Ornstein-Uhlenbeck semigroup maximal operator on weighted $L^p$ space

Abstract

Let $(\mathcal{H}\_t)_{t \geq 0}$ denote the standard Ornstein-Uhlenbeck semigroup over ambient space $\mathbb R^d$ equipped with Gaussian measure $dγ(x) = e^{-|x|^2} dx$. Further let $\mathcal{H}^*(f)(x) = \sup_{t > 0} \mathcal{H}_t(|f|)(x)$ denote the associated maximal operator. Since $(\mathcal{H}_t)_{t \geq 0}$ is a Markovian semigroup over $(\mathbb R^d,dγ(x))$, it is well known that $\mathcal{H}^*$ is bounded on $L^p(\mathbb R^d,dγ(x))$ for all $1 < p \leq \infty$. We show that $\mathcal{H}^*$ is also bounded on the weighted space $L^p(\mathbb{R}^d,ω(x)e^{-\frac{p}{2}|x|^2}dx)$ for $1 < p <\infty$, for weights $ω\,: \mathbb R^d \to (0,\infty)$ belonging to a certain Muckenhoupt type class $A^α_p$, where $α\in [0,1)$. Essentially, a weight $ω$ belongs to $A^α_p$ if and only if $ω(Q) ω^{-\frac{1}{p-1}}(Q)^{p-1} \leq C |Q|^p$ for all cubes $Q$ contained in any reference cube $N_α(R_x) \subseteq \mathbb R^d$, where the well chosen family $(N_α(R_x))_{x \in \mathbb R^d}$ covers $\mathbb R^d$, $N_α(R_x)$ contains $x$ and the side length of $N_α(R_x)$ is comparable to $1/\max(1,|x|)^α$, with constants depending only on $d$. The class $A^α_p$ contains the usual Muckenhoupt class. We also use the weight class $A_p^{loc}$ from \cite[Definition 2.2]{B} defined in a similar way as $A^α_p$ but with dyadic subcubes contained in some $N(R)$ built over the Gaussian dyadic grid (see below), and we show that the local part $\sup_{t >0}\mathcal{H}_t(|f|χ_{N(R_x)})(x)$ is bounded on $L^p(\mathbb{R}^d,ω(x)e^{-\frac{p}{2}|x|^2}dx)$ if and only if $ω\in A^{loc}_p$.