  We study fine boundary regularity properties of solutions to fully nonlinear
elliptic integro-differential equations of order $2s$, with $s\in(0,1)$.
  We consider the class of nonlocal operators $\mathcal L_*\subset \mathcal
L_0$, which consists of infinitesimal generators of stable L\'evy processes
belonging to the class $\mathcal L_0$ of Caffarelli-Silvestre. For fully
nonlinear operators $I$ elliptic with respect to $\mathcal L_*$, we prove that
solutions to $I u=f$ in $\Omega$, $u=0$ in $\mathbb R^n\setminus\Omega$,
satisfy $u/d^s\in C^{s+\gamma}(\overline\Omega)$, where $d$ is the distance to
$\partial\Omega$ and $f\in C^\gamma$.
  We expect the class $\mathcal L_*$ to be the largest scale invariant subclass
of $\mathcal L_0$ for which this result is true. In this direction, we show
that the class $\mathcal L_0$ is too large for all solutions to behave like
$d^s$.
  The constants in all the estimates in this paper remain bounded as the order
of the equation approaches 2. Thus, in the limit $s\uparrow1$ we recover the
celebrated boundary regularity result due to Krylov for fully nonlinear
elliptic equations.

Analysis of PDEs