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

Basic estimates for the generalized $\partial$-complex

Pages: 583 – 597

DOI: https://dx.doi.org/10.4310/PAMQ.2022.v18.n2.a10


Friedrich Haslinger (Faculty of Mathematics, University of Vienna, Austria)


We study certain densely defined unbounded operators on the Segal–Bargmann space, related to the annihilation and creation operators of quantum mechanics. We consider the corresponding D-complex and study properties of the complex Laplacian $\tilde{\Box}_D = DD^\ast + D^\ast D$, where $D$ is a differential operator of polynomial type, in particular we discuss the corresponding basic estimates, where we express a commutator term as a sum of squared norms.


$\partial$-complex, Segal–Bargmann space, sum of squared norms

2010 Mathematics Subject Classification

Primary 30H20, 32A36, 32W50. Secondary 47B38.

The author is supported by the Austrian Science Fund (FWF) project P28154.

Received 12 March 2021

Received revised 12 July 2021

Accepted 22 July 2021

Published 13 May 2022