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Fourier transform of Schwartz functions on the Heisenberg group. (English) Zbl 1277.43012
The authors consider the Schwartz space \(\mathcal S(H_1)\) on the (three-dimensional) Heisenberg group \(H_1\). They characterise \(\mathcal S (H_1)\) via sequences of Schwartz functions defined on \(\mathbb R^2\). The definition of these sequences relies on the decomposition given by the representation theory of the group \(H_1\) (Plancherel formula) as well as certain special features of the action of the torus \(\mathbb T_1\) on functions of \(H_1\). This generalises the results in [J. Funct. Anal. 251, No. 2, 772–791 (2007; Zbl 1128.43009); ibid. 256, No. 5, 1565–1587 (2009; Zbl 1167.43008)] by the same authors in the case of \(\mathbb T_1\)-invariant functions in \(\mathcal S(H_1)\). The proofs are based on the results in [loc. cit.] as well as Whitney’s extension properties and a deep understanding of the Gelfand spectrum in this context.

43A80 Analysis on other specific Lie groups
22E25 Nilpotent and solvable Lie groups
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