Integral representations on weakly pseudoconvex domains

“…For instance, it would be interesting to know if a more general result can be established for the ∂-equation on a family of Stein manifolds. We notice a remarkable construction of an integral ∂-solution operator by Michel [21] for a smoothly bounded weakly pseudoconvex domain in C n . It would be interesting to know if the assertion in Theorem 1.1 (ii) remains true when the given domains are only weakly pseudoconvex.…”

Section: Introductionmentioning

confidence: 99%

The $$\overline{\partial }$$ ∂ ¯ -equation on variable strictly pseudoconvex domains

Gong

Kim

2017

Math. Z.

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“…The plan of this paper is as follows: In section I we derive a homotopy formula for ∂ on a domain with a piecewise smooth boundary using the barrier functions constructed in Michel [21]. In section II we construct a homotopy formula on an annulus such that Ω 2 is the union of finitely many smooth pseudoconvex domains.…”

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confidence: 99%

“…Let U i be an open neighborhood of D i . It follows from Michel [21] that for each D i there exist an increasing sequence 0 < t 0 < · · · and a C 3 mapping w (i) :…”

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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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Integral representations on weakly pseudoconvex domains

Cited by 20 publications

References 8 publications

<i>C</i><sup>∞</sup>-REGULARITY OF THE TANGENTIAL CAUCHY-RIEMANN EQUATIONS ON LEVI-FLAT SUBMANIFOLDS OF C<sup><i>n</sup></i>

<i>C</i><sup>∞</sup>-REGULARITY OF THE TANGENTIAL CAUCHY-RIEMANN EQUATIONS ON LEVI-FLAT SUBMANIFOLDS OF C<sup><i>n</sup></i>

The $$\overline{\partial }$$ ∂ ¯ -equation on variable strictly pseudoconvex domains

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

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Integral representations on weakly pseudoconvex domains

Cited by 20 publications

References 8 publications

<i>C</i><sup>&infin;</sup>-REGULARITY OF THE TANGENTIAL CAUCHY-RIEMANN EQUATIONS ON LEVI-FLAT SUBMANIFOLDS OF C<sup><i>n</sup></i>

<i>C</i><sup>&infin;</sup>-REGULARITY OF THE TANGENTIAL CAUCHY-RIEMANN EQUATIONS ON LEVI-FLAT SUBMANIFOLDS OF C<sup><i>n</sup></i>

The $$\overline{\partial }$$ ∂ ¯ -equation on variable strictly pseudoconvex domains

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

Contact Info

Product

Resources

About

<i>C</i><sup>∞</sup>-REGULARITY OF THE TANGENTIAL CAUCHY-RIEMANN EQUATIONS ON LEVI-FLAT SUBMANIFOLDS OF C<sup><i>n</sup></i>

<i>C</i><sup>∞</sup>-REGULARITY OF THE TANGENTIAL CAUCHY-RIEMANN EQUATIONS ON LEVI-FLAT SUBMANIFOLDS OF C<sup><i>n</sup></i>