An Elementary Integral Solution Operator for the Cauchy-Riemann Equations on Pseudoconvex Domains in C n

1982 ◽  
Vol 274 (2) ◽  
pp. 809
Author(s):  
R. Michael Range
Keyword(s):  
Integral Solution ◽  
Solution Operator ◽  
Pseudoconvex Domains ◽  
Cauchy Riemann

The $\bar \partial $-problem on q-pseudoconvex domains with applications

2013 ◽  
Vol 63 (3) ◽  
Author(s):  
S. Saber
Keyword(s):  
Pseudoconvex Domain ◽  
Smooth Boundary ◽  
Pseudoconvex Domains ◽  
Lipschitz Boundary ◽  
Cauchy Riemann

AbstractFor a q-pseudoconvex domain Ω in ℂn, 1 ≤ q ≤ n, with Lipschitz boundary, we solve the $\bar \partial $-problem with exact support in Ω. Moreover, we solve the $\bar \partial $-problem with solutions smooth up to the boundary over Ω provided that it has smooth boundary. Applications are given to the solvability of the tangential Cauchy-Riemann equations on the boundary.

scholarly journals L2-Sobolev theory for the complex Green operator

2017 ◽  
Vol 28 (09) ◽  
pp. 1740006 ◽  
Author(s):  
Séverine Biard ◽  
Emil J. Straube
Keyword(s):  
Sobolev Spaces ◽  
Pseudoconvex Domains ◽  
Green Operator ◽  
Cr Geometry ◽  
Basic Formula ◽  
Cauchy Riemann ◽  
Cr Submanifolds

These notes are concerned with the [Formula: see text]-Sobolev theory of the complex Green operator on pseudoconvex, oriented, bounded and closed Cauchy–Riemann (CR)-submanifolds of [Formula: see text] of hypersurface type. This class of submanifolds generalizes that of boundaries of pseudoconvex domains. We first discuss briefly the CR-geometry of general CR-submanifolds and then specialize to this class. Next, we review the basic [Formula: see text]-theory of the tangential CR operator and the associated complex Green operator(s) on these submanifolds. After these preparations, we discuss recent results on compactness and regularity in Sobolev spaces of the complex Green operator(s).

scholarly journals On Sharp Hölder Estimates of the Cauchy-Riemann Equation on Pseudoconvex Domains inCnwith One Degenerate Eigenvalue

2015 ◽  
Vol 2015 ◽  
pp. 1-6
Author(s):  
Sanghyun Cho ◽  
Young Hwan You
Keyword(s):  
Pseudoconvex Domain ◽  
Pseudoconvex Domains ◽  
Hölder Regularity ◽  
Holomorphic Curve ◽  
Riemann Equation ◽  
Degenerate Eigenvalue ◽  
Cauchy Riemann ◽  
Maximal Gain ◽  
Holder Regularity ◽  
Hölder Estimates

LetΩbe a smoothly bounded pseudoconvex domain inCnwith one degenerate eigenvalue and assume that there is a smooth holomorphic curveVwhose order of contact withbΩatz0∈bΩis larger than or equal toη. We show that the maximal gain in Hölder regularity for solutions of the∂¯-equation is at most1/η.

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