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

Abstract

AbstractLet $$(\lambda _k)$$ ( λ k ) be a strictly increasing sequence of positive numbers such that $${\sum _{k=1}^{\infty } \frac{1}{\lambda _k} < \infty }$$ ∑ k = 1 ∞ 1 λ k < ∞ . Let f be a bounded smooth function and denote by $$u= u^f$$ u = u f the bounded classical solution to $$\begin{aligned} u(x) - \frac{1}{2}\sum _{k=1}^m D^2_{kk} u(x) + \sum _{k =1}^m \lambda _k x_k D_k u(x) = f(x),\quad x \in {{\mathbb {R}}}^m . \end{aligned}$$ u ( x ) - 1 2 ∑ k = 1 m D kk 2 u ( x ) + ∑ k = 1 m λ k x k D k u ( x ) = f ( x ) , x ∈ R m . It is known that the following dimension-free estimate holds: $$\begin{aligned} \displaystyle \int _{{{\mathbb {R}}}^m}\! \left[ \sum _{k=1}^m \lambda _k \, (D_k u (y))^2 \right] ^{p/2} \!\! \!\!\!\! \mu _m (\textrm{d}y) \le (c_p)^p \!\! \int _{{{\mathbb {R}}}^m} \!\! |f( y)|^p \mu _m (\textrm{d}y),\;\;\; 1< p < \infty \end{aligned}$$ ∫ R m ∑ k = 1 m λ k ( D k u ( y ) ) 2 p / 2 μ m ( d y ) ≤ ( c p ) p ∫ R m | f ( y ) | p μ m ( d y ) , 1 < p < ∞ where $$\mu _m$$ μ m is the “diagonal” Gaussian measure determined by $$\lambda _1, \ldots , \lambda _m$$ λ 1 , … , λ m and $$c_p > 0$$ c p > 0 is independent of f and m. This is a consequence of generalized Meyer’s inequalities [4]. We show that, if $$\lambda _k \sim k^2$$ λ k ∼ k 2 , then such estimate does not hold when $$p= \infty $$ p = ∞ . Indeed we prove $$\begin{aligned} \sup _{\begin{array}{c} f \in C^{ 2}_b({{\mathbb {R}}}^m),\;\;\; \Vert f\Vert _{\infty } \le 1 \end{array}} \left\{ \sum _{k=1}^m \lambda _k \, (D_k u^f (0))^2 \right\} \rightarrow \infty \;\; \text{ as } \; m \rightarrow \infty . \end{aligned}$$ sup f ∈ C b 2 ( R m ) , ‖ f ‖ ∞ ≤ 1 ∑ k = 1 m λ k ( D k u f ( 0 ) ) 2 → ∞ as m → ∞ . This is in contrast to the case of $$\lambda _k = \lambda >0$$ λ k = λ > 0 , $$k \ge 1$$ k ≥ 1 , where a dimension-free bound holds for $$p =\infty $$ p = ∞ .