Continuous Dependence of Szegő Kernel on a Weight of Integration

The weighted Szegő kernel was investigated in a few papers (see Nehari in J d’Analyse Mathématique 2:126–149, 1952; Alenitsin in Zapiski Nauchnykh Seminarov LOMI 24:16–28, 1972; Uehara and Saitoh in Mathematica Japonica 29:887–891, 1984; Uehara in Mathematica Japonica 42:459–469, 1995). In all of these, however, only continuous weights were considered. The aim of this paper is to show that the Szegő kernel depends in a continuous way on a weight of integration in the case when the weights are not necessarily continuous. A topology on the set of admissible weights will be constructed and Pasternak’s theorem (see Pasternak-Winiarski in Studia Mathematica 128:1, 1998) on the dependence of the orthogonal projector on a deformation of an inner product will be used in the proof of the main theorem.