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Original Article | Open Access

Uniform Estimates of the Cauchy–Riemann Equation and Kerzman’s Question

Department of Mathematics, Syracuse University, Syracuse, NY 13244, USA
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Abstract

We answer a question raised by Kerzman in 1971. More precisely, we show that the canonical solution of the ∂ ¯ -equation satisfies the L p estimate on the polydisc for p ∈ [ 1 , ∞ ]. Moreover, the L p estimates for p ∈ [ 1 , ∞ ] of ∂ ¯ can also be obtained on the product of bounded C 2 planar domains by an observation based on the method developed in [Dong et al. arXiv:2006.14484].

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Peking Mathematical Journal
Pages 629-642

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Cite this article:
Yuan, Y. Uniform Estimates of the Cauchy–Riemann Equation and Kerzman’s Question. Peking Math J 9, 629-642 (2026). https://doi.org/10.1007/s42543-025-00096-0

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Received: 22 October 2023
Revised: 18 December 2024
Accepted: 13 January 2025
Published: 24 March 2025
© The Author(s) 2025

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