A counterexample to $L^{\infty }$-gradient type estimates for Ornstein-Uhlenbeck operators

Abstract

Let $({\lambda }_k)$ be a strictly increasing sequence of positive numbers such that ${\sum }_{k=1}^{\infty }\frac{1}{{\lambda }_k}<\infty .$ Let $f$ be a bounded smooth function and denote by $u=u^f$ the bounded classical solution to $u(x)-\frac{1}{2}{\sum }_{k=1}^m{D}_{kk}^{2}u(x)+{\sum }_{k=1}^m{\lambda }_k{x}_k{D}_{k}u(x)=f(x),$ $x\in {\mathbb{R}}^m$. It is known that the following dimension-free estimate holds: $${\int }_{{\mathbb{R}}^m}(\sum _{k=1}^m{\lambda }_k(D_{k}u(y){)}^2{)}^{p/2}{\mu }_m(dy)\le (c_p{)}^p{\int }_{{\mathbb{R}}^m}|f(y){|}^p{\mu }_m(dy),1<p<\infty ;$$ here ${\mu }_m$ is the "diagonal" Gaussian measure determined by ${\lambda }_1,\dots ,{\lambda }_m$ 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 ${\lambda }_k\sim k^2$, then such estimate does not hold when $p=\infty$. Indeed we prove $$\underset{\begin{array}{c}f\in C_b^2({\mathbb{R}}^m),\Vert f{\Vert }_{\infty }\le 1\end{array}}{\sup }\{\sum _{k=1}^m{\lambda }_k(D_k{u}^f(0){)}^2\}\to \infty \text{as}m\to \infty .$$ This is in contrast to the case of ${\lambda }_k=\lambda >0$, $k\ge 1$, where a dimension-free bound holds for $p=\infty$.