论文标题
$ l^{\ infty} $的反例 - Ornstein-uhlenbeck操作员的渐变类型估计值
A counterexample to $L^{\infty}$-gradient type estimates for Ornstein-Uhlenbeck operators
论文作者
论文摘要
令$（λ_k）$是严格增加的正数顺序，以便$ \ sum_ {k = 1}^{\ infty} \ frac {1} {1} {λ_k} <\ infty。 \ frac {1} {2} \ sum_ {k = 1}^m d^2_ {kk} u（x） + \ sum_ {k = 1}^mλ_kx_k x_k d_k d_k d_k d_k u（x）= f（x），$ x \ in \ r^m $。众所周知，以下无维估计值保持：$$ \ displayStyle \ int _ {\ r^m} \ big（\ sum_ {k = 1}^mλ_k\，（d_k u（y） \ le（c_p）^p \，\ int _ {\ r^m} | f（y）|^pμ_m（dy），\; \; \; 1 <p <\ infty; $$这里$μ_m$是由$λ_1，\ ldots，λ_m$和$ c_p> 0 $确定的“对角线”高斯措施，独立于$ f $和$ m $。这是迈耶（Meyer）普遍不平等的结果[Chojnowska-Michalik，Goldys，J。Funct。肛门。 182（2001）]。我们表明，如果$λ_k\ sim k^2 $，那么当$ p = \ infty $时，此估计不会成立。实际上，我们证明$$ \ sup _ {\ ordack {f \ in c^{2} _b（\ r^m），\; \; \ | f \ | _ {\ infty} \ leq 1}}}} \ big \ {\ sum_ {k = 1}^mλ_k\ \，（d_k u^f（0）^f（0）^2 \ big \} \ \ text {as} \; m \ to \ infty。 $$这与$λ_k=λ> 0 $，$ k \ ge 1 $的情况形成鲜明对比，其中无尺寸限制为$ p = \ infty $。
Let $(λ_k)$ be a strictly increasing sequence of positive numbers such that $\sum_{k=1}^{\infty} \frac{1}{λ_k} < \infty.$ Let $f $ be a bounded smooth function and denote by $u= u^f$ the bounded classical solution to $u(x) - \frac{1}{2}\sum_{k=1}^m D^2_{kk} u(x) + \sum_{k =1}^m λ_k x_k D_k u(x) = f(x), $ $ x \in \R^m$. It is known that the following dimension-free estimate holds: $$ \displaystyle \int_{\R^m} \Big (\sum_{k=1}^m λ_k \, (D_k u (y))^2 \Big)^{p/2} μ_m (dy) \le (c_p)^p \, \int_{\R^m} |f( y)|^p μ_m (dy),\;\;\; 1 < p < \infty; $$ here $μ_m$ is the "diagonal" Gaussian measure determined by $λ_1, \ldots, λ_m$ and $c_p > 0$ is independent of $f$ and $m$. This is a consequence of generalized Meyer's inequalities [Chojnowska-Michalik, Goldys, J. Funct. Anal. 182 (2001)]. We show that, if $λ_k \sim k^2$, then such estimate does not hold when $p= \infty$. Indeed we prove $$ \sup_{\substack{f \in C^{ 2}_b(\R^m),\;\; \|f\|_{\infty} \leq 1}} \Big \{ \sum_{k=1}^m λ_k \, (D_k u^f (0))^2 \Big \} \to \infty \;\; \text {as} \; m \to \infty. $$ This is in contrast to the case of $λ_k = λ>0$, $k \ge 1$, where a dimension-free bound holds for $p =\infty$.
