Existence theorems for the dbar equation and Sobolev estimates on $ q $-convex domains
In this paper, we study a sufficient condition for subelliptic estimates in the weak $ \operatorname{Z}(k) $ domain with $ \operatorname{\operatorname{C}}^{3} $ boundary in an $ n $-dimentionsl $ \operatorname{Stein\, manifold} $ $ \operatorname{X} $. Consequently, the compactness of the $ \overline...
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| Vydané v: | AIMS mathematics Ročník 8; číslo 12; s. 31141 - 31157 |
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| Hlavní autori: | , , , |
| Médium: | Journal Article |
| Jazyk: | English |
| Vydavateľské údaje: |
AIMS Press
01.01.2023
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| Predmet: | |
| ISSN: | 2473-6988, 2473-6988 |
| On-line prístup: | Získať plný text |
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| Shrnutí: | In this paper, we study a sufficient condition for subelliptic estimates in the weak $ \operatorname{Z}(k) $ domain with $ \operatorname{\operatorname{C}}^{3} $ boundary in an $ n $-dimentionsl $ \operatorname{Stein\, manifold} $ $ \operatorname{X} $. Consequently, the compactness of the $ \overline\partial $-Neumann operator $ \operatorname{N} $ on $ { M } $ is obtained and the closedness ranges of $ \overline\partial $ and $ \overline\partial^* $ are presented. The $ \operatorname{L}^2 $-setting and the Sobolev estimates of $ \operatorname{N} $ on $ { M } $ are proved. We study the $ \overline\partial $ problem with support conditions in $ { M } $ for $ \Xi $-valued $ (\operatorname{p}, \operatorname{k}) $ forms, where $ \Xi $ is the $ m $-times tensor product of holomorphic line bundle $ \Xi^{\otimes \operatorname{m}} $ for positive integer $ m $. Moreover, the compactness of the weighted $ \overline\partial $-Neumann operator is studied on an annular domain in a $ \operatorname{Stein\, manifold} $ $ { M } = { M }_{1}\backslash\overline{ M }_{2} $, between two smooth bounded domains $ { M }_{1} $ and $ { M }_{2} $ satisfy $ \overline{ M }_{2}\Subset{ M }_{1} $, $ { M }_{1} $ is weak $ Z(k) $, $ { M }_{2} $ is weak $ Z(n-1-k) $, $ 1 \leqslant k\leqslant n -2 $ with $ n \geqslant 3 $. In all cases, the closedness of $ \overline\partial $ and $ \overline\partial^* $, global boundary regularity for $ \overline\partial $ and $ \overline\partial_b $ are studied. |
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| ISSN: | 2473-6988 2473-6988 |
| DOI: | 10.3934/math.20231594 |