Linear Analysis of Quadrature Domains. IV

The positive definiteness of the exponential transform of a planar domain is proved by elementary means. This direct approach avoids the heavy machinery of the theory of hyponormal operators and leads to a better understanding of the linear data associated in previous works to a quadrature domain. Version: September 10, 2003. 1. The exponential transform Let Ω be a bounded open subset of the complex plane and let dA stand for the Lebesgue planar measure. The exponential transform of the set Ω is the function: (1.1) EΩ(z, w) = exp[− 1 π ∫ Ω dA(ζ) (ζ − z)(ζ − w) ]. The integral is convergent for all values of z, w ∈ C avoiding the diagonal ∆ = {(z, w); z = w ∈ Ω}. In case (z, w) ∈ ∆ and the integral is divergent (necessarily to infinity) we adopt the convention exp(−∞) = 0. Thus EΩ(z, w) is defined everywhere on C and one proves that the resulting function is uniformly bounded and separately continuous in each variable, see [10]. We shall occasionally use the notation (1.1) also when the set Ω is not open. The above exponential transform has appeared in operator theory as a determining function for a class of hyponormal operators ([18], [20], [2], [3], [4]). Later it was analyzed in purely function theoretic terms and was used in proving the regularity of certain free boundaries ([10]) or in image reconstruction ([8]). More Received by the editors today. 1991 Mathematics Subject Classification. Primary 65D32; Secondary 47B20, 31A10.