Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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
${\mathcal L}f = 0.$

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The categories of this paper are: plasm-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: econ
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The categories of this paper are: cs
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The categories of this paper are: solv-int
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The categories of this paper are: q-alg
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The categories of this paper are: physics
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The categories of this paper are: atom-ph
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The categories of this paper are: funct-an
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The categories of this paper are: alg-geom
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The categories of this paper are: q-bio
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The categories of this paper are: nucl-th
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The categories of this paper are: stat
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The categories of this paper are: nlin
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The categories of this paper are: patt-sol