On the $L^p$ boundedness of the non-centered Gaussian Hardy-Littlewood maximal function

The purpose of this paper is to prove the $L^p(\mathcal{R}^n, d\gamma)$ boundedness, for $p>1$, of the non-centered Hardy-Littlewood maximal operator associated with the Gaussian measure $d\gamma=e^{-\vert x\vert^2} dx$.