On boundary behavior of the Bergman kernel function and related domain functionals (Q2535424)

Generalizing the notion of a strictly pseudoconvex domain in \(\mathbb C^n\) the author considers domains which are strictly \((p,q)\) pseudoconvex at the given boundary point \(Q\). The relevant definition is given in terms of the equation of the boundary in a neighborhood of this point. This enables to use Reinhardt domain \[ R_p(n,s) = \left\{ r_{1\,(s-1)}^{2/p} + r_{s\,n}^2 < 1;\ r_{u\,v}^2 = \sum_{k=u}^v \vert z_k\vert^2\right\} \] as an inner domain of comparison (instead of the unit ball \(\{\vert z\vert <1\})\) in the study of the boundary behavior of the Bergman function at the point \(Q\). The author obtains sufficient conditions for the Bergman function to become infinite of order \(p(n-q+1)+q\) as \(Q\) is approached in a ``nontangential'' way. These conditions generalize the corresponding theorems for strictly pseudoconvex domains which were obtained by Bergman and Hörmander. 1. S. Bergman, Über die Kernfunktion eines Bereiches und ihr Verhalten am Rande, J. Reine Angew. Math. 169 (1933), 1--42; and 172 (1934), 89--128. 2. S. Bergman, The behaviour of the kernel function at boundary points of the second order, Am. J. Math. 65 (1943), 679--700. 3. S. Bergman, Sur les fonctions orthogonales de plusieurs variables complexes, Mém. des Sciences Math. 106 (1947). 4. S. Bergman, Sur la fonction-noyau d'un domaine et ses applications dans la théorie des transformations pseudo-conformes, Mém. des Sciences Math. 108 (1948). 5. S. Bergman, The kernel function and conformal mapping, Math. Surveys 5, Amer. Math. Soc, New York, 1950. 6. S. Bergman, Zur Theorie von pseudokonformen Abbildungen, Recueil Math. 1 (1936), 79--96. 7. H. J. Bremermann, Holomorphic continuation of the kernel function and the Bergman metric in several complex variables, in: Lectures on functions of a complex variable, ed. by W. Kaplan, Univ. of Michigan Press, 1955. 8. K. Diederich, Das Rand verhalten der Bergmanschen Kernfunktion und Metrik in streng pseudokonvexen Gebieten (to appear) 9. B. A. Fuks, Special chapters in the theory of analytic functions of several complex variables, (Russian) Moscow, 1963 (Engl. Transl., A.M.S. 1965). 10. L. Hörmander, Existence theorem for the \(\delta\)-operator by \(L^2\) methods, Acta Math. 113 (1965), 89--152. 11. H. Meschkowski, Hilbertsche Räume mit Kernfunktion, Berlin, 1962.