Harnack inequality for non-local Schrödinger operators

Abstract

Let $x\in {\mathbb{R}}^d$, $d\ge 3,$ and $f:{\mathbb{R}}^d\to \mathbb{R}$ be a twice differentiable function with all second partial derivatives being continuous. For $1\le i,j\le d$, let $a_{ij}:{\mathbb{R}}^d\to \mathbb{R}$ be a differentiable function with all partial derivatives being continuous and bounded. We shall consider the Schrödinger 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 $J:{\mathbb{R}}^d\times {\mathbb{R}}^d\to \mathbb{R}$ is a symmetric measurable function. Let $q:{\mathbb{R}}^d\to \mathbb{R}.$ 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 $\mathcal{L}f=0.$

38 pages, added proof of BHP. Note that in Section 3, after necessary modifications the proofs of the results follow from the results in arXiv:0908.1559. We have reproduced them here (verbatim) for completeness