Harnack inequality for non-local Schr\"odinger operators

Let $x \in \mathbb{R}^d$, $d \geq 3,$ and $f: \mathbb{R}^d \rightarrow \mathbb{R}$ be a twice differentiable function with all second partial derivatives being continuous. For $1\leq i,j \leq d$, let $a_{ij} : \mathbb{R}^d \rightarrow \mathbb{R}$ be a differentiable function with all partial derivatives being continuous and bounded. We shall consider the Schr\"odinger operator associated to \begin{eqnarray*} \mathcal{L}f(x) &=& \frac12 \sum_{i=1}^d \sum_{j=1}^d \frac{\partial}{\partial x_i} \left(a_{ij}(\cdot) \frac{\partial f}{\partial x_j}\right)(x) + \int_{\mathbb{R}^d\setminus{\{0\}}} [f(y) - f(x) ]J(x,y)dy. \end{eqnarray*} where $J: \mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{R}$ is a symmetric measurable function. Let $q: \mathbb{R}^d \rightarrow \mathbb{R}.$ We specify assumptions on $a,q,$ and $J$ so that non-negative bounded solutions to $${\mathcal L}f + qf = 0$$ satisfy a Harnack inequality. As tools we also prove a Carleson estimate, a Uniform Boundary Harnack Principle and a 3G inequality for solutions to ${\mathcal L}f = 0.$