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

$q$-effectiveness for holomorphic subelliptic multipliers

Pages: 617 – 637

DOI: https://dx.doi.org/10.4310/PAMQ.2022.v18.n2.a12


Sung-Yeon Kim (Center for Complex Geometry, Institute for Basic Science, Daejeon, South Korea)

Dmitri Zaitsev (School of Mathematics, Trinity College Dublin, Ireland)


We provide a solution to the effectiveness problem in Kohn’s algorithm for generating holomorphic subelliptic multipliers for $(0, q)$ forms for arbitrary $q$. As application, we obtain subelliptic estimates for $(0, q)$ forms with effectively controlled order $\varepsilon \gt 0$ (the Sobolev exponent) for domains given by sums of squares of holomorphic functions (J.J. Kohn called them “special domains” in [K79]). These domains are of particular interest due to their relation with complex and algebraic geometry. Our methods include triangular resolutions introduced by the authors in [KZ20].

This work was supported by the Institute for Basic Science (IBS-R032-D1-2021-a00).

Received 6 May 2021

Accepted 29 December 2021

Published 13 May 2022