Quadratic expansions and partial regularity for fully nonlinear uniformly parabolic equations

For a parabolic equation associated to a uniformly elliptic operator, we obtain a W 3,ε estimate, which provides a lower bound on the Lebesgue measure of the set on which a viscosity solution has a quadratic expansion. The argument combines parabolic W 2,ε estimates with a comparison principle argument. As an application, we show, assuming the operator is C1, that a viscosity solution is C2,α on the complement of a closed set of Hausdorff dimension ε less than that of the ambient space, where the constant ε > 0 depends only on the dimension and the ellipticity.