A Hardy-Littlewood Maximal Operator Adapted to the Harmonic Oscillator

Abstract

This paper constructs a Hardy-Littlewood type maximal operator adapted to the Schrödinger operator $\mathcal{L}:=-\Delta +|x{|}^2$ acting on $L^2({\mathbb{R}}^d)$. It achieves this through the use of the Gaussian grid ${\Delta }_0^{\gamma }$, constructed by J. Maas, J. van Neerven and P. Portal 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 ${\Delta }_0^{\gamma }$ 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 $A_p$.