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We consider the non-local operator $\\mathcal{L}^{b}=\\Delta^{\\alpha/2}+\\mathcal{S}^{b}$, where $$\\mathcal{S}^{b}f(x):=\\lim_{\\varepsilon\\to 0}\\mathcal{A}(d,-\\beta)\\int_{|z|>\\varepsilon}\\left(f(x+z)-f(x)\\right)\\frac{b(x,z)}{|z|^{d+\\beta}}\\,dy.$$ Here $b(x,z)$ is a bounded measurable function on $\\mathbb{R}^{d}\\times\\mathbb{R}^{d}$ that is symmetric in $z$, and $\\mathcal{A}(d,-\\beta)$ is a normalizing constant so that when $b(x, z)\\equiv 1$, $\\mathcal{S}^{b}$ becomes the fractional Laplacian $\\Delta^{\\beta/2}:=-(-\\Delta)^{\\beta/2}$. 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