Abstract
One of the big contributions of E. M. Stein is the development of harmonic analysis on the Heisenberg group. In a fundamental joint paper with G. B. Folland, Stein laid all the groundwork for this study. In this chapter we reproduce and develop some of that groundwork. We contrast classical analysis on Euclidean space with the new analysis on the Heisenberg group.
In particular we treat the noncommutativity of the Heisenberg group. We develop the idea of homogeneous dimension. We prove results about fractional integrals. And we learn about commutators of vector fields.
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Notes
- 1.
Suffice it to say that a group is nilpotent of step m if all mth order commutators in the group equal the identity—and if m is the least such integer. In particular, an abelian group is nilpotent of step 1.
- 2.
It must be noted, however, that the rotations on the disc and the translations on the upper half plane do not “correspond” in any natural way; certainly the Cayley transform does not map the one group to the other. This anomaly is explored in the fine text [HOF, Ch. 3, 8].
- 3.
We use here the classical terminology of Siegel upper half spaces. Such an upper half space is defined with an inequality using a quadratic form. The resulting space is unbounded. However, when the quadratic form is positive definite then the domain has a bounded realization—that is to say, it is biholomorphically equivalent to a bounded domain. See [KAN, Ch. 1, 7] for details of this theory.
- 4.
In some sense it is more natural to consider commutators in the Lie algebra of the group. By way of the exponential map and the Campbell–Baker–Hausdorff formula (see [SER, Ch. 4]), the two different points of view are equivalent. We shall describe some of the Lie algebra approach in the material below. For now, the definition of commutators in the context of the group is a quick-and-dirty way to get at the idea we need to develop right now.
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Krantz, S.G. (2017). The Heisenberg Group. In: Harmonic and Complex Analysis in Several Variables. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-63231-5_3
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