A note on Gaussian maximal functions
Let \(\gamma\) be the Gaussian measure \(d\gamma(x)=\pi^{-\frac{d}2}e^{-|x|^2}\,d\lambda(x)\), where \(\lambda\) is the Lebesgue measure, and let \(L:=\frac12\Delta-\langle x,\nabla\rangle\) be the Ornstein-Uhlenbeck operator on the Gaussian measure space \((\mathbb{R}^d,d\gamma)\). In this paper, it is proved that the non-tangential maximal function of the Ornstein-Uhlenbeck semigroup is bounded pointwise by the Gaussian Hardy-Littlewood maximal function, namely, \[ \sup_{(y,t)\in \Gamma^{(A,a)}_x}|e^{t^2L}u(y)|\lesssim \sup_{r>0}\frac1{\gamma(B(x,r))}\int_{B(x,r)}|u|\,d\gamma, \] where \(\Gamma^{(A,a)}_x\) denotes the set \[ \{(y,t)\in \mathbb{R}^{d+1}_+:\;\;|x-y|<AT,\;\;t\leq am(x)\} \] with \(A,a>0\) and \(m(x)=\min\{1, 1/|x|\}\). This generalizes a former result by \textit{E. Pineda} and \textit{W. Urbina R.} [Divulg. Mat. 16, No. 1, 107--124 (2008; Zbl 1217.60028)].
scientific article