A Hardy-Littlewood maximal operator adapted to the harmonic oscillator

This paper constructs a Hardy-Littlewood type maximal operator adapted to the Schrödinger operator L := -δ + |x|² acting on L²(ℝ^d). It achieves this through the use of the Gaussian grid δ ₀ ^γ, 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 δ ₀ ^γ and weighted appropriately. Through this maximal function, a new class of weights is defined, A _p ⁺, with the property that for any w ∈ A _p ⁺ the heat maximal operator associated with 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.