The Heisenberg Group and Its Relatives in the Work of Elias M. Stein

Abstract

We survey the work of Elias M. Stein in the field of analysis on the Heisenberg group and other nilpotent Lie groups, together with its applications to complex analysis in several variables and partial differential equations.

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Notes

  1. 1.

    In the literature of sub-Riemannian geometry, stratified groups are commonly called Carnot groups.

  2. 2.

    In complex geometry, boundaries of domains are denoted by b rather than \(\partial \) to avoid confusion with the holomorphic exterior derivative.

  3. 3.

    More precisely, the mean-zero condition means that K acts on test functions by a principal-value integral, \(\langle K,\phi \rangle =\text {p.v.}\int K\phi \), which satisfies \(\langle K,\phi \circ \delta _r\rangle =\langle K,\phi \rangle \) for all \(r>0\).

  4. 4.

    The second term in (9) is absent when \(q=0\).

  5. 5.

    The inclusion of the minus sign is a matter of taste. It has the advantage of making \({{\mathscr {L}}}\) a positive operator in the sense of Hilbert space theory.

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Correspondence to Gerald B. Folland.

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Folland, G.B. The Heisenberg Group and Its Relatives in the Work of Elias M. Stein. J Geom Anal (2019). https://doi.org/10.1007/s12220-019-00309-w

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Keywords

  • Harmonic analysis
  • Heisenberg group
  • Homogeneous group
  • Nilpotent Lie group

Mathematics Subject Classification

  • Primary 43A80
  • Secondary 32V20
  • 35B65
  • 42B20
  • 42B37