Hypoellipticity on the Heisenberg Group-Representation-Theoretic Criteria

Abstract

A representation-theoretic characterization is given for hypoellipticity of homogeneous (with respect to dilations) left-invariant differential operators P on the Heisenberg group ${H_n}$; it is the precise analogue for ${H_n}$ of the statement for ${{\mathbf {R}}^n}$ that a homogeneous constant-coefficient differential operator is hypoelliptic if and only if it is elliptic. Under these representation-theoretic conditions a parametrix is constructed for P by means of the Plancherel formula. However, these conditions involve all the irreducible representations of ${H_n}$, whereas only the generic, infinite-dimensional representations occur in the Plancherel formula. A simple class of examples is discussed, namely $P = \Sigma _{i = 1}^nX_i^{2m} + Y_i^{2m}$, where ${X_i},{Y_i},i = 1, \ldots ,n$, and Z generate the Lie algebra of ${H_n}$ via the commutation relations $[{X_i},{Y_j}] = {\delta _{ij}}Z$, and where m is a positive integer. In the course of the proof a connection is made between homogeneous left-invariant operators on ${H_n}$ and a class of degenerate-elliptic operators on ${{\mathbf {R}}^{n + 1}}$ studied by Grušin. This connection is examined in the context of localization in enveloping algebras.