Behavior of the Bergman kernel near smooth convex boundary points (Q1404810)

This article is motivated by a joint result of \textit{E.~J. Straube, J.~Yu}, and the reviewer [Mich. Math. J. 42, 449--461 (1995; Zbl 0853.32028)] (see also a paper of \textit{K.~Diederich and G.~Herbort} [Aspects Math. E 26, 127--161 (1994; Zbl 0845.32019)]). Namely, let \(D\) be a bounded pseudoconvex domain in \(\mathbb{C}^{n}\), let \(d(z)\) denote the distance from \(z\) to the boundary of~\(D\), let \(z_{0}\) be a smooth boundary point of finite type at which the Catlin multitype \((m_{1}, \dots, m_{n})\) agrees with the D'Angelo types \((\Delta_{n}, \dots, \Delta_{1})\) (such a boundary point is called h-extendible or semiregular), let \(m\) denote \(2\sum_{q=1}^{n} 1/\Delta_ {q}(z_ {0})\), and let \(K(z)\) denote the Bergman kernel function for \(D\) on the diagonal; then \(K(z)d(z)^{m}\) has a positive, finite, non-tangential limit at \(z_{0}\). The authors on the one hand specialize by restricting to the case of a convex boundary point \(z_{0}\) and on the other hand generalize by allowing \(z_{0}\) to be a point of infinite type. They show that the nontangential limit of \(K(z) d(z)^{\beta}\) at \(z_{0}\) is infinite when \(\beta<m\) and is equal to~\(0\) when \(\beta> m\). In the borderline case when \(\beta=m\), the key issue is whether every analytic disc (equivalently, every complex line) having infinite order of contact with the boundary of \(D\) at \(z_{0}\) also belongs locally to the boundary near \(z_{0}\): if so, then \(K(z) d(z)^{m}\) has a positive \(\liminf\) and a finite \(\limsup\) as \(z\to z_{0}\) within a nontangential cone; if not, then \(K(z) d(z)^{m}\) has an infinite nontangential limit at \(z_{0}\).