scholarly journals The L 2 $$\overline \partial $$ -Cauchy problem on weakly q-pseudoconvex domains in Stein manifolds

2015 ◽  
Vol 65 (3) ◽  
pp. 739-745 ◽  
Author(s):  
Sayed Saber
Keyword(s):  
Cauchy Problem ◽  
Pseudoconvex Domains ◽  
Stein Manifolds

The L2 ∂¯-Cauchy problem on pseudoconvex domains and applications

2018 ◽  
Vol 11 (02) ◽  
pp. 1850025
Author(s):  
Sayed Saber
Keyword(s):  
Cauchy Problem ◽  
Positive Integer ◽  
Line Bundle ◽  
Tensor Product ◽  
Smooth Boundary ◽  
Pseudoconvex Domains ◽  
Holomorphic Line Bundle ◽  
Dimension Formula ◽  
Weakly Pseudoconvex Domain ◽  
Support Conditions

Let [Formula: see text] be a complex manifold of dimension [Formula: see text] and [Formula: see text] be a weakly pseudoconvex domain with smooth boundary in [Formula: see text]. Let [Formula: see text] be a holomorphic line bundle over [Formula: see text] which is positive on a neighborhood of [Formula: see text]. Let [Formula: see text] be the [Formula: see text]-times tensor product of [Formula: see text] for positive integer [Formula: see text]. The purpose of this paper is to study the [Formula: see text]-problem with support conditions in [Formula: see text] for forms of type [Formula: see text], [Formula: see text] with values in [Formula: see text]. Applications to the [Formula: see text]-problem for smooth forms on boundaries of [Formula: see text] are given.

scholarly journals Semi-continuity of the Diederich–Fornaess and Steinness indices

2020 ◽  
Vol 31 (13) ◽  
pp. 2050107
Author(s):  
Young-Jun Choi ◽  
Jihun Yum
Keyword(s):  
Pseudoconvex Domains ◽  
Stein Manifolds ◽  
Continuity Theorem ◽  
Smooth Deformation

In this paper, we prove the semi-continuity theorem of Diederich–Fornaess index and Steinness index under a smooth deformation of pseudoconvex domains in Stein manifolds.

Cauchy Problem for Equations with Fractional Differentiation Bessel Operator in the Space of Temperate Distributions

2003 ◽  
Vol 8 (1) ◽  
pp. 61-75
Author(s):  
V. Litovchenko
Keyword(s):  
Functional Analysis ◽  
Cauchy Problem ◽  
Fundamental Solution ◽  
Initial Function ◽  
Well Posedness ◽  
Fractional Differentiation ◽  
Basic Tool ◽  
Conjugate Space ◽  
Analysis Technique ◽  
The Cauchy Problem

The well-posedness of the Cauchy problem, mentioned in title, is studied. The main result means that the solution of this problem is usual C∞ - function on the space argument, if the initial function is a real functional on the conjugate space to the space, containing the fundamental solution of the corresponding problem. The basic tool for the proof is the functional analysis technique.

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