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 {H_n} ; it is the precise analogue for H n {H_n} of the statement for R n {{\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 {H_n} , whereas only the generic, infinite-dimensional representations occur in the Plancherel formula. A simple class of examples is discussed, namely P = Σ i = 1 n X i 2 m + Y i 2 m P = \Sigma _{i = 1}^nX_i^{2m} + Y_i^{2m} , where X i , Y i , i = 1 , … , n {X_i},{Y_i},i = 1, \ldots ,n , and Z generate the Lie algebra of H n {H_n} via the commutation relations [ X i , Y j ] = δ i j Z [{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 {H_n} and a class of degenerate-elliptic operators on R n + 1 {{\mathbf {R}}^{n + 1}} studied by Grušin. This connection is examined in the context of localization in enveloping algebras.