  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 $A_{p}$.
