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Transformée de Fourier adaptée et convoluteurs de Schwartz sur les groupes de Lie nilpotents

Abstract : The adapted Fourier transform, so-called nilpotent Fourier transform, was first introduced by D. Arnal and J.C. Cortet as a generalisation of the usual abelian Fourier transform. This definition was limited at the orbits of the group under the coadjont action. We define in this thesis new adapted Fourier transforms on the dual of the Lie algebra and the product of this dual space with the set of all Malcev bases. Then, we study Schwartz multipliers for nilpotent Lie groups and we give an idea to prove Howe conjecture that characterizes the bi-invariant Schwartz multipliers on nilpotent Lie groups. Such characterization is given as the following : a tempered distribution on a nilpotent group Lie is a bi-invariant Schwartz multiplier if and only if its Fourier transform as a distribution is a smooth, Ad*-invariant function on the dual of the Lie algebra and all of its derivatives have polynomial bounds. Finally, we define Schwartz multipliers for variable nilpotent Lie groups and we characterize them as a bove
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Semi Dhieb. Transformée de Fourier adaptée et convoluteurs de Schwartz sur les groupes de Lie nilpotents. Mathématiques générales [math.GM]. Université Paul Verlaine - Metz, 1995. Français. ⟨NNT : 1995METZ010S⟩. ⟨tel-01776898⟩

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