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Description

Let $J$ be the Lévy density of a symmetric Lévy process in $\bf{R}^d$ with its Lévy exponent satisfying a weak lower scaling condition at infinity. Consider the non-symmetric and non-local operator \[ {\cal L}^{\kappa}f(x):= \lim_{\epsilon \downarrow 0} \int_{\{z \in \bf{R}^d: |z|>\epsilon\}}(f(x+z)-f(z))\kappa(x,z)J(z)\, dz\, , \] where $\kappa(x,z)$ is a Borel measurable function on $\bf{R}^d\times \bf{R}^d$ satisfying $0