Hypoellipticity on the Heisenberg group

Let P be a left-invariant differential operator on the Heisenberg group Hⁿ, P homogeneous with respect to the dilations on Hⁿ. We show that a necessary and sufficient condition for the hypoellipticity of P is that π(P) be an injective operator for every irreducible unitary representation π of Hⁿ (except the trivial representation). Furthermore, hypoellipticity is preserved if the homogeneous operator P is perturbed by terms of lower order of homogeneity. (Homogeneity means homogeneity with respect to dilations of Hⁿ.) It is also shown that if P is homogeneous, left-invariant and hypoelliptic on Hⁿ, then its formal adjoint is hypoelliptic.