A decomposition problem on weakly pseudoconvex domains

Michel, Joachim; Shaw, Mei-Chi

doi:10.1007/pl00004685

Cited by 14 publications

(16 citation statements)

References 15 publications

(14 reference statements)

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“…It is essential to remark that a quite similar result has been obtained previously by Joachim Michel and Mei-Chi Shaw [8]. However, the proof in [8], using the powerful machinery from [7], seems much more technical than ours. It also requires the boundary of Ω to be C 2 -smooth, while we require only Lip 1 regularity.…”

Section: Introduction and Statement Of Resultssupporting

confidence: 81%

“…It also requires the boundary of Ω to be C 2 -smooth, while we require only Lip 1 regularity. Moreover, the article [8] yelds C k functions w j (z, ζ) having their growth controlled by dist(z, ∂Ω) −(ank 2 +bn) (for suitable a n , b n ) instead of dist(z, ∂Ω) −(ank+bn) as in statement (3). So our version of the theorem improves the main result in [8] with respect to both aspects.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…. , w n ) has many applications, for which we refer the reader to [8], [9], [10], [11]. Of course, our improved barrier allows one to obtain, more or less mechanically, some improvements of these applications too (see e.g.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

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Hilbert-valued forms and barriers on weakly pseudoconvex domains

Thilliez¹

1998

Publicacions Matemàtiques

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We introduce an alternative proof of the existence of certain C k barrier maps, with polynomial explosion of the derivatives, on weakly pseudoconvex domains in C n. Barriers of this sort have been constructed very recently by J. Michel and M.-C. Shaw, and have various applications. In our paper, the adaptation of Hörmander's L 2 techniques to suitable vector-valued functions allows us to give a very simple approach of the problem and to improve some aspects of the result of Michel and Shaw, regarding the explosion of the barrier and the regularity assumption on the domain. n j=1 w j (z, ζ)(z j − ζ j) = 1.

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Section: Introduction and Statement Of Resultssupporting

confidence: 81%

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

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Hilbert-valued forms and barriers on weakly pseudoconvex domains

Thilliez¹

1998

Publicacions Matemàtiques

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“…It was first proved by Hortmann [14] that one can construct a homotopy formula for the annulus between two strictly pseudoconvex domains. Recently Michel-Shaw [25] have extended this result to the case when Ω = Ω 1 \ Ω 2 and the boundary of Ω 2 is pseudoconvex with only C 2 boundary. When Ω 1 , Ω 2 have piecewise smooth strongly pseudoconvex boundaries, Equation (0.1) was studied in Michel-Perotti [22], [23].…”

mentioning

confidence: 89%

“…In section II we construct a homotopy formula on an annulus such that Ω 2 is the union of finitely many smooth pseudoconvex domains. The proof depends on the barrier functions constructed recently in Michel-Shaw [25]. We then use induction to construct a solution for Equation (0.1) when Ω 2 is the transversal intersection of finitely many smooth pseudoconvex domains.…”

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confidence: 99%

The $\overline {\partial }$ problem on domains with piecewise smooth boundaries with applications

Michel

Shaw

1999

Trans. Amer. Math. Soc.

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Abstract. Let Ω be a bounded domain in C n such that Ω has piecewise smooth boudnary. We discuss the solvability of the Cauchy-Riemann equationwhere α is a smooth ∂-closed (p, q) form with coefficients C ∞ up to the bundary of Ω, 0 ≤ p ≤ n and 1 ≤ q ≤ n. In particular, Equation (0.1) is solvable with u smooth up to the boundary (for appropriate degree q) if Ω satisfies one of the following conditions: i) Ω is the transversal intersection of bounded smooth pseudoconvex domains. ii) Ω = Ω 1 \ Ω 2 where Ω 2 is the union of bounded smooth pseudoconvex domains and Ω 1 is a pseudoconvex convex domain with a piecewise smooth boundary. iii) Ω = Ω 1 \ Ω 2 where Ω 2 is the intersection of bounded smooth pseudoconvex domains and Ω 1 is a pseudoconvex domain with a piecewise smooth boundary. The solvability of Equation (0.1) with solutions smooth up to the boundary can be used to obtain the local solvability for ∂ b on domains with piecewise smooth boundaries in a pseudoconvex manifold.Let Ω be a bounded domain in C n such that Ω has a piecewise smooth boundary. In this paper we study the solvability of the Cauchy-Riemann equationwhere α is a smooth ∂-closed (p, q) form with coefficients C ∞ up to the boundary of Ω, 0 ≤ p ≤ n and 1 ≤ q ≤ n. In particular, we prove that Equation (0.1) is solvable with u smooth up to the boundary (for appropriate degree q) if Ω satisfies one of the following conditions: i) Ω is the transversal intersection of bounded smooth pseudoconvex domains. ii) Ω = Ω 1 \Ω 2 , where Ω 2 is the union of bounded smooth pseudoconvex domains and Ω 1 is a pseudoconvex domain with a piecewise smooth boundary.iii) Ω = Ω 1 \ Ω 2 , where Ω 2 is the intersection of bounded smooth pseudoconvex domains and Ω 1 is a pseudoconvex domain with a piecewise smooth boundary.The boundary regularity problem for ∂ has been studied extensively by two very different methods: by L 2 a priori estimates and by the integral kernel approach.

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$$ \bar{\partial } $$ and $$ {\bar{\partial }_b} $$ Problems on Nonsmooth Domains

Michel¹,

Shaw

1999

Analysis and Geometry in Several Complex Variables

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A decomposition problem on weakly pseudoconvex domains

Cited by 14 publications

References 15 publications

Hilbert-valued forms and barriers on weakly pseudoconvex domains

Hilbert-valued forms and barriers on weakly pseudoconvex domains

The $\overline {\partial }$ problem on domains with piecewise smooth boundaries with applications

$$ \bar{\partial } $$ and $$ {\bar{\partial }_b} $$ Problems on Nonsmooth Domains

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