Higher-order estimates for fully nonlinear difference equations. II

The purpose of this work is to establish a priori C-2,C-alpha estimates for mesh function solutions of nonlinear difference equations of positive type in fully nonlinear form on a uniform mesh, where the fully nonlinear finit e difference operator F-h is concave in the second-order variables. The est imate is an analogue of the corresponding estimate for solutions of concave fully nonlinear elliptic partial differential equations. We use the result s for the special case that the operator does not depend explicitly upon th e independent variables (the so-called frozen case) established in part I t o approach the general case of explicit dependence upon the independent var iables. We make our approach for the diagonal case via a discretization of the approach of Safonov for fully nonlinear elliptic partial differential e quations using the discrete linear theory of Kuo and Trudinger and an espec ially agreeable mesh function interpolant provided by Kunkle. We generalize to non-diagonal operators using an idea which, to the author's knowledge, is never. In this paper we establish the desired Holder estimate in the lar ge, that is, on the entire mesh n-plane. In a subsequent paper a truly inte rior estimate will be established in a mesh n-box.