Polynomial decay in $W^{2,\epsilon }$ estimates for viscosity supersolutions of fully nonlinear elliptic equations

We prove $W^{2,\epsilon }$ estimates for viscosity supersolutions of fully nonlinear, uniformly elliptic equations where $\epsilon$ decays polynomially with respect to the ellipticity ratio of the equations. Our result is related to a conjecture of Armstrong-Silvestre-Smart [Comm. Pure Appl. Math. 65 (2012), no. 8, 1169–1184] which predicts a linear decay for $\epsilon$ with respect to the ellipticity ratio of the equations.