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On a class of ¯ -equations without solutions

Telemachos Hatziafratis (1998)

Commentationes Mathematicae Universitatis Carolinae

In this note we construct ¯ -equations (inhomogeneous Cauchy-Riemann equations) without solutions. The construction involves Bochner-Martinelli type kernels and differentiation with respect to certain parameters in appropriate directions.

On a Monge-Ampère type equation in the Cegrell class χ

Rafał Czyż (2010)

Annales Polonici Mathematici

Let Ω be a bounded hyperconvex domain in ℂn and let μ be a positive and finite measure which vanishes on all pluripolar subsets of Ω. We prove that for every continuous and strictly increasing function χ:(-∞,0) → (-∞,0) there exists a negative plurisubharmonic function u which solves the Monge-Ampère type equation - χ ( u ) ( d d c u ) = d μ . Under some additional assumption the solution u is uniquely determined.

On a space of smooth functions on a convex unbounded set in ℝn admitting holomorphic extension in ℂn

Il’dar Musin, Polina Yakovleva (2012)

Open Mathematics

For some given logarithmically convex sequence M of positive numbers we construct a subspace of the space of rapidly decreasing infinitely differentiable functions on an unbounded closed convex set in ℝn. Due to the conditions on M each function of this space admits a holomorphic extension in ℂn. In the current article, the space of holomorphic extensions is considered and Paley-Wiener type theorems are established. To prove these theorems, some auxiliary results on extensions of holomorphic functions...

On canonical homotopy operators for ∂ in Fock type spaces in Cn.

Jörgen Boo (2001)

Publicacions Matemàtiques

We show that a certain solution operator for ∂ in a space of forms square integrable against e-|z|2 is canonical, i.e., that it gives the minimal solution when applied to a ∂-closed form, and gives zero when applied to a form orthogonal to Ker ∂.As an application, we construct a canonical homotopy operator for i∂∂.

On iterations of Green type integrals for matrix factorizations of the Laplace operator

Alexandre A. Shlapunov (1994)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Convergence of special Green integrals for matrix factorization of the Laplace operator in R n is proved. Explicit formulae for solutions of ¯ -equation in strictly pseudo-convex domains in C n are obtained.

On ∂̅-problems on (pseudo)-convex domains

R. Range (1995)

Banach Center Publications

In this survey we shall tour the area of multidimensional complex analysis which centers around ∂̅-problems (i.e., the Cauchy-Riemann equations) on pseudoconvex domains. Along the way we shall highlight some of the classical milestones as well as more recent landmarks, and we shall discuss some of the major open problems and conjectures. For the sake of simplicity we will only consider domains in n ; intriguing phenomena occur already in the simple setting of (Euclidean) convex domains. We will not...

On splitting up singularities of fundamental solutions to elliptic equations in ℂ2

T. Savina (2007)

Open Mathematics

It is known that the fundamental solution to an elliptic differential equation with analytic coefficients exists, is determined up to the kernel of the differential operator, and has singularities on characteristics of the equation in ℂ2. In this paper we construct a representation of fundamental solution as a sum of functions, each of those has singularity on a single characteristic.

On subextension and approximation of plurisubharmonic functions with given boundary values

Hichame Amal (2014)

Annales Polonici Mathematici

Our aim in this article is the study of subextension and approximation of plurisubharmonic functions in χ ( Ω , H ) , the class of functions with finite χ-energy and given boundary values. We show that, under certain conditions, one can approximate any function in χ ( Ω , H ) by an increasing sequence of plurisubharmonic functions defined on strictly larger domains.

On the absence of the one-sided Poincaré lemma in Cauchy-Riemann manifolds

Fabio Nicola (2005)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Given an embeddable C R manifold M and a non-characteristic hypersurface S M we present a necessary condition for the tangential Cauchy-Riemann operator ¯ M on M to be locally solvable near a point x 0 S in one of the sidesdetermined by S .

On the Dirichlet problem in the Cegrell classes

Rafał Czyż, Per Åhag (2004)

Annales Polonici Mathematici

Let μ be a non-negative measure with finite mass given by φ ( d d c ψ ) , where ψ is a bounded plurisubharmonic function with zero boundary values and φ L q ( ( d d c ψ ) ) , φ ≥ 0, 1 ≤ q ≤ ∞. The Dirichlet problem for the complex Monge-Ampère operator with the measure μ is studied.

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