Non-vanishing of the Bergman kernel function at boundary points of certain domains in ? n

Abstract

Let f2 be a smooth bounded domain in (t TM whose associated Bergman kernel function K(w, z) extends continuously to O as a function ofw for each z ~ f~. We shall prove that for any fixed we bf2, K(w, z) cannot vanish identically in z for a wide class of domains O. Our theorem applies to any domain for which the L2(O) orthogonal projection P onto holomorphic functions is bounded in Sobolev norms ; in particular, if f2 is 1) smooth bounded strictly pseudoconvex, or 2) bounded pseudoconvex with smooth real analytic boundary (Kohn [7], Diederich and Fornaess [,2]), or 3) any smooth bounded domain for which subelliptic estimates hold for the ~-Neumann opcrator (Kohn [7]). Webster [8] has pointed out that the non-vanishing of K(w, z) for strongly pseudoconvex points of domains for which the c~-Neumann operator is pseudo-local follows easily from the asymptotic expansion for K(w, z) given by Fefferman [-3]. Our proof has no recourse to the asymptotic expansion, is elementary in spirit, and extends Webster's result to include certain weakly pseudoconvex domains. The non-vanishing question arises in connection with the problem of boundary behavior of biholomorphic mappings (see Webster [-8]). As a corollary to the non-vanishing theorem, we prove that if (2 is strictly pseudoconvex and Dis a set of determinacy for holomorphic functions in fL then the linear span of {K(., z) :ze ID} is dense in the subspace of holomorphic functions in w+(Q).