Weighted generalization of the Ramadanov's theorem and further considerations

Abstract

We study the limit behavior of weighted Bergman kernels on a sequence of domains in a complex space C N , and show that under some conditions on domains and weights, weighed Bergman kernels converge uniformly on compact sets. Then we give a weighted generalization of the theorem given in [S, p. 38], highlighting some special property of the domains, on which the weighted Bergman kernels converge uniformly. Moreover we will show that convergence of weighted Bergman kernels implies this property, which will give a characterization of the domains, for which the inverse of Ramadanov's theorem holds.

Equivalent Bergman Spaces with Inequivalent Weights

“…Moreover, for each n ∈ N the weight ν n (z) := min(n, z −2N ) · µ(z) is equivalent to µ(z): each ν n is simply the product of µ and a bounded function that is uniformly bounded away from zero (recall we are assuming D is bounded). Since ν n increases to ν as n → ∞, we may apply a weighted generalization of the Ramadanov theorem [7] to see that K νn → K ν uniformly on compact subsets of D × D. Regarding ζ ∈ D as fixed, the function K ν ( · , ζ) vanishes at zero, so a variant of Hurwitz's theorem shows that for large n we have K νn (0, ζ) = 0, completing the proof. The general case requires a more delicate approach, as a general measurable function may have extremely pathlogical behavior near every point in its domain, but the main idea remains the same.…”

“…We also require a weighted generalization of the Ramadanov theorem [7], whose statement is included for the convenience of the reader. We now have all the necessary tools to prove the main result of this section.…”

“…Moreover, for each n ∈ N the weight ν n (z) := min(n, z −2N ) • µ(z) is equivalent to µ(z): each ν n is simply the product of µ and a bounded function that is uniformly bounded away from zero (recall we are assuming D is bounded). Since ν n increases to ν as n → ∞, we may apply a weighted generalization of the Ramadanov theorem [7] to see that…”

The reduced Bergman kernel and its properties

“…). Therefore equation (7) holds true for the domain D. Now using the transformation formula proved in Theorem 1.11, we get…”

“…Ramadanov [10] showed that for a sequence of domains D j ⊂ C such that D j ⊂ D j+1 for all j ∈ Z + , and D := ∞ j=1 D j , the Bergman kernel K j (•, •) corresponding to the domain D j converges uniformly on compacts of D × D to the Bergman kernel K(•, •) corresponding to the domain D. Over the years, different versions of Ramadanov type theorems have been proved for the Bergman kernels and the weighted Bergman kernels (see, for instance, [7], [14]), for different types of convergence of the sequence of domains. Let us prove some of these Ramadanov type theorems for the n-th order weighted reduced Bergman kernels.…”