Estimates of kernels on three-dimensional CR manifolds
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 theorem. Under the above assumptions 1. K is of non-isotropic order zero, 2. H is of non-isotropic order one, 3. R is of non-isotropic order two. These results are contained in \textit{M. Christ}, J. Am. Math. Soc. 1, No.3, 587-646 (1988; Zbl 0671.35017); \textit{M. Christ}, Proc. Am. Math. Soc. 104, 787-792 (1988); \textit{A. Nagel, J.-P. Rosay, E. M. Stein} and \textit{S. Wainger}, Ann. Math., II. Ser. 129, No.1, 113-149 (1989; Zbl 0667.32016) and are applied to the understanding of how the considered operators act on function spaces.
three-dimensional compact pseudoconvex CR manifold
scientific article; zbMATH DE number 4140163