$L^2$ estimates and existence theorems for the $\bar{\partial}$ operators in infinite dimensions, I

Jiayang Yu, Xu Zhang

Published 2020-02-06Version 1

In this paper we introduce a probability measure on $\ell^p$ ($p\in [1,\infty)$) by which we define the $(s,t)$-forms on $\ell^p$ and the $\bar{\partial}$ operator from $(s,t)$-forms to $(s,t+1)$-forms. Then we can employ the $L^2$-method to establish the exactness of the $\bar{\partial}$ operator. As a consequence, the counterexamples given by Coeur\'{e} and Lempert which are not solvable on any open set are solvable for the $\bar{\partial}$ operator defined in this paper.