Tower multitype and global regularity of the $\bar\partial$-Neumann operator
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866913341675405312 |
|---|---|
| author | Zaitsev, Dmitri |
| author_facet | Zaitsev, Dmitri |
| contents | A new approach is given to property $(P_q)$ defined by Catlin for $q=1$ in a global and by Sibony in a local context, subsequently extended by Fu-Straube for $q>1$. This property is known to imply compactness and global regularity in the $\bar\partial$-Neumann problem by a result of Kohn-Nirenberg, as well as condition $R$ by a result of Bell-Ligocka. In particular, we provide a self-contained proof of property $(P_q)$ for pseudoconvex hypersurfaces of finite D'Angelo $q$-type, the case originally studied by Catlin. Moreover, our proof covers more general classes of hypersurfaces inspired by a recent work of Huang-Yin. Proofs are broken down into isolated steps, some of which do not require pseudoconvexity.
Our tools include: a new multitype invariant based on distinguished nested sequences of $(1,0)$ subbundles, defined in terms of derivatives of the Levi form; real and complex formal orbits; $k$-jets of functions relative to pairs of formal submanifolds; relative contact orders generalizing the usual contact orders; a new notion of supertangent vector fields having higher than expected relative contact orders; and a formal variant of a result by Diederich-Fornæss arising as a key step in their proof of Kohn's ideal termination in the real-analytic case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_02836 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tower multitype and global regularity of the $\bar\partial$-Neumann operator Zaitsev, Dmitri Complex Variables 32W05, 32T25, 32V15, 32V35, 32F10, 35N15 A new approach is given to property $(P_q)$ defined by Catlin for $q=1$ in a global and by Sibony in a local context, subsequently extended by Fu-Straube for $q>1$. This property is known to imply compactness and global regularity in the $\bar\partial$-Neumann problem by a result of Kohn-Nirenberg, as well as condition $R$ by a result of Bell-Ligocka. In particular, we provide a self-contained proof of property $(P_q)$ for pseudoconvex hypersurfaces of finite D'Angelo $q$-type, the case originally studied by Catlin. Moreover, our proof covers more general classes of hypersurfaces inspired by a recent work of Huang-Yin. Proofs are broken down into isolated steps, some of which do not require pseudoconvexity. Our tools include: a new multitype invariant based on distinguished nested sequences of $(1,0)$ subbundles, defined in terms of derivatives of the Levi form; real and complex formal orbits; $k$-jets of functions relative to pairs of formal submanifolds; relative contact orders generalizing the usual contact orders; a new notion of supertangent vector fields having higher than expected relative contact orders; and a formal variant of a result by Diederich-Fornæss arising as a key step in their proof of Kohn's ideal termination in the real-analytic case. |
| title | Tower multitype and global regularity of the $\bar\partial$-Neumann operator |
| topic | Complex Variables 32W05, 32T25, 32V15, 32V35, 32F10, 35N15 |
| url | https://arxiv.org/abs/2405.02836 |