Degenerate Sobolev spaces and regularity of subelliptic equations

We develop a notion of degenerate Sobolev spaces naturally associated with nonnegative quadratic forms that arise from a large class of linear subelliptic equations with rough coefficients. These Sobolev spaces allow us to make the widest possible definition of a weak solution that leads to local Hölder continuity of solutions, extending our results in an earlier work, where we studied regularity of classical weak solutions. In cases when the …