  Established in the 30's, Schauder {\it a priori} estimates are among the most
classical and powerful tools in the analysis of problems ruled by 2nd order
elliptic PDEs. Since then, a central problem in regularity theory has been to
understand Schauder type estimates fashioning particular borderline scenarios.
In such context, it has been a common accepted aphorism that the continuity of
the Hessian of a solution could never be superior than the continuity of the
medium. Notwithstanding, in this article we show that solutions to uniformly
elliptic, linear equations with $C^{0,\bar{\epsilon}}$ coefficients are of
class $C^{2,\alpha}$, for any $0 < \bar{\epsilon} \ll \alpha < 1$, at Hessian
degenerate points, $\mathscr{H}(u):=\{X \suchthat D^2u(X) = 0\}$. In fact we
develop a more general regularity result at such Hessian degenerate points,
featuring into the theory of fully nonlinear equations. Insofar as the optimal
modulus of continuity for the Hessian is concerned, the result of this paper is
the first one in the literature to surpass the inborn obstruction from the
sharp Schauder {\it a priori} regularity theory.
