论文标题
Littlewood's on Random Power系列的高斯版本
A Gaussian version of Littlewood's theorem on random power series
论文作者
论文摘要
我们证明，对于不一定是独立的高斯过程的随机分析函数的Littlewood型定理。我们表明，如果我们通过高斯流程在Hardy Space $ H^2（\ dd）$中随机化功能，其协方差矩阵$ k $在$ l^2 $上诱导了一个有界的操作员，那么所得的随机函数几乎可以肯定在$ h^p（\ dd）$中，对于任何$ p> 0 $。 $ k = \ text {id} $，身份操作员恢复了Littlewood的定理。我们证明中的一种新成分是将会员问题重新阐述为操作员的界限。这种重新启动使我们能够在功能分析中使用工具，并且适用于其他情况。讨论了新条件的清晰度和几个后果。
We prove a Littlewood-type theorem on random analytic functions for not necessarily independent Gaussian processes. We show that if we randomize a function in the Hardy space $H^2(\dd)$ by a Gaussian process whose covariance matrix $K$ induces a bounded operator on $l^2$, then the resulting random function is almost surely in $H^p(\dd)$ for any $p>0$. The case $K=\text{Id}$, the identity operator, recovers Littlewood's theorem. A new ingredient in our proof is to recast the membership problem as the boundedness of an operator. This reformulation enables us to use tools in functional analysis and is applicable to other situations. The sharpness of the new condition and several ramifications are discussed.
