Some Lipschitz Regularity for Integral Kernels on Subvarieties of Pseudoconvex Domains in C2 Hongraecho

Let D be a smoothly bounded pseudoconvex domain in c2. Let M be a one-dimensional subvariety of D which has no singularities on bM and intersects bD transversally. If bA4 consists of the points of finite type, then we can construct an integral kernel CM (C, z) for iM which satisfies the reproducing property of holomorphic functions f E O(A4) nC(a) from their boundary values. Furthermore, we get a Lipschitz estimate of the operator induced by the integral kernel, which depends on the type of the boundary bM. 1. INTR~DUOTI~N The Cauchy kernel C(C, 2) (see [7], [lo], [12]) for a strongly pseudoconvex domain D in @” satisfies the reproducing property of holomorphic functions from their boundary values, that is, for all f E A(D) = O(D) n C(D) one has (1.1) f(t) = 6, f(<)c(rY z)dS(<) for z E D, where &S(c) is the surface measure on bD. If Cf(z) denotes the holomorphic function obtained by plugging in an arbitrary function f E L’(bD) in the integral in (l.l), then for 0 < cy < 1, the operator C : R,(bD) + O(D)nA,(D) is bounded (see [2], [% 1% 1121). In [Ill, R an g e introduced a new method for constructing integral kernels on bounded pseudoconvex domains in C”. By using the integral kernel, he obtained Holder estimates for a on pseudoconvex domains of finite type in UJ2. In this paper, we consider an integral kernel for a one-dimensional subvariety M of a smoothly bounded pseudoconvex domain D in C2. With the finite type condition only on bM we construct an integral kernel C”(C,z) for 1991 Mathematics Subject Classification. 32A25.