Global $W^{2,\delta}$ estimates for singular fully nonlinear elliptic equations with $L^n$ right hand side terms

We establish in this paper \emph{a priori} global $W^{2,\delta}$ estimates for singular fully nonlinear elliptic equations with $L^n$ right hand side terms. The method is to slide paraboloids and barrier functions vertically to touch the solution of the equation, and then to estimate the measure of the contact set in terms of the measure of the vertex point set. To derive global estimates from $L^n$ data, the Hardy-Littlewood maximal functions, appropriate localizations and a new type of covering argument are adopted. These methods also provide us a more direct proof of the $W^{2,\delta}$ estimates for (nonsingular) fully nonlinear elliptic equations established by L. A. Caffarelli and X. Cabr\'{e}.