Estimates of kernels on three-dimensional CR manifolds
type: Item
instance of: scholarly article
title: Estimates of kernels on three-dimensional CR manifolds
zbMATH Open document ID: 0696.32009
DOI: 10.4171/RMI/78
author: J. J. Kohn
author: Charles Fefferman
published in: Revista Matemática Iberoamericana
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/39369
review text: The authors consider the three-dimensional smooth pseudoconvex boundary \(M=b\Omega\) of a bounded domain \(\Omega \subset {\mathbb{C}}^ 2\) and the boundary Cauchy-Riemann operator \({\bar \partial}_ b\) on M. More generally, let M be a three-dimensional compact pseudoconvex CR manifold with linearly independent vector fields L, \(\bar L,\) T. Suppose the range of \(\bar L\) is closed. A point \(P\in M\) is called of finite type not exceeding m if the vector fields L, \(\bar L\) and their commutators of order not exceeding m span the tangent space at P. The following three operators are treated. 1. The Szegö projection \(S_ b\), the orthogonal projection from \(L^ 2(M)\) onto the subspace \(K_ b\subset L^ 2(M)\) of functions annihilated by \({\bar \partial}_ b\). 2. The operator which gives a solution to the inhomogeneous Cauchy-Riemann equations \(\partial_ bu=f\) in the following (well-posed) sense: find \(u\perp K_ b\) such that \({\bar \partial}_ bu=f-S_ b^{1,0}f\), where \(S_ b^{1,0}\) is the orthogonal projection (with respect to a suitable metric) from \(L^ 2\) sections of the form bundle to the subspace annihilated by \({\bar \partial}^*_ b\). 3. The operator which solves the equation \(\square_ bu={\bar \partial}^*_ b{\bar \partial}_ bu=f-S_ bf\) with \(u\perp K_ b\). These operators can be written as integral operators on M with kernels K(x,y), H(x,y), R(x,y) (x,y\(\in M)\), respectively. The authors treat the pointwise behaviour of these kernels near a point \(P\in M\) of finite type. This is done in terms of a family of non- Euclidean balls B(P,\(\delta)\) which correspond to an intrinsic metric on M. This metric is naturally connected with the finite type condition. They define the non-isotropic order of a distribution Q on \(M\times M\) in terms of the behaviour of Q on B(P,\(\delta\) /3)\(\times \{B(P,\delta)\setminus B(P,\delta /3)\}.\) The main purpose of the paper is to give a self-contained and relatively simple direct proof of the following theo…
Mathematics Subject Classification ID: 32W05
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 32T99
zbMATH DE Number: 4140163
zbMATH Keywords: three-dimensional compact pseudoconvex CR manifold
zbMATH Keywords: kernels
MaRDI profile type: Publication
OpenAlex ID: W1972724947
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description: scientific article; zbMATH DE number 4140163