CR Invariants of Weight Five in the Bergman Kernel
CR Invariants of Weight Five in the Bergman Kernel
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Bergman 内核中权重五的 CR 不变量
DOI：
10.1006/aima.1998.1794
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发表时间：
1999
影响因子：
1.7
通讯作者：
Noriyuki Nakazawa
中科院分区：
文献类型：
--
作者：
K. Hirachi;G. Komatsu;Noriyuki Nakazawa
Fefferman的程序（1979，C. Fefferman，数学学报，31,131-262）在二维光滑有界严格伪凸区域上得到Bergman核的一个生物全纯不变渐近展开，并给出了普遍常数的辨识。根据程序，展开式是用一个近似不变的光滑定义域函数来表示的，我们称之为Fefferman定义函数，而系数是用Fefferman定义函数的导数构造的定义域内的函数。因此，不变展开式必然是一个有余项的有限和，模糊估计是问题的关键。我们得到一个展开式，使得系数的边值是权值为5的CR不变量。这改进了C. R. Graham（1987）在《数学讲义》（Lecture Notes in mathematics）中的早期结果。，第1276卷，第108-135页，斯普林格出版社，纽约/柏林)和作者(1993年，K. Hirachi， G. Komatsu和N. Nakazawa，在“纯和应用的课堂笔记”。数学。，第143卷，第77-96页，Dekker，纽约)。通过在权值为4的CR不变量的域中进行适当的扩展，细化成为可能。由于这些扩展的模糊估计，只要使用Fefferman的定义函数，我们的扩展是最优的。对于Szego内核也得到了类似的结果。
Abstract Fefferman's program (1979, C. Fefferman, Adv. Math. 31 , 131–262) of getting a biholomorphically invariant asymptotic expansion of the Bergman kernel for smoothly bounded strictly pseudoconvex domains is realized in dimension 2 with the identification of universal constants. According to the program, the expansion is in terms of an approximately invariant smooth defining function of the domain, which we refer to as Fefferman's defining function, and the coefficients are functions in the domain constructed by using derivatives of Fefferman's defining function. Consequently, the invariant expansion is necessarily a finite sum with a remainder term and the ambiguity estimate is crucial in the problem. We get an expansion such that the boundary values of the coefficients are CR invariants of weight ⩽5. This refines earlier results of C. R. Graham (1987, in “Lecture Notes in Math.,” Vol. 1276, pp. 108–135, Springer–Verlag, New York/Berlin) and the authors (1993, K. Hirachi, G. Komatsu, and N. Nakazawa, in “Lecture Notes in Pure and Appl. Math.,” Vol. 143, pp. 77–96, Dekker, New York). The refinement becomes possible by appropriate extensions inside the domain of the CR invariants of weight 4. Due to the ambiguity estimate of these extensions, our expansion is optimal as far as Fefferman's defining function is used. A similar result for the Szego kernel is also obtained.