PDE Seminar Abstracts

Hardy spaces and Schrödinger operators

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


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) :={f L1(d): sup t>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 V 0 on d an atom is a function a for which there exist a ball B d 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 .