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The Helgason Fourier transform on a class of nonsymmetric harmonic spaces. (English) Zbl 0894.43003
This paper is about harmonic analysis on a family of solvable Lie groups of the form $$AN$$, where $$A$$ is $${\mathbb{R}}^+$$, acting by dilations on a generalised Heisenberg-type group $$N$$. This $$AN$$ group includes the Iwasawa $$AN$$ components of split rank on real semisimple Lie groups. A “Fourier transformation” for these Iwasawa groups defined and studied by S. Helgason is extended to the whole family. Since less structure is available in the general case, some new ideas are needed to generalise many of Helgason’s theorems; a number of these are provided in this paper.
Reviewer: M.Cowling (Sydney)

##### MSC:
 43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc. 43A80 Analysis on other specific Lie groups 22E25 Nilpotent and solvable Lie groups
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