On the continuity of solutions to degenerate elliptic equations

The local behavior of solutions to a degenerate elliptic equation where andw(x)ξ2 A(x)ξ,ξ v(x)ξ2 for weights w(x) 0 and v(x), has been studied by Chanillo and Wheeden. In Chanillo and Wheeden (1986) [7], they generalize the results of Fabes, Kenig, and Serapioni (1961) [8] relative to the case v(x)=Λw(x). We consider the case where and v(x)=K(x). The assumption that v A2, the Muckenhoupt class, is not sufficient as it was in the case v(x)=Λw(x) to obtain the continuity of local solutions. However, if v Gn, the Gehring class, and if Sv is the domain of the maximal function of v,Sv={x Ω:Mv(x)<∞}, then the restriction to Sv of the representative of any non-negative solution u is continuous.