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Transactions of the American Mathematical Society

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Hypoellipticity on the Heisenberg group-representation-theoretic criteria

Author: Charles Rockland
Journal: Trans. Amer. Math. Soc. 240 (1978), 1-52
MSC: Primary 22E30; Secondary 35H05, 58G05
MathSciNet review: 0486314
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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.

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Keywords: Heisenberg group, nilpotent Lie group, unitary representations, Plancherel formula, dilations, homogeneous left-invariant differential operator, hypoellipticity, parametrix
Article copyright: © Copyright 1978 American Mathematical Society