Area Integral Characterization of Hardy space H1L related to Degenerate Schrödinger Operators

Let $$\begin{array}{} \displaystyle Lf(x)=-\frac{1}{\omega(x)}\sum_{i,j}^{}\partial_{i}(a_{ij}(\cdot)\partial_{j}f)(x)+V(x)f(x) \end{array}$$ be the degenerate Schrödinger operator, where ω is a weight from the Muckenhoupt class A2, V is a nonnegative potential that belongs to a certain reverse Hölder class with respect to the measure ω(x)dx. For such an operator we define the area integral $\begin{array}{} \displaystyle S^{L}_h \end{array}$ associated with the heat semigroup and obtain the area integral characterization of $\begin{array}{} \displaystyle H^{1}_{L} \end{array}$, which is the Hardy space associated with L.