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Semi-global solutions of ∂b with Lp (1 ≤ p ≤ ∞) bounds on strongly pseudoconvex real hypersurfaces in Cn (n ≥ 3).

C. H. Chang, H. P. Lee (1999)

Publicacions Matemàtiques

Let M be an open subset of a compact strongly pseudoconvex hypersurface {ρ = 0} defined by M = D × Cn-m ∩ {ρ = 0}, where 1 ≤ m ≤ n-2, D = {σ(z1, ..., zm) < 0} ⊂ Cm is strongly pseudoconvex in Cm. For ∂b closed (0, q) forms f on M, we prove the semi-global existence theorem for ∂b if 1 ≤ q ≤ n-m-2, or if q = n - m - 1 and f satisfies an additional “moment condition”. Most importantly, the solution operator satisfies Lp estimates for 1 ≤ p ≤ ∞ with p = 1 and ∞ included.

Seperable maximal pluriharmonic functions in two complex variables.

Ulf Backlund, Leif Persson (1998)

Mathematica Scandinavica

Sharp Hölder Estimates for ... on Ellipsoides.

J.E. Fornaess, K. Diederich, ... (1986)

Manuscripta mathematica

Singularity and regularity -- local and global.

Christ, Michael (1998)

Documenta Mathematica

Sobolev cohomology for locally polycylindrical domains.

Darko, Patrick W. (2002)

International Journal of Mathematics and Mathematical Sciences

Sobolev Estimates for the Lewy Operator on Weakly Pseudo-Convex Boundaries.

Harold P. Boas, Mei-Chi Shaw (1986)

Mathematische Annalen

Sobolev estimates for the ?-Neumann operator on domains in Cn admitting a defining function that is plurisubharmonic on the boundary.

Harold P. Boas, Emil J. Straube (1991)

Mathematische Zeitschrift

Sobolev space estimates for solutions of the equation $\bar{\partial} u = f$ on polycylinders.

Darko, Patrick W., Lutterodt, Clement H. (2004)

International Journal of Mathematics and Mathematical Sciences

Solution of the ?-Equation with Uniform Estimates on Strictly q-Convex Domains with Non-Smooth Boundary.

Gerd Schmalz (1989)

Mathematische Zeitschrift

Solution operators for the ...-equation on non-transversal intersections.

Klaus Peters (1991)

Mathematische Annalen

Solutions de l’équation $\bar{\partial}$ et zéros de la classe de Nevanlinna dans certains domaines faiblement pseudo-convexes

Aline Bonami, Philippe Charpentier (1982)

Annales de l'institut Fourier

Il est montré que la condition de Blaschke est nécessaire et suffisante pour qu’un sous-ensemble analytique du domaine $D = \{z \in C^{n}; \sum_{1}^{n} | z_{i} |^{2 ρ_{i}} < 1\}$ soit l’ensemble des zéros d’une fonction de la classe de Nevanlinna.

Solutions to $\bar{\partial}$ -equations on strongly pseudo-convex domains with $L^{p}$ -estimates.

Abdelkader, Osama, Khidr, Shaban (2004)

Electronic Journal of Differential Equations (EJDE) [electronic only]

Some applications of a new integral formula for $\partial ̅_{b}$

Moulay-Youssef Barkatou (1998)

Annales Polonici Mathematici

Let M be a smooth q-concave CR submanifold of codimension k in $ℂ^{n}$ . We solve locally the $\partial ̅_{b}$ -equation on M for (0,r)-forms, 0 ≤ r ≤ q-1 or n-k-q+1 ≤ r ≤ n-k, with sharp interior estimates in Hölder spaces. We prove the optimal regularity of the $\partial ̅_{b}$ -operator on (0,q)-forms in the same spaces. We also obtain $L^{p}$ estimates at top degree. We get a jump theorem for (0,r)-forms (r ≤ q-2 or r ≥ n-k-q+1) which are CR on a smooth hypersurface of M. We prove some generalizations of the Hartogs-Bochner-Henkin extension...

Some applications of the trace condition for pluriharmonic functions in Cn.

Alessandro Perotti (2000)

Publicacions Matemàtiques

In this paper we investigate some applications of the trace condition for pluriharmonic functions on a smooth, bounded domain in Cn. This condition, related to the normal component on ∂D of the ∂-operator, permits us to study the Neumann problem for pluriharmonic functions and the ∂-problem for (0,1)-forms on D with solutions having assigned real part on the boundary.

Some characterizations of the class $_{m} (Ω)$ and applications

Hai Mau Le, Hong Xuan Nguyen, Hung Viet Vu (2015)

Annales Polonici Mathematici

We give some characterizations of the class $_{m} (Ω)$ and use them to establish a lower estimate for the log canonical threshold of plurisubharmonic functions in this class.

Some open problems in higher dimensional complex analysis and complex dynamics.

John Erik Fornaess, Nessim Sibony (2001)

Publicacions Matemàtiques

We present a collection of problems in complex analysis and complex dynamics in several variables.

Some sufficient conditions for solvability of the Dirichlet problem for the complex Monge-Ampère operator

Sławomir Kołodziej (1996)

Annales Polonici Mathematici

We find a bounded solution of the non-homogeneous Monge-Ampère equation under very weak assumptions on its right hand side.

Spectral Asymptotics for the $\bar{\partial}$ -Neumann Problem

Guy Métivier (1981/1982)

Publications mathématiques et informatique de Rennes

Spectrum of certain Banach algebras and ∂̅-problems

Linus Carlsson, Urban Cegrell, Anders Fällström (2007)

Annales Polonici Mathematici

We study the spectrum of certain Banach algebras of holomorphic functions defined on a domain Ω where ∂̅-problems with certain estimates can be solved. We show that the projection of the spectrum onto ℂⁿ equals Ω̅ and that the fibers over Ω are trivial. This is used to solve a corona problem in the special case where all but one generator are continuous up to the boundary.

Splitting of the ∂-complex in weighted spaces of square integrable functions.

Michael Langenbruch (1992)

Revista Matemática de la Universidad Complutense de Madrid

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