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Sharp pointwise and uniform estimates for $\bar\partial$

Robert Xin Dong, Song-Ying Li and John N. Treuer

Vol. 16 (2023), No. 2, 407–431
Abstract

We use weighted L2-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 Ω of rank 2 the maximum blow-up order is greater than log δΩ(z), which was obtained for the unit ball case by Berndtsson. For example, for Ω of type IV (n) with n 3, the maximum blow-up order is δ(z)1n2 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.

Keywords
$L^2$ minimal solution, canonical solution, $\bar\partial$ equation, Cartan classical domain, Bergman kernel, Bergman metric
Mathematical Subject Classification
Primary: 32A25
Secondary: 32A36, 32M15, 32W05
Milestones
Received: 26 May 2020
Revised: 29 March 2021
Accepted: 10 June 2021
Published: 3 May 2023
Authors
Robert Xin Dong
Department of Mathematics
University of Connecticut
Storrs, CT
United States
Song-Ying Li
Department of Mathematics
University of California
Irvine, CA
United States
John N. Treuer
Department of Mathematics
University of California
Irvine, CA
United States

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