Schrodinger Operators and the Kato Square Root Problem

Abstract

The general theme of this thesis is the harmonic analysis of Schr¨odinger operators and its applications. We will focus on two distinct but related open problems in this field. The first problem is the construction of potential dependent averaging operators and will be primarily considered in the first part of this thesis. Here, a Hardy-Littlewood type maximal operator adapted to the Schrodinger operator L := −∆ + |x| 2 and acting on L 2 (R n ) is constructed. This is achieved through the use of the Gaussian grid ∆γ 0 , constructed in [42] with the Ornstein-Uhlenbeck operator in mind. At the scale of this grid, the maximal operator will resemble the classical Hardy-Littlewood operator. At a larger scale, the constituent averaging operators of the maximal function are decomposed over the cubes from ∆γ 0 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 and contains weights of exponential growth. In particular, it is strictly larger than Ap. The second problem that we consider is the Kato square root problem for divergence form elliptic operators with potential V : R n → C. This is the equivalence statement (L + V ) 1 2 u L2(Rn) ' k∇ukL2(Rn) + V 1 2 u L2(Rn) , where L + V := −div (A∇) + V and the perturbation A is an L ∞ complex matrix-valued function satisfying an ellipticity condition. One possible path to a solution for this problem is by proving square function estimates for perturbations of associated non-homogeneous Dirac-type operators. At present, there is no general method to obtain such square function estimates other than for potentials bounded both from above and below (cf. [10]). We develop such a method by adapting the homogeneous framework introduced by A. Axelsson, S. Keith and A. McIntosh in their seminal paper [11]. Two distinct approaches will be considered when adapting this framework. The second such approach will yield a satisfying solution to the potential dependent Kato problem for a large class of potentials. This class will include any potential V with range contained in some sector of angle ωV ∈ [0, π 2 ) and for which |V | belongs either to the reverse H¨older class RH2 in any dimension or L n 2 (R n ) for n > 4.