A note on Gaussian maximal functions

This note presents a proof that the non-tangential maximal function of the Ornstein–Uhlenbeck semigroup is bounded pointwise by the Gaussian Hardy–Littlewood maximal function. In particular this entails an extension on a result by Pineda and Urbina [1] who proved a similar result for a ‘truncated’ version with fixed parameters of the non-tangential maximal function. We actually obtain boundedness of the maximal function on non-tangential cones of arbitrary aperture.