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

Abstract

Let

$$Lf\left(x\right)=-\frac{1}{\omega \left(x\right)}{\sum }_{i,j}{\partial }_i\left(a_{ij}\right(\cdot \left){\partial }_{j}f\right)\left(x\right)+V\left(x\right)f\left(x\right)$$ be the degenerate Schrödinger operator, where $\omega$ is a weight from the Muckenhoupt class $A_2$ and $V$ is a nonnegative potential that belongs to a certain reverse Hölder class with respect to the measure $\omega \left(x\right)dx$. Based on some smoothness estimates of the Poisson semigroup $e^{-t\sqrt{L}}$, we introduce the area function $S_P^L$ associated with $e^{-t\sqrt{L}}$ to characterize the Hardy space associated with $L$.