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• W418857350 abstract "The study of boundary regularity for solutions of the system of Cauchy-Riemann equations •f =g on strictly pseudoconvex domains D has been a central theme in the theory of several complex variables for many years. Classically, two different approaches have been used : A) the abstract L2-theory of the O-Neumann problem (Kohn [12]), which leads to the study of the solution f perpendicular to ker~ with respect to some Hermitian metric, and B) the construction of rather explicit integral solution operators for c~, in analogy to the Cauchy transform in IF 1. The latter approach, first introduced by Grauert and Lieb [6] and by Henkin [9], has led to more refined estimates not directly available in the L 2 theory. As a consequence, there has been much interest in obtaining such estimates also for the abstractly defined canonical solution of 8f = g. In order to deal with this problem, E.M. Stein and his collaborators introduced the method of osculating the boundary of D by the Heisenberg group and using the precise analytic information available in this group setting. This was first carried out by Folland and Stein [-5] for the boundary Laplacian ff]b and the ~b-complex. Building upon these results, Greiner and Stein [7] then treated the 0-Neumann problem on D, obtaining a parametrix as a product of several operators whose symbols were known precisely. Although this parametrix allows to prove optimal estimates for the 0-Neumann operator N and the canonical solution operator ~*N for 0 in various norms, its representation is far from explicit. Moreover the 0-Neumann problem in [7] is defined with respect to a metric carefully adapted to the boundary a so-called Levi metric and hence the results have to be understood accordingly; in addition, they refer only to (0, 1) forms. Independently, Ovrelid [16] announced results for the O-Neumann operator N in a more general setting; this work, the details of which have not been published, also relies heavily on the symbol calculus for pseudodifferential Operators. More recently, Phong [17] has announced results for N based on the" @default.
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• W418857350 date "1983-01-01" @default.
• W418857350 modified "2023-09-26" @default.
• W418857350 title "On Integral Representations and a priori Lipschitz Estimates for the Canonical Solution of the ~-Equation*" @default.
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