Weighted regularity estimates in Orlicz spaces for fully nonlinear elliptic equations

In this paper we develop a global W2,p estimate for the viscosity solution of the Dirichlet problem of fully nonlinear elliptic equations F(D2u,Du,u,x)=f(x) in Ω,u=0 on ∂Ωto a more general function space. Given an N-functionΦ and a Muckenhoupt weight w, we prove that if f belongs to the associated weighted Orlicz space LwΦ(Ω), then D2u∈LwΦ(Ω) and u satisfies a global Ww2,Φ estimate, under a minimal regularity requirement on F in the variable x and a basic geometric assumption on ∂Ω. The correct condition on the couple, Φ and w, is also addressed. This result generalizes the W2,p estimate (Caffarelli, 1989, Escauriaza, 1993, Winter, 2009) of Calderón and Zygmund as well as an analogous one (Byun et al., 2016) in the weighted Lp setting.