Weighted Bergman kernel: asymptotic behavior, applications and comparison results

Abstract

Abstract. Inspired by the work of Engliš, we study the asymptotic behavior of the weighted Bergman kernel together with an application to the Lu Qi-Keng conjecture. Some comparison results between the weighted and the classical Bergman kernel are also obtained.

On some extremal problems in Bergman spaces in weakly pseudoconvex domains

“…Similar condition on defining function of a general pseudoconvex domain can be seen in [5]. Theorem 1.…”

Lu Qi-Keng's problem on some complex ellipsoids

“…When m = q = 1 and 2p 1 is not an integer, as an application of a theorem due to M. Englis [9] and an improvement by B. Chen, Chen [10] proved there exists a constant n(p) depending on p such that for all n > n(p), the domain Ω p,1 1,n = {|w| 2p + |z 1 | 2 + · · · + |z n | 2 < 1} is not Lu Qi-Keng. A similar argument as in [6] immediately shows that {|w| 2p + |z 1 | + · · · + |z n | < 1} is not Lu Qi-Keng iff n [n(p)/2] + 1, where [n(p)/2] produces the integer part of n(p)/2.…”