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In this paper we obtain interior regularity estimates for viscosity solutionsof nonlocal Dirichlet problems that degenerate when the gradient of thesolution vanishes. Interior H\"older estimates are obtained when the order ofthe fractional diffusion is less or equal than one, and Lipschitz estimateswhen it is bigger than one. In the latter case, the estimates are robust enoughto conclude interior $C^{1, \alpha}$ regularity by an improvement of theflatness procedure, which is possible when the nonlocal term is close enough toa second-order diffusion.

Interior regularity results for fractional elliptic equations that degenerate with the gradient