The Bergman and Szegő kernels near points of infinite type
Let \(b\in C^\infty (\mathbb R)\) be a smooth function. The Bergman and Szegő kernels \({\mathcal B}_\Omega\) and \({\mathcal S}_\Omega\) for the domain \[ \Omega= \{ z=(z_1,z_2)\in \mathbb C^2\mid \Im (z_2) >b(\Re z_1)\} \] are shown to admit a smooth extension to a certain large part of \(\overline \Omega\times \overline \Omega\) provided that the function \(b\) satisfies certain basic assumptions. These assumptions are: (1) \(b\) is convex and is normalized at \(0 \in\mathbb R: b(0)=b'(0)=0\); (2) \(b\) is even; (3) \(b'\) and \(b''\) are convex on the positive real axis; (4) Let \(B(r)=\int_0^r b(t)\,dt\). Then there exists a constant \(C>0\) such that \[ 0< \frac{B^{(j+1)}}{B^{(j)}} (r) \leq C \frac{B^{(k+1)}}{B^{(k)}} (r),\quad \text{for } r>0 \text{ and any } j,k \in \{0,1,2,3\}. \] (5) There is a convex polynomial \(P\) of degree \(N\geq 2\) such that \(b=P\) on \(\{|r| \geq 1\}\). Let \(\Theta (r) = \frac{b(r)}{rb'(r)}\). Then \(b\) is said to be ``subcritical'' at \(r=0\), if, for some constants \(c>0\) and \(\gamma \in (0,1)\), one has \(\Theta (r) >c |r|^\gamma \); ``critical'' at \(r=0\), if, for some constant \(C>0\), one has \(\Theta (r) \leq C|r|\) for small \(|r|\); ``supercritical'' at \(r=0\), if, for some constants \(C>0\) and \(a>1\), one has \(\Theta (r) \leq C|r|^a\) for small \(|r|\). For \(0 \leq \rho \leq 1\), let \(B_\rho^2(\Omega):=\{F \in{\mathcal O}( \Omega)\mid \|f\|_\rho <\infty\}\), where \[ \|f\|_\rho^2 = \int_\Omega |f(x,y,t,s)|^2 (s-b(x))^{-\rho} \,dx\,dy\,dt\,ds. \] Then \(B_\rho^2(\Omega)\) is a Hilbert space, and the orthogonal projector from \(L_\rho^2(\Omega):=\{f \mid\|f\|_\rho<\infty\}\) onto \(B_\rho^2(\Omega)\) has an integral kernel \({\mathcal B}_\rho\). For \(\rho=0\), we obtain the Bergman kernel, and, for \(\rho=1\), we get the Szegő kernel. The authors obtain the following results: {\parindent=7mm \begin{itemize}\item[(A)] If \(b\) is subcritical at \(0\), then \({\mathcal B}_\rho\) extends smoothly to \(\overline \Omega\times \overline \Omega \setminus \Delta\), where \(\Delta\) is the diagonal. \item[(B)] If \(b\) is supercritical at \(0\), then \({\mathcal B}_\rho\) extends smoothly to \(\overline \Omega\times \overline \Omega \setminus (\Delta \cup \Sigma)\), where \(\Sigma=\{(z,w) \in \partial \Omega\times \partial \Omega\mid \Re z_1=\Re w_1=0\), \(\Re z_2=\Re w_2 \}\). \item[(C)] The kernels \({\mathcal B}_\rho\) have actually singularities off the diagonal, namely, let \(t >0\). \item[(i)] If \(b\) is critical at 0, then there exists \(\varepsilon >0\), such that for \(|y-v|<\varepsilon\) \[ \lim_{h+k\to 0, \,h,k>0} |{\mathcal B}_\rho (\,(iy,t+ih),\;(iv, t+ik))|= \infty. \] \item[(ii)] In case that \(b\) is supercritical at 0, one has \[ \lim_{h+k\to 0, \,h,k>0} |{\mathcal B}_\rho (\,(iy,t+ih),\;(iv, t+ik))|= \infty, \] for all \(y,v \in\mathbb R\). \end{itemize}}
scientific article; zbMATH DE number 5704132