CR Invariants of Weight Five in the Bergman Kernel

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 Szegö kernel is also obtained.