A Hardy-Littlewood Maximal Operator Adapted to the Harmonic Oscillator

TL;DR

This paper develops a new maximal operator adapted to the harmonic oscillator, defining a broader class of weights for which the associated heat maximal operator is bounded on weighted L^p spaces.

Contribution

It introduces a novel maximal operator and a new weight class, $A_{p}^{+}$, extending the boundedness results for the heat maximal operator related to the harmonic oscillator.

Findings

The new maximal operator resembles the classical Hardy-Littlewood operator at small scales.

The weight class $A_{p}^{+}$ is strictly larger than the classical $A_{p}$ class.

The heat maximal operator is bounded on $L^{p}(w)$ for weights in $A_{p}^{+}$.

Abstract

This paper constructs a Hardy-Littlewood type maximal operator adapted to the Schr\"{o}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 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 $\Delta^{\gamma}_{0}$ 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 $A_{p}$.