PDE Seminar Abstracts

Hardy spaces and Schrödinger operators

Marcin Preisner
Macquarie University, Australia
Mon 20th May 2019, 2-3pm, Carslaw Room 829 (AGR)

Abstract

On Rd we consider the Schrödinger operator

Lf(x)=-Δf(x)+V(x)f(x),

where Δ=x12++xd2 and V(x)0 is a positive function (“potential”).

Let Tt=exp(-tL) be the heat semigroup associated with to L. In the talk we shall consider the Hardy space

H1(L):={fL1(d):supt>0Ttf(x)L1(d)}

which is a natural substitute of L1(d) in harmonic analysis associated with L. Our main interest will be in showing that elements H1(L) have decompositions of the type f(x)=kλkak(x), where k|λk|< and ak (“atoms”) have some nice properties.

In the classical case V0 on d an atom is a function a for which there exist a ball Bd such that

supp(a)B,a|B|-1,a(x)dx=0.

We shall see that for L=-Δ+V we can still prove some atomic decompositions, but the properties of atoms depend on the dimension d and the potential V.