretic property of strict singularity for optimal Sobolev embeddings within the general framework of rearrangement-invariant function spaces (r.i. spaces).
More specifically, we focus on studying the ``quality'' of non-compactness for optimal Sobolev embeddings $V^m_0X(Ω)\to Y_X(Ω)$, where $X$ is a given r.i. space and $Y_X$ is the corresponding optimal target r.i. space (i.e., the smallest among all r.i. spaces).
For the class of sub-limiting norms (i.e., the norms whose fundamental function satisfies $φ_{Y_X}(t)\approx t^{-m/n}φ_X(t)$ as $t\to0^+$), we construct suitable spike-function sequences that establish a general framework for proving non-strict singularity of optimal (and thus non-compact) sublimiting Sobolev embeddings.
As an application, we show that optimal sublimiting Sobolev embeddings are not strictly singular in a rather large subclass of r.i. spaces, namely weighted Lambda spaces $X=Λ^q_w$, $q\in[1, \infty)$. Except for the endpoint case $X=L^{n/m,1}$, our spike-function construction enables us to construct a subspace of $V^m_0X$ that is isomorphic to $\ell_q$, which we then leverage to prove the non-strict singularity of the corresponding optimal Sobolev embedding. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_19981 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-strict singularity of optimal Sobolev embeddings Lang, Jan Mihula, Zdeněk Functional Analysis 46E35, 46E30, 47B60 We investigate the operator-theoretic property of strict singularity for optimal Sobolev embeddings within the general framework of rearrangement-invariant function spaces (r.i. spaces). More specifically, we focus on studying the ``quality'' of non-compactness for optimal Sobolev embeddings $V^m_0X(Ω)\to Y_X(Ω)$, where $X$ is a given r.i. space and $Y_X$ is the corresponding optimal target r.i. space (i.e., the smallest among all r.i. spaces). For the class of sub-limiting norms (i.e., the norms whose fundamental function satisfies $φ_{Y_X}(t)\approx t^{-m/n}φ_X(t)$ as $t\to0^+$), we construct suitable spike-function sequences that establish a general framework for proving non-strict singularity of optimal (and thus non-compact) sublimiting Sobolev embeddings. As an application, we show that optimal sublimiting Sobolev embeddings are not strictly singular in a rather large subclass of r.i. spaces, namely weighted Lambda spaces $X=Λ^q_w$, $q\in[1, \infty)$. Except for the endpoint case $X=L^{n/m,1}$, our spike-function construction enables us to construct a subspace of $V^m_0X$ that is isomorphic to $\ell_q$, which we then leverage to prove the non-strict singularity of the corresponding optimal Sobolev embedding. |
| title | Non-strict singularity of optimal Sobolev embeddings |
| topic | Functional Analysis 46E35, 46E30, 47B60 |
| url | https://arxiv.org/abs/2505.19981 |