Explicit formulas for the Szegö kernel for some domains in \({\mathbb{C}}^ 2\)
As is well-known from the work of \textit{L. Boutet de Monvel} and \textit{J. Sjöstrand} [Asterisque 34-35, 123-164 (1976; Zbl 0344.32010)], the Szegö kernel for bounded strictly pseudoconvex domains in \({\mathbb{C}}^ n\) can be expressed as a Fourier integral operator which is determined by a complex phase \(\psi\) and a classical symbol h. The author gives an explicit algorithm for computing the asymptotic expansion of h for a class of strictly pseudoconvex domains in \({\mathbb{C}}^ 2\). More explicitly, if \((z,w)\) are coordinates near \(0\in {\mathbb{C}}^ 2\) then the class under consideration consists of domains \(\Omega\) in \({\mathbb{C}}^ 2\) whose boundary near 0 is given by Im w\(=\phi (z,\bar z)\), where \(\phi\) is a real analytic function of \((z,\bar z)\) with \(\phi (0,0)=0\). In this case, the above complex phase \(\psi\) is given by \(\psi (z,w,\bar z',\bar w')=w-\bar w'-2i\phi (z,\bar z')\) where \((z,w,\bar z',\bar w')\)varies near \(0\in {\mathbb{C}}^ 4\). Denote by K the Szegö kernel of a domain \(\Omega\) in the class. It follows from the above mentioned work of Boutet de Monvel and Sjöstrand that there exists a symbol h of order 1 such that \[ K(z,w,z',w'-\int^{\infty}_{0}e^{i\psi (z,w,\bar z',\bar w')\tau} h(z,\bar z',\tau)d\tau \] is a smooth function near \((0,0)\in \partial \Omega \times \partial \Omega\). Here h is a smooth function in all arguments such that for each \(\tau\in (0,\infty)\), \(h(\cdot,\cdot,\tau)\) holomorphic, and, moreover, h admits an asymptotic expansion of the form \[ h(z,\bar z',\tau)\sim \sum^{\infty}_{k=- 1}h_{-k}(z,\bar z')\tau^{-k}. \] In this paper, the first three terms \(h_ 1\), \(h_ 0\), \(h_{-1}\) are computed explicitly in terms of \(\phi\) and its derivatives. In particular, this gives a complete description of the logarithmic term \(h_{-1}\) for the asymptotic expansion of K near \((0,0)\in \partial \Omega \times \partial \Omega\).
scientific article; zbMATH DE number 4148532