Hardy Spaces Associated with Generalized Degenerate Schrödinger Operators with Applications to Carleson Measure

Abstract

Let $L:=-\frac{1}{\omega }{\textrm{div}}(A\,\nabla \cdot )+\mu$ be the generalized degenerate Schrödinger operator in $L^2_\omega (\mathbb {R}^n)$ ($n\ge 3$) with suitable weight $\omega$ and nonnegative Radon measure $\mu$. In this article, the authors first introduce the Hardy spaces $H_{L,S_h}^p(\mathbb {R}^n,\omega )$ and $H_{L}^p(\mathbb {R}^n,\omega )$, respectively, via the Lusin area function and the maximal function associated with L, where $p\in (0,1]$, and then, show that, when $p\in (\frac{n}{n+\theta },1]$, $H_{L,S_h}^p(\mathbb {R}^n,\omega )=H_{L}^p(\mathbb {R}^n,\omega )$ with equivalent quasi-norms, where $\theta \in (0,1]$ is the critical index of Hölder continuity for the heat kernels $\{k_{t}\}_{t>0}$ generated by L. As an application, the authors further obtain that the BMO type space $\textrm{BMO}_L(\mathbb {R}^n,\omega )$ associated with L can be characterized by the Carleson measure. This result is also new even when $L:=-\frac{1}{\omega }\mathrm{{div}}(A\,\nabla \cdot )+V$, where $V\ge 0$ belongs to the reverse Hölder class with respect to the measure $\omega (x)\textrm{d}x$.