Ali Baklouti

The Lp –Lq version of Hardy's Theorem on nilpotent Lie groups

Abstract Let p, q be such that 2≤ p, q ≤ + ∞. We prove in this paper the Lp – Lq version of Hardys Theorem for an arbitrary nilpotent Lie group G extending then earlier cases and the classical Hardy theorem proved recently by E. Kaniuth and A. Kumar. The case where 1 ≤ p, q ≤ + ∞ is studied for a restricted class of nilpotent Lie groups.

On the local rigidity of discontinuous groups for exponential solvable Lie groups

Abstract. Let H be an arbitrary closed connected subgroup of an exponential solvable Lie group. Then a deformation of a discontinuous subgroup Γ of G for the homogeneous space G/H may be locally rigid. When G is nilpotent, connected and simply connected, the question whether the Weil–Kobayashi local rigidity fails to hold is posed by Baklouti [Proc. Japan Acad. Ser. A Math. Sci. 87 (2011), no. 9, 173–177]. A positive answer is only provided for some very few cases by now. This note aims to positively answer this question for some new settings. Our study goes even farther to exponential groups. In this case, the local rigidity fails to hold if the automorphism group of the Lie algebra of the syndetic hull of Γ is not solvable. In addition, any deformation of an abelian discontinuous subgroup is shown to be continuously deformable outside the setting of the affine group .

WHEN IS THE DEFORMATION SPACE

Let G = H2n + 1 be the 2n + 1-dimensional Heisenberg group and H be a connected Lie subgroup of G. Given any discontinuous subgroup Γ ⊂ G for G/H, a precise union of open sets of the resulting deformation space of the natural action of Γ on G/H is derived since the paper of Kobayshi and Nasrin [Deformation of Properly discontinuous action of ℤk and ℝk+1, Internat. J. Math.17 (2006) 1175–1190]. We determine in this paper when exactly this space is endowed with a smooth manifold structure. Such a result is only known when the Clifford–Klein form Γ\G/H is compact and Γ is abelian. When Γ is not abelian or H meets the center of G, the parameter and deformation spaces are shown to be semi-algebraic and equipped with a smooth manifold structure. In the case where Γ is abelian and H does not meet the center of G, then splits into finitely many semi-algebraic smooth manifolds and fails to be a Hausdorff space whenever Γ is not maximal, but admits a manifold structure otherwise. In any case, it is shown that admits an open smooth manifold as its dense subset. Furthermore, a sufficient and necessary condition for the global stability of all these deformations to hold is established.

\mathscr{T}(\Gamma, H_{2n+1}, H)

Abstract So far, the uncertainty principles for solvable non-exponential Lie groups have been treated only in few cases. The first author and Kaniuth produced an analogue of Hardys theorem for a diamond Lie group, which is a semi-direct product of ℝd with the Heisenberg group ℍ 2d+1

A SMOOTH MANIFOLD?

{\mathbb {H}_{2d+1}}

Some uncertainty principles for diamond Lie groups

In this setting, we formulate and prove in this paper some other uncertainty principles (Miyachi, Cowling–Price and Lp-Lq Morgan). This allows us to provide a refined version of Hardys theorem and to study the sharpness problems.

Estimate of the Lp -Fourier transform norm for connected nilpotent Lie groups

Abstract Let G be a connected and simply connected nilpotent Lie group, and m the dimension of the generic coadjoint orbits of G. Then it is proved in [Baklouti, Smaoui, Ludwig, J. Funct. Anal. 199: 508–520, 2003] that the operator value Fourier transform norm satisfies , where and q being the conjugate of p. When the assumption on G to be simply connected is removed, we prove an analogue estimate of type , where H designates the maximal compact central subgroup of G and m the dimension of generic coadjoint orbits of the universal covering of G. The case of non-simply connected Heisenberg groups is treated as an example.

On the generalized moment separability theorem for type 1 solvable Lie groups

Abstract Let G be a type 1 connected and simply connected solvable Lie group. The generalized moment map for π in G ^ {\widehat{G}} , the unitary dual of G, sends smooth vectors of the representation space of π to 𝒰 ⁢ ( 𝔤 ) * {{\mathcal{U}(\mathfrak{g})}^{*}} , the dual vector space of 𝒰 ⁢ ( 𝔤 ) {\mathcal{U}(\mathfrak{g})} . The convex hull of the image of the generalized moment map for π is called its generalized moment set, denoted by J ⁢ ( π ) {J(\pi)} . We say that G ^ {\widehat{G}} is generalized moment separable when the generalized moment sets differ for any pair of distinct irreducible unitary representations. Our main result in this paper provides a second proof of the generalized moment separability theorem for G.

Variants of stability of discontinuous groups for Euclidean motion groups

Let G be a Lie group, H a closed subgroup of G and Γ a discontinuous group for the homogeneous space 𝒳 = G/H. Given a deformation parameter φ ∈Hom(Γ,G), the deformed subgroup φ(Γ) may fail to act properly discontinuously on 𝒳. To understand this phenomenon in the case when G stands for an Euclidean motion group On(ℝ) ⋉ ℝn, we compare the notion of stability for discontinuous groups (cf. [T. Kobayashi and S. Nasrin, Deformation of properly discontinuous action of ℤk on ℝk+1, Int. J. Math. 17 (2006) 1175–1193]) with its variants. We prove that the defined stability variants hold when Γ turns out to be a crystallographic subgroup of G.

A Stability Theorem for Non-Abelian Actions on Threadlike Homogeneous Spaces

Let \(G = \mathbb {G}_n^r\) be the \((n+1)\)-dimensional reduced threadlike Lie group, H an arbitrary closed subgroup of G and \(\Gamma \subset G\) a non-abelian discontinuous group for G / H. Unlike the setting where \(\Gamma \) is abelian, we show that the stability property holds on the related parameter space.