Sobolev Regularity of the Bergman Projection on a Smoothly Bounded Stein Domain that is not Hyperconvex
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866915007958089728 |
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| author | Harrington, Phillip S. |
| author_facet | Harrington, Phillip S. |
| contents | For every $0<r<\frac{1}{2}$, we will construct a flat Kähler manifold $M$ and a relatively compact domain with smooth boundary $Ω\subset M$ that is Stein but not hyperconvex such that the Bergman projection $P$ on $Ω$ is regular in the $L^2$ Sobolev space $W^s(Ω)$ for all $0\leq s<r$ but irregular in $W^r(Ω)$. On these domains, we will also construct $f\in C^\infty(\overlineΩ)$ such that $Pf\notin C^\infty(\overlineΩ)$. We will prove the same result for the invariant Bergman projection on $(2,0)$-forms. These domains are modelled on a construction of Diederich and Ohsawa. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_14519 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sobolev Regularity of the Bergman Projection on a Smoothly Bounded Stein Domain that is not Hyperconvex Harrington, Phillip S. Complex Variables 32A36, 32T20, 32Q02 For every $0<r<\frac{1}{2}$, we will construct a flat Kähler manifold $M$ and a relatively compact domain with smooth boundary $Ω\subset M$ that is Stein but not hyperconvex such that the Bergman projection $P$ on $Ω$ is regular in the $L^2$ Sobolev space $W^s(Ω)$ for all $0\leq s<r$ but irregular in $W^r(Ω)$. On these domains, we will also construct $f\in C^\infty(\overlineΩ)$ such that $Pf\notin C^\infty(\overlineΩ)$. We will prove the same result for the invariant Bergman projection on $(2,0)$-forms. These domains are modelled on a construction of Diederich and Ohsawa. |
| title | Sobolev Regularity of the Bergman Projection on a Smoothly Bounded Stein Domain that is not Hyperconvex |
| topic | Complex Variables 32A36, 32T20, 32Q02 |
| url | https://arxiv.org/abs/2401.14519 |