THE &lt;TEX&gt;${\bar{\partial}}$&lt;/TEX&gt;-PROBLEM WITH SUPPORT CONDITIONS AND PSEUDOCONVEXITY OF GENERAL ORDER IN KÄHLER MANIFOLDS

Saber, Sayed

doi:10.4134/jkms.j140768

Cited by 3 publications

(3 citation statements)

References 24 publications

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“…In [5], Saber proved that the operators N, ∂ N and P are regular in W m r,s (D) for some m on a smooth weakly q-convex domain in C n . Similar results can be found in [7][8][9][10][11][12][13][14][15][16].…”

Section: Introductionsupporting

confidence: 87%

Sobolev Estimates for the ∂¯ and the ∂¯-Neumann Operator on Pseudoconvex Manifolds

Adam,

Ahmed,

Saber

et al. 2023

Mathematics

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Let D be a relatively compact domain in an n-dimensional Kähler manifold with a C2 smooth boundary that satisfies some “Hartogs-pseudoconvexity” condition. Assume that Ξ is a positive holomorphic line bundle over X whose curvature form Θ satisfies Θ≥Cω, where C>0. Then, the ∂¯-Neumann operator N and the Bergman projection P are exactly regular in the Sobolev space Wm(D,Ξ) for some m, as well as the operators ∂¯N, ∂¯⋇N.

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Section: Introductionsupporting

confidence: 87%

Sobolev Estimates for the ∂¯ and the ∂¯-Neumann Operator on Pseudoconvex Manifolds

Adam,

Ahmed,

Saber

et al. 2023

Mathematics

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“…By defining u as in (2.4), then supp u ⊂ Ω and u vanishes on bΩ. As in Saber [20], one can prove that the extended form u satisfies the equation ∂u = f in the distribution sense in X. □…”

Section: The L 2 ∂ Cauchy Problem On Piecewise Smooth Strongly Pseudoconvex Domainsmentioning

confidence: 93%

“…Now, we extend u to P n by defining u = 0 in P n \ Ω. As in Saber [20], the extended form u satisfies the equation ∂u = f in the distribution sense in P n . □ Corollary 6.1.…”

mentioning

confidence: 99%

On the Applications of Bochner-Kodaira-Morrey-Kohn Identity

Saber¹,

Baljurashi²

2021

KgJMat

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This paper is devoted to studying some applications of the Bochner-Kodaira-Morrey-Kohn identity. For this study, we deﬁne a condition which is called (Hq) condition which is related to the Levi form on the complex manifold. Under the (Hq) condition and combining with the basic Bochner-Kodaira-Morrey-Kohn identity, we study the L2 ∂ Cauchy problems on domains in ℂn, Kähler manifold and in projective space. Also, we study this problem on a piecewise smooth strongly pseudoconvex domain in a complex manifold. Furthermore, the weighted L2 ∂ Cauchy problem is studied under the same condition in a Kähler manifold with semi-positive holomorphic bisectional curvature. On the other hand, we study the global regularity and the L2 theory for the ∂-operator with mixed boundary conditions on an annulus domain in a Stein manifold between an inner domain which satisfy (Hn−q−1) and an outer domain which satisfy (Hq).

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Existence theorems for the dbar equation and Sobolev estimates on $ q $-convex domains

Adam,

Saber

et al. 2023

MATH

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<abstract><p>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.</p></abstract>

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THE <TEX>${\bar{\partial}}$</TEX>-PROBLEM WITH SUPPORT CONDITIONS AND PSEUDOCONVEXITY OF GENERAL ORDER IN KÄHLER MANIFOLDS

Cited by 3 publications

References 24 publications

Sobolev Estimates for the ∂¯ and the ∂¯-Neumann Operator on Pseudoconvex Manifolds

Sobolev Estimates for the ∂¯ and the ∂¯-Neumann Operator on Pseudoconvex Manifolds

On the Applications of Bochner-Kodaira-Morrey-Kohn Identity

Existence theorems for the dbar equation and Sobolev estimates on $ q $-convex domains

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