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##### 1936-1944
A Hardy–Littlewood maximal operator adapted to the harmonic oscillator
Volume 59, no. 2 (2018), pp. 339–373

https://doi.org/10.33044/revuma.v59n2a07

### Abstract

This paper constructs a Hardy–Littlewood type maximal operator adapted to the Schr ö dinger operator $\mathcal{L} := - \Delta + |x|^2$ acting on $L^ {2} ( \mathbb{R} ^ {d} )$. It achieves this through the use of the Gaussian grid $\Delta^{\gamma} _ {0}$, constructed by Maas, van Neerven, and Portal [Ark. Mat. 50 (2012), no. 2, 379-395] with the Ornstein-Uhlenbeck operator in mind. At the scale of this grid, this maximal operator will resemble the classical Hardy–Littlewood operator. At a larger scale, the cubes of the maximal function are decomposed into cubes from $\Delta^{\gamma} _ {0}$ and weighted appropriately. Through this maximal function, a new class of weights is defined, $A_ {p} ^ {+}$, with the property that for any $w \in A_{p} ^ {+}$ the heat maximal operator associated with $\mathcal{L}$ is bounded from $L^ {p} (w)$ to itself. This class contains any other known class that possesses this property. In particular, it is strictly larger than $A_ {p}$.