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