Fourier transforms of smooth functions on certain nilpotent Lie groups

L. Corwin, F. P. Greenleaf

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The authors consider irreducible representations π ε{lunate} N ̂ of a nilpotent Lie group and define a Fourier transform for Schwartz class (and other) functions φ on N by forming the kernels Kφ(x, y) of the trace class operations πφ = ∝N φ(n)πn dn, regarding the π as modeled in L2(Rk) for all π in general position. For a special class of groups they show that the models, and parameters λ labeling the representations in general position, can be chosen so the joint behavior of the kernels Kφ(x, y, λ) can be interpreted in a useful way. The variables (x, y, λ) run through a Zariski open set in Rn, n = dim N. The authors show there is a polynomial map u = A(x, y, λ) that is a birational isomorphism A: Rn → Rn with the following properties. The Fourier transforms F1φ = Kφ(x, y, λ) all factor through A to give "rationalized" Fourier transforms Fφ(u) such that Fφ {ring operator} A = F1φ. On the rationalized parameter space a function f(u) is of the form Fφ = f ⇔ f is Schwartz class on Rn. If polynomial operators T ε{lunate} P(N) are transferred to operators T ̃ on Rn such that F(Tφ) = T ̃(Fφ), P(N) is transformed isomorphically to P(Rn).

Original languageEnglish (US)
Pages (from-to)203-217
Number of pages15
JournalJournal of Functional Analysis
Issue number2
StatePublished - Jun 15 1980

ASJC Scopus subject areas

  • Analysis


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