The Bernstein Technique for Integro-Differential Equations

We extend the classical Bernstein technique to setting of integro-differential operators. As a consequence, we provide first and one-sided second derivative estimates for solutions fractional equations, including some convex fully nonlinear equations order smaller than two—for which prove uniform as their approaches two. Our method is robust enough be applied Pucci-type extremal obstacle problems operators, although several results are new even in linear case. also raise intriguing open questions, one them concerning “pure” Laplacian, another being validity associated operators with general kernels.