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?



Maximal function related to the Ornstein-Uhlenbeck operator.
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.


