Poisson semigroup, area function, and the characterization of Hardy space associated to degenerate Schrödinger operators

Abstract

Let Lf(x)=−1ω(x)∑i,j∂i(aij(⋅)∂jf)(x)+V(x)f(x) be the degenerate Schrodinger operator, where ω is a weight from the Muckenhoupt class A2 and V is a nonnegative potential that belongs to a certain reverse Holder class with respect to the measure ω(x)dx. Based on some smoothness estimates of the Poisson semigroup e−tL, we introduce the area function SPL associated with e−tL to characterize the Hardy space associated with L.