For any open hyperbolic Riemann surface X, the Bergman kernel K, the logarithmic capacity c β , and the analytic capacity c B satisfy the inequality chain πK ≥ c 2 β ≥ c 2 B ; moreover, equality holds at a single point between any two of the three quantities if and only if X is biholomorphic to a disk possibly less a relatively closed polar set. In this paper, we extend the inequality chain by showing that c 2 B ≥ πv −1 (X) on planar domains, where v(•) is the Euclidean volume, and characterize the extremal cases when equality holds at one point. Similar rigidity theorems concerning the Szegö kernel, the higher-order Bergman kernels, and the sublevel sets of the Green's function are also developed. Additionally, we explore rigidity phenomena related to the multi-dimensional Suita conjecture for domains in C n , n ≥ 1.
For any open hyperbolic Riemann surface $X$, the Bergman kernel $K$, the logarithmic capacity $c_{\beta }$, and the analytic capacity $c_{B}$ satisfy the inequality chain $\pi K \geq c^2_{\beta } \geq c^2_B$. Moreover, equality holds at a single point between any two of the three quantities if and only if $X$ is biholomorphic to a disk possibly less a relatively closed polar set. We extend the inequality chain by showing that $c_{B}^2 \geq \pi v^{-1}(X)$ on planar domains, where $v(\cdot )$ is the Euclidean volume, and characterize the extremal cases when equality holds at one point. Similar rigidity theorems concerning the Szegö kernel, the higher-order Bergman kernels, and the sublevel sets of the Green’s function are also developed. Additionally, we explore rigidity phenomena related to the multi-dimensional Suita conjecture.
We use weighted L 2 -methods to obtain sharp pointwise estimates for the canonical solution to the equation ∂u = f on smoothly bounded strictly convex domains and the Cartan classical domains when f is bounded in the Bergman metric g. We provide examples to show our pointwise estimates are sharp. In particular, we show that on the Cartan classical domains of rank 2 the maximum blow-up order is greater than − log δ (z), which was obtained for the unit ball case by Berndtsson. For example, for of type IV(n) with n ≥ 3, the maximum blow-up order is δ(z) 1−n/2 because of the contribution of the Bergman kernel. Additionally, we obtain uniform estimates for the canonical solutions on the polydiscs, strictly pseudoconvex domains and the Cartan classical domains under stronger conditions on f .
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