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The Rigidity Theorems and Pointwise ∂-estimates


By Guan and Zhou’s resolution of the Suita Conjecture, it is known that for any open, hyperbolic Riemann surface X, the Bergman kernel K, the logarithmic capacity c_β, and the analytic capacity c_B, are related by πK ≥ c_β ≥ c_B. When X is a domain in C, we show that c_B ≥ π(Vol(X))^{-1} where Vol is the volume, and determine the conditions for when there exists a point z_0 such that c_B(z_0) = π(Vol(X))^{−1}, c_β(z_0) = π(Vol(X))^{−1}, and πK(z_0) = π(Vol(X))^{−1}. For open Riemann surfaces, we also determine equality conditions for c_B ≤ c_β. A significant portion of this part of the thesis is based on joint work with Dong and Zhang.

The second part of the thesis is motivated by Henkin and Leiterer’s question of whether uniform estimates for the ∂-operator hold on the Cartan classical bounded symmetric domains. Using weighted L^2-methods initiated by Berndtsson, we obtain a pointwise estimate for the canonical solutions to the equation ∂u = f when f is bounded in a Bergman-type L∞-norm. This part is based on joint work with Dong and Li.

In the third part of the thesis, we extend a theorem of M. Christ and S.-Y. Li on the ∂-equation ∂u = f. Let D ⊂ C^n be a bounded pseudoconvex domain with C∞ boundary which has a Stein neighborhood basis. We show that if f is a (p,q) form defined on Ω whose coefficients lie in a quasi-analytic class C^L(Ω), then there exists a solution u to ∂u = f such that the coefficients of u belong to the same quasi-analytic class.

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