论文标题
高斯尾部空间上Ornstein-Uhlenbeck操作员的急剧生长
Sharp growth of the Ornstein-Uhlenbeck operator on Gaussian tail spaces
论文作者
论文摘要
令$ x $为标准的高斯随机变量。对于任何$ p \（1，\ infty）$中的任何$ p \，我们证明存在通用常数$ c_ {p}> 0 $ 0 $，使得不等式$$（\ Mathbb {e} | h'（x）|^{p} |^{p}） | h（x）|^{p}）^{1/p} $$所有$ d \ geq 1 $和所有多项式$ h：\ mathbb {r} \ to \ mathbb {c} $的频谱至少在$ d $ $ d $上支持频率，$ d $，$ d $，$ \ mathbbbbbbbbbbbbbbbbbbbbbbbbbbbb bbbbbbbbbbbbbbbbbbb {e e} $ k = 0,1，\ ldots，d-1 $。作为此最佳估计值的应用，我们对Mendel和Naor（2014）问题的高斯类似物获得了肯定的答案，该问题是关于Ornstein-Uhlenbeck操作员在实际线路上的尾部空间上的生长。我们还显示了在任意维度中分析多项式梯度的相同界限。
Let $X$ be a standard Gaussian random variable. For any $p \in (1, \infty)$, we prove the existence of a universal constant $C_{p}>0$ such that the inequality $$(\mathbb{E} |h'(X)|^{p})^{1/p} \geq C_{p} \sqrt{d} (\mathbb{E} |h(X)|^{p})^{1/p}$$ holds for all $d\geq 1$ and all polynomials $h : \mathbb{R} \to \mathbb{C}$ whose spectrum is supported on frequencies at least $d$, that is, $\mathbb{E} h(X) X^{k}=0$ for all $k=0,1, \ldots, d-1$. As an application of this optimal estimate, we obtain an affirmative answer to the Gaussian analogue of a question of Mendel and Naor (2014) concerning the growth of the Ornstein-Uhlenbeck operator on tail spaces of the real line. We also show the same bound for the gradient of analytic polynomials in an arbitrary dimension.
