Revista de la
Unión Matemática Argentina
A Hardy–Littlewood maximal operator adapted to the harmonic oscillator
Julian Bailey
Volume 59, no. 2 (2018), pp. 339–373


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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} $.