Complex Analysis and CR Geometry

Zampieri, Giuseppe

doi:10.1090/ulect/043

Cited by 30 publications

(42 citation statements)

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“…The boundary term had been computed in [12]; by combining the Morrey-Kohn technique on the boundary with non-trivial weight function. One combines the results of [16] and [27] with the interior formulae discussed above, one can prove that (2.1) holds for the general case with a weight function e −φ and the curvature term. For the case φ = 0, the stated formula was proved in Siu [24].…”

Section: The ∂-Problem With Support Conditionsmentioning

confidence: 85%

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

Saber¹

2016

Journal of the Korean Mathematical Society

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Abstract. Let M be an n-dimensional Kähler manifold with positive holomorphic bisectional curvature and let Ω ⋐ M be a pseudoconvex domain of order n − q, 1 ≤ q ≤ n, with C 2 smooth boundary. Then, we study the (weighted) ∂-equation with support conditions in Ω and the closed range property of ∂ on Ω. Applications to the ∂-closed extensions from the boundary are given. In particular, for q = 1, we prove that there exists a number ℓ 0 > 0 such that the ∂-Neumann problem and the Bergman projection are regular in the Sobolev space W ℓ (Ω) for ℓ < ℓ 0 .

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Section: The ∂-Problem With Support Conditionsmentioning

confidence: 85%

“…This formula is known (cf. [2], [8], [16], [24] and [27]) for some special cases, although it has not been stated in the literature in the form (2.1). The boundary term had been computed in [12]; by combining the Morrey-Kohn technique on the boundary with non-trivial weight function.…”

Section: The ∂-Problem With Support Conditionsmentioning

confidence: 99%

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

Saber¹

2016

Journal of the Korean Mathematical Society

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“…As shown in Theorem 1.9.9 in [18], q-pseudoconvexity implies that for L 2 0,s+1 -forms f in the kernel of ∂, there exists an L 2 0,s -form u solving the ∂-problem ∂u = f . It has been proved recently, by several authors including Harrington-Raich [8], that N exists on q-forms in a q-pseudoconvex domain.…”

Section: Introductionmentioning

confidence: 91%

The L 2 $$\overline \partial $$ -Cauchy problem on weakly q-pseudoconvex domains in Stein manifolds

Saber

2015

Czech Math J

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“…We refer for instance to [20] for the proof of Proposition 1.1; some ideas of the proof can also be found in [17]. We denote by T 1,0 M and T 0,1 M the holomorphic and antiholomorphic tangent bundles to M respectively; they are both isomorphic to the complex tangent bundle T C M := T M ∩ iT M .…”

Section: Regularity Of The∂-system Tangential To a Hypersurfacementioning

confidence: 99%

“…Note that the definition differs from [19] and [20] where (1.4) defines (q − 1)-pseudoconvexity ((q − 1)-pseudoconcavity). There is a basic relation between pseudoconvexity/concavity of the two sides D ± of M .…”

Section: Wwwmn-journalcommentioning

confidence: 99%

Estimates for regularity of the tangential \documentclass{article}\usepackage{amssymb,amsmath}\begin{document}\pagestyle{empty}$\bar{\partial }$\end{document}‐system

Khanh

Zampieri

2011

Mathematische Nachrichten

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Key words∂-Neumann problem,∂ b -system, q-pseudoconvex/concave manifolds. MSC (2010) 32F10, 32F20, 32N15, 32T25We study ellipticity in a weak sense, such as fractional or logarithmic, of the system ∂ b ,∂ * b tangential to a hypersurface or a generic higher codimensional submanifold M ⊂ C n . The geometric setting which assures the estimates is the q-pseudoconvexity/concavity of M in addition to the existence of a suitable family of weights in a strip or a tube around M . The basic estimates for the∂-Neumann problem on q-pseudoconvex/concave domains is related to the classical work by Shaw [17] and more recent by Zampieri [19]. The method of the weights is due to Catlin [3] and the relation between the tangential and the ambient∂ system on pseudoconvex domains is inspired to Kohn [14]. Both these techniques are adapted here to a general Levi signature.Let M be a real smooth hypersurface of C n defined by r = 0 with ∂r = 0. We denote by D = D + and D = D − the two sides of M in a neighborhood V of a point z o ∈ M ; we assume r < 0 in D + and r > 0 in D − . Let ω 1 , . . . , ω n be an orthonormal basis of (1, 0) forms in a neighborhood of z o with ω n = ∂r, and let ∂ ω1 , . . . , ∂ ωn be the dual basis of (1, 0) vector fields. For 0 ≤ k ≤ n, we write a general k-form u on V aswhere denotes summation restricted to ordered multiindices J = {j 1 , . . . , j k } and whereω J =ω j1 ∧ · · · ∧ ω j k . When the multiindex is no more ordered, it is understood that the coefficient u J is antisymmetric with respect to J; in particular, if J decomposes into jK, then u j K = sign J j K u J . We define a scalar product and a norm by u, u = |u| 2 = |J |=k |u J | 2 ; this definition is independent of the choice of the orthonormal basis ω 1 , . . . , ω n . The coefficients of our forms are taken in various spaces such asand the corresponding spaces of k-forms are denoted by C ∞ (D ∩ V ) k and so on. All our discussion is local; sometimes, we omit this specification. Though our a priori estimates are proved over smooth forms, they are stated in Hilbert norms. Thus, let u H 0 or u 0 be the H 0 = L 2 norm and, for a real function ϕ, let the weighted L 2 -norm be defined bywhere dv is the element of volume in C n .

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Complex Analysis and CR Geometry

Cited by 30 publications

References 0 publications

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

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

The L 2 $$\overline \partial $$ -Cauchy problem on weakly q-pseudoconvex domains in Stein manifolds

Estimates for regularity of the tangential \documentclass{article}\usepackage{amssymb,amsmath}\begin{document}\pagestyle{empty}$\bar{\partial }$\end{document}‐system

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