A HARDY-LITTLEWOOD MAXIMAL OPERATOR ADAPTED TO THE HARMONIC OSCILLATOR

This paper constructs a Hardy-Littlewood type maximal operator adapted to the Schrodinger operator L := -Delta + |x|(2) acting on L-2(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 is an element of 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).