Before revision:  With the aid of weighted Sobolev spaces, we prove supremum bounds, a Harnack inequality, and Holder continuity near the boundary for solutions to elliptic variational equations defined by the Heston partial differential operator, as well as Holder continuity up to the boundary for solutions to elliptic variational inequalities defined by the  Heston operator.
After revision:  With the aid of weighted Sobolev spaces, we prove supremum bounds, a Harnack inequality, and Holder continuity near the boundary for solutions to elliptic variational equations defined by the elliptic Heston operator, as well as Holder continuity up to the boundary for solutions to elliptic variational inequalities defined by the Heston operator.