论文标题
$ l^2 $估计和存在定理的$ \ bar {\ partial} $ operators in Infinite Dimensions，i
$L^2$ estimates and existence theorems for the $\bar{\partial}$ operators in infinite dimensions, I
论文作者
论文摘要
$ \ overline {\ partial} $运算符的经典$ l^2 $估计是对几个变量的复杂分析中的基本工具。自然地，预计该估计值将扩展到无限的维数复杂分析，但这是一个长期以来的未解决问题，因为在无限维度的设置中没有非平凡的翻译不变性度量。这一系列工作的主要目的是为上述问题提供肯定的解决方案，并将估计值应用于无限尺寸$ \ operline {\ partial} $方程的溶解度。在第一部分中，我们专注于最简单的情况，即$ \ overline {\ partial} $方程的$ l^2 $估计和存在定理，全部$ \ ell^p $ for $ p $ for $ p \ in [1，\ infty）$。我们方法的关键是要引入一个合适的工作空间，即希尔伯特（Hilbert）空间$（s，t）$ - $ \ ell^p $上的形式（对于每个非负整数$ s $ and $ t $），并通过$ \ overline {\ partial} $从$（s，t）$ - $ - $ - $ - $ - $ - $ 1）定义$ \ overline {\ partial} $（s，t）$ - t+1（s），这个案例我们解决了一个已经开放已有近40年的问题。
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.
