Harnack Inequality for Non-Local Schrödinger Operators

Let x∈Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x \in \mathbb {R}^{d}$\end{document}, d ≥ 3, and f:Rd→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f: \mathbb {R}^{d} \rightarrow \mathbb {R}$\end{document} be a twice differentiable function with all second partial derivatives being continuous. For 1 ≤ i, j ≤ d, let aij:Rd→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a_{ij} : \mathbb {R}^{d} \rightarrow \mathbb {R}$\end{document} be a differentiable function with all partial derivatives being continuous and bounded. We shall consider the Schrödinger operator associated to 𝓛f(x)=12∑i=1d∑j=1d∂∂xiaij(·)∂f∂xj(x)+∫Rd\{0}[f(y)-f(x)]J(x,y)dy\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{L}f(x) = \frac12 \sum\limits_{i=1}^{d} \sum\limits_{j=1}^{d} \frac{\partial}{\partial x_{i}} \left( a_{ij}(\cdot) \frac{\partial f}{\partial x_{j}}\right)(x) + {\int}_{\mathbb{R}^{d}\setminus{\{0\}}} [f(y) - f(x) ]J(x,y)dy $$\end{document}where J:Rd×Rd→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$J: \mathbb {R}^{d} \times \mathbb {R}^{d} \rightarrow \mathbb {R}$\end{document} is a symmetric measurable function. Let q:Rd→R.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$q: \mathbb {R}^{d} \rightarrow \mathbb {R}.$\end{document} We specify assumptions on a, q, and J so that non-negative bounded solutions to 𝓛f+qf=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{L}f + qf = 0 $$\end{document}satisfy a Harnack inequality. As tools we also prove a Carleson estimate, a uniform Boundary Harnack Principle and a 3G inequality for solutions to 𝓛f=0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {L}f = 0.$\end{document}