The formula for the singularity of Szegö kernel. II (Q1769522)

Let \(M\) be a strongly pseudoconvex hypersurface in \(\mathbb C^{n+1}\), i.e. the boundary of a domain \(\Omega\subset\mathbb C^{n+1}\). The Szegö kernel \(K^S(x,y)\), \(x,y\in M\) is smooth outside of the diagonal \(x=y\). The singularity at \((x, x)\) is determined by the local datum at \(x\) of \(M\), even though \(K^S\) itself is a global object. The problem is to write down the singularity at \((x,x)\) in terms of the local equation of \(M\) in \(\mathbb C^{n+1}\). We fix a reference point, say \(p_\ast\), in \(M\) and only consider the germ of \(M\) at \(p_\ast\). Let a strongly pseudoconvex inside tubular neighborhood \(N\) of \(M\) in \(\mathbb C^{n+1}\) be defined by \(r > 0\). Let \(r(x,x')\) be the function on \(N\times N\) satisfying the conditions: (i) \(r(x,x) = r(x)\), (ii) \(r(x,x')\) is holomorphic in \(x\), (iii) \(r(x,x') =r(x',x)\). As is shown by Fefferman and Boutet de Monvel-Sjöstrand, the singularity of \(K^S\) at \((p_\ast,p_\ast)\) is of the form: \(K^S = Fr^{-(n+1)}+G\log r\), where \(F\) and \(G\) are smooth functions. In part I [J. Korean Math. Soc. 40, No. 4, 641--666 (2003; Zbl 1047.32001)] of this series of papers the author developed a procedure to write down the above \(F\), \(G\) near \((p_\ast,p_\ast)\) for a specific choice of \(r\). The author used the method developed by Boutet de Monvel and Sjöstrand. The method is based on a symplectic transformation \(\chi\) which transforms a conic neighbourhood of the characteristics of the symbols of \({\overline\partial}_\flat\) operators of the model structure to that of the structure. The symplectic transformation is defined as the solution of an ordinary differential equation of Hamilton type. However, when one tries to carry out the construction explicitly in a straightforward way, the formula becomes rather cumbersome. In this paper the author develops a way of calculation so that it becomes more accessible. The author also constructs the inverse map of \(\chi\) by the similar method.