A Class of Nonlocal Hypoelliptic Operators and their Extensions

In this paper we study nonlocal equations driven by the fractional powers of hypoelliptic operators in the form Ku = Au -partial_t u = tr(Q nabla^2 u) + < BX, nabla u > - partial_t u, introduced by Hormander in his 1967 hypoellipticity paper. We show that the nonlocal operators (-K)^s , (-A)^s can be realized as the Dirichlet-to-Neumann map of doubly-degenerate extension problems. We solve such problems in L^infty, in L^p for 1 <= p < infty when tr(B) >= 0. In forthcoming works we use such calculus to establish some new Sobolev and isoperimetric inequalities.