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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 boundary behaviour of the Bergman projection on pseudoconvex domains

M. Jasiczak (2005)

Studia Mathematica

It is shown that on strongly pseudoconvex domains the Bergman projection maps a space L v k of functions growing near the boundary like some power of the Bergman distance from a fixed point into a space of functions which can be estimated by the consecutive power of the Bergman distance. This property has a local character. Let Ω be a bounded, pseudoconvex set with C³ boundary. We show that if the Bergman projection is continuous on a space E ⊃ L ∞ ( Ω ) defined by weighted-sup seminorms and equipped with the topology...

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 Hardy spaces on worm domains

Alessandro Monguzzi (2016)

Concrete Operators

In this review article we present the problem of studying Hardy spaces and the related Szeg˝o projection on worm domains. We review the importance of the Diederich–Fornæss worm domain as a smooth bounded pseudoconvex domain whose Bergman projection does not preserve Sobolev spaces of sufficiently high order and we highlight which difficulties arise in studying the same problem for the Szeg˝o projection. Finally, we announce and discuss the results we have obtained so far in the setting of non-smooth...

On locally convex extension of H ∞ in the unit ball and continuity of the Bergman projection

M. Jasiczak (2003)

Studia Mathematica

We define locally convex spaces LW and HW consisting of measurable and holomorphic functions in the unit ball, respectively, with the topology given by a family of weighted-sup seminorms. We prove that the Bergman projection is a continuous map from LW onto HW. These are the smallest spaces having this property. We investigate the topological and algebraic properties of HW.

On some inequalities in holomorphic function theory in polydisk related to diagonal mapping

Romi F. Shamoyan, Olivera R. Mihić (2010)

Czechoslovak Mathematical Journal

We present a description of the diagonal of several spaces in the polydisk. We also generalize some previously known contentions and obtain some new assertions on the diagonal map using maximal functions and vector valued embedding theorems, and integral representations based on finite Blaschke products. All our results were previously known in the unit disk.

On the Cauchy problem in a class of entire functions in several variables

Eugeni Leinartas (1996)

Banach Center Publications

We study the integral representation of solutions to the Cauchy problem for a differential equation with constant coefficients. The Cauchy data and the right-hand of the equation are given by entire functions on a complex hyperplane of ℂ n + 1 . The Borel transformation of power series and residue theory are used as the main methods of investigation.

On the dependence of the Bergman function on deformations of the Hartogs domain

Zbigniew Pasternak-Winiarski (1991)

Annales Polonici Mathematici

We apply the Rudin idea to represent the Bergman kernel of the Hartogs domain as the sum of a series of weighted Bergman functions in the study of the dependence of this kernel on deformations of the domain. We prove that the Bergman function depends smoothly on the function defining the Hartogs domain.

On the Djrbashian kernel of a Siegel domain

Elisabetta Barletta, Sorin Dragomir (1998)

Studia Mathematica

We establish an inversion formula for the M. M. Djrbashian A. H. Karapetyan integral transform (cf. [6]) on the Siegel domain Ω n = ζ ∈ ℂ n : ϱ ( ζ ) > 0 , ϱ ( ζ ) = I m ( ζ 1 ) - | ζ ' | 2 . We build a family of Kähler metrics of constant holomorphic curvature whose potentials are the ϱ α -Bergman kernels, α > -1, (in the sense of Z. Pasternak-Winiarski [20] of Ω n . We build an anti-holomorphic embedding of Ω n in the complex projective Hilbert space ℂ ℙ ( H α 2 ( Ω n ) ) and study (in connection with work by A. Odzijewicz [18] the corresponding transition probability amplitudes....

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