We consider a class of fully nonlinear integro-differential operators where the nonlocal integral has two components: the non-degenerate one corresponds to the $alpha$-stable operator and the second one (possibly degenerate) corresponds to a class of textit{lower order} Levy measures. Such operators do not have a global scaling property. We establish H{o}lder regularity, Harnack inequality and boundary Harnack property of solutions of these operators.
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