Analytic Functionals and the Bergman Projection on Circular Domains

A property of the Bergman projection associated to a bounded circular domain containing the origin in C^ is proved: Functions which extend to be holomorphic in large neighborhoods of the origin are characterized as Bergman projections of smooth functions with small support near the origin. For certain circular domains D, it is also shown that functions which extend holomorphically to a neighborhood of D are precisely the Bergman projections of smooth functions whose supports are compact subsets of D. Two applications to proper holomorphic mappings are given. This paper treats properties of the Bergman projection on certain domains in CN. We denote by L2(D) the space of functions which are square integrable with respect to Lebesgue measure on a domain 7?, and by H2(D) the (Bergman) space of holomorphic functions in L2(D). The orthogonal projection P: L2(D) —» H2(D) is called the Bergman projection. The Bergman kernel is the integral kernel k(z, w) such that, for all / in L2{D), Pf(z) = / f(w)k(z,w)dw. Jd An analytic functional on a domain D in CN is a continuous linear functional on O(D), the space of holomorphic functions on D, with the topology of uniform convergence on compact subsets of D. The connection with the Bergman projection is LEMMA 1. Let D be a domain in CN, f in H2(D), and U an open, relatively compact subset of D. There exists u in Co°(/7) with Pu = / if and only if, for some compact K C U and some constant C > 0, (1) 1/ fg\ C given by Tf(g) = (gJ)= f lg. Jd If (1) holds, then Tf extends to a continuous linear functional on C(7f), the space of continuous functions on K. Hence T¡ is represented by a measure p supported Received by the editors April 26, 1984 and, in revised form, February 25, 1985. 1980 Mathematics Subject Classification. Primary 32A07, 32H10; Secondary 46E20.