  We consider the fully nonlinear equation with variable-exponent double phase
type degeneracies $$ \big[|Du|^{p(x)}+a(x)|Du|^{q(x)}\big]F(D^2u)=f(x). $$
Under some appropriate assumptions, by making use of geometric tangential
methods and combing a refined improvement-of-flatness approach with compactness
and scaling techniques we obtain the sharp local $C^{1,\alpha}$ regularity of
viscosity solutions to such equations.
