La transformation de Fourier et son inverse sur le groupe des ax+b d'un corps local

Lp-Fourier transforms on nilpotent Lie groups and solvable Lie groups acting on Siegel domains

“…On the ax+b group, Eymard and Terp [7] and Russo [20] obtained similar results. The former is based on the Plancherel theorem of Duflo and Moore, but the latter is based on that of Kleppner and Lipsman [12].…”

“…But we will give an explicit realization of the LP -Fourier transform based on the Plancherel theorem of Duflo and Moore [5]. For each irreducible representation π corresponding to one of the generic coadjoint orbits, which are open, we modify the map φ -• π{φ) using the operator called the formal degree of π [5] [7] and Russo [20] developed their LP -Fourier analysis for the ax + b group (the group of all affine transformations of the real line), and we are generalizing their results to our G.…”

The Hausdorff-Young theorem for the matricial groups Gnm = ax + b

“…In our case, that is G = G nm , like Eymard-Terp's case [3] which helped us as a model, we are led to temper the Fourier transformation of L 1 (G) by the unbounded operators δ 1 q . G nm is a locally compact group on which we fix the left Haar measure to be d(b, a) = dbda | det(a)| n+m , where db (resp.…”

Sur le groupe affine d'un corps local

“…Par la suite, et sauf mention contraire, on désigne par G le groupe affine d'un corps local K [3]. Comme l'application f →f = F −1 (f ) est une isométrie d'espaces de Banach de A(G) sur L 1 (H), le Théorème 4 qui suit généralise le cas abélien (Théorème 1) au groupe affine qui n'est ni abélien ni unimodulaire.…”