Pure and Applied Mathematics Quarterly

Volume 18 (2022)

Number 2

Special issue in honor of Joseph J. Kohn on the occasion of his 90th birthday

Guest Editors: J.E. Fornaess, Stanislaw Janeczko, Duong H. Phong, and Stephen S.T. Yau

Sup-norm estimates for $\overline{\partial}$

Pages: 531 – 571

DOI: https://dx.doi.org/10.4310/PAMQ.2022.v18.n2.a8

Authors

Dusty Grundmeier (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)

Lars Simon (Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway)

Berit Stensønes (Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway)

Abstract

We develop a method for proving sup-norm and Hölder estimates for $\overline{\partial}$ on wide class of finite type pseudoconvex domains in $\mathbb{C}^n$. A fundamental obstruction to proving sup-norm estimates is the possibility of singular complex curves with exceptionally high order of contact with the boundary. Our method handles this problem, and in $\mathbb{C}^3$, we prove sup-norm and Hölder estimates for all bounded, pseudoconvex domains with real-analytic boundary.

Keywords

finite type, bumping, Hölder estimates, sup-norm estimates, $\overline{\partial}$-equation

2010 Mathematics Subject Classification

32A26, 32T25

Received 22 March 2021

Received revised 8 October 2021

Accepted 9 November 2021

Published 13 May 2022