Geodesic
The Bergman kernel near the diagonal for domains close to ellipsoids
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 19, Tome 190 (1991), pp. 163-172

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Asymptotics of the Bergman kernel near the diagonal of a pseudoconvex domain with mild boundary restrictions is investigated. For strictly pseudoconvex domains with $C^4$-boundary the principal term has the Ch. Fefferman form, and the error estimate coincides with that of L. Boutet de Monvel. 
@article{ZNSL_1991_190_a8,
     author = {N. A. Shirokov},
     title = {The {Bergman} kernel near the diagonal for domains close to ellipsoids},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {163--172},
     publisher = {mathdoc},
     volume = {190},
     year = {1991},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1991_190_a8/}
}
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N. A. Shirokov. The Bergman kernel near the diagonal for domains close to ellipsoids. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 19, Tome 190 (1991), pp. 163-172. http://geodesic.mathdoc.fr/item/ZNSL_1991_190_a8/