Compactness Estimate for the ∂-Neumann Problem on a Q-Pseudoconvex Domain in a Stein Manifold

Saber, Sayed; Alahmari, Abdullah

doi:10.46793/kgjmat2304.627s

KgJMat

2003

DOI: 10.46793/kgjmat2304.627s

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Compactness Estimate for the ∂-Neumann Problem on a Q-Pseudoconvex Domain in a Stein Manifold

Sayed Saber¹,

Abdullah Alahmari²

Abstract: Abstract. We consider a smoothly bounded q-pseudoconvex domain Ω in an ndimensional Stein manifold X and suppose that the boundary bΩ of Ω satisfies (q − P) property, which is the natural variant of the classical P property. Then, one prove the compactness estimate for the ∂-Neumann operator Nr,s in the Sobolev kspace. Applications to the boundary global regularity for the ∂-Neumann operator Nr,s in the Sobolev k-space are given. Moreover, we prove the boundary global regularity of the ∂-operator on Ω.

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2023

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

Adam,

Saber

et al. 2023

MATH

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Compactness Estimate for the ∂-Neumann Problem on a Q-Pseudoconvex Domain in a Stein Manifold

Cited by 1 publication

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

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

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