  We obtain a new general extension theorem in Banach spaces for operators
which are not required to be symmetric, and apply it to obtain Harnack
estimates and a priori regularity for solutions of fractional powers of several
second order differential operators. These include weighted elliptic and
subellitptic operators in divergence form (nonnecessarily self-adjoint), and
nondivergence form operators with rough coefficients. We utilize the reflection
extension technique introduced by Caffarelli and Silvestre.
