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

Abstract

The classical $L^2$ estimate for the $\overline{\partial}$ operators is a basic tool in complex analysis of several variables. Naturally, it is expected to extend this estimate to infinite dimensional complex analysis, but this is a longstanding unsolved problem, due to the essential difficulty that there exists no nontrivial translation invariance measure in the setting of infinite dimensions. The main purpose in this series of work is to give an affirmative solution to the above problem, and apply the estimates to the solvability of the infinite dimensional $\overline{\partial}$ equations. In this first part, we focus on the simplest case, i.e., $L^2$ estimates and existence theorems for the $\overline{\partial}$ equations on the whole space of $\ell^p$ for $p\in [1,\infty)$. The key of our approach is to introduce a suitable working space, i.e., a Hilbert space for $(s,t)$-forms on $\ell^p$ (for each nonnegative integers $s$ and $t$), and via which we define the $\overline{\partial}$ operator from $(s,t)$-forms to $(s,t+1)$-forms and establish the exactness of these operators, and therefore in this case we solve a problem which has been open for nearly forty years.