Maximal function related to the Ornstein-Uhlenbeck operator.

On $\mathbf R^d$ the Ornstein-Uhlenbeck operator is defined as ($\partial_i = \frac{\partial}{\partial x_i}$).

$$L = \frac12 \sum_i \partial_i^* \partial_i$$

where $\partial_i^* = -\partial_i + 2 x_i$.

Now we can form the semigroup $e^{-t L}$ and the corresponding maximal function

$$M^* f(x) = \sup_{t > 0} \left |e^{-t L} f(x) \right |, \quad f \in L^1(\gamma)$$

where $\gamma$ is the Gaussian measure $\gamma(x) = \pi^{-d/2} e^{-|x|^2} dx$.

Now the claim is that $M^*$ is of weak $(p, p)$ type for $1 < p < \infty$. How do I show this? Any references?

-

There is a considerable literature about such operators. See, for example,

P. Sj\"ogren, Operators associated with the Hermite semigroup - a survey. J. Fourier Anal. Appl. 3, Spec. Iss., 813-823 (1997).

S. Pérez and F. Soria, Operators associated with the Ornstein-Uhlenbeck semigroup. J. Lond. Math. Soc., 61, No.3, 857-871 (2000)

and references therein.

-
@Anatoly: Thanks! – Jonas Teuwen May 16 '11 at 20:22

There is a very simple probabilistic proof of this fact that uses only basic properties of Ornstein-Uhlenbeck process (its symmetry) and Doob's maximal inequality

-