The behavior of Fourier transforms for nilpotent Lie groups

We study weak analogues of the Paley-Wiener Theorem for both the scalar-valued and the operator-valued Fourier transforms on a nilpotent Lie group . Such theorems should assert that the appropriate Fourier transform of a function or distribution of compact support on extends to be ``holomorphic' on an appropriate complexification of (a part of) . We prove the weak scalar-valued Paley-Wiener Theorem for some nilpotent Lie groups but show that it is false in general. We also prove a weak operator-valued Paley-Wiener Theorem for arbitrary nilpotent Lie groups, which in turn establishes the truth of a conjecture of Moss. Finally, we prove a conjecture about Dixmier-Douady invariants of continuous-trace subquotients of when is two-step nilpotent.

     

Radon transforms on Siegel-type nilpotent Lie groups

Let \mathfrak{N}:={{\displaystyle H}}_n\times {ℂ}^n be the Siegel-type nilpotent group, which can be identified as the Shilov boundary of Siegel domain of type II, where {{\displaystyle H}}_n denotes the set of all n\times n Hermitian matrices. In this article, we use singular convolution operators to define Radon transform on \mathfrak{N} and obtain the inversion formulas of Radon transforms. Moveover, we show that Radon transform on \mathfrak{N} is a unitary operator from Sobolev space W^n;2 into L²(\mathfrak{N}):     

Fourier transforms of functions in Herz spaces on certain groups

G— - {G _n } _n =– , G _n =G G _n ={0}. G. K(,p,q;G) K(,p,q;) - G , . , G - (. . sup {order (G _n /G _n +1):=0, ± 1, ...<), K(.,p,q;G) L ^{p/(p–p–1),q} () L ^{p/(p–p–1),q} () K(-,p,q; ), 1<p2, 0<<1/p=1–1/p, 0<q. . . . .     

Littlewood-Paley and Lusin functions on nilpotent Lie groups

In this paper, we define the Littlewood-Paley and Lusin functions associated to the sub-Laplacian operator on nilpotent Lie groups. Then we prove the Lp (1<p<∞) boundedness of Littlewood-Paley and Lusin functions.     

Fourier transforms and bent functions on finite groups

Let G be a finite nonabelian group. Bent functions on G are defined by the Fourier transforms at irreducible representations of G. We introduce a dual basis \({\widehat{G}}\), consisting of functions on G determined by its unitary irreducible representations, that will play a role similar to the dual group of a finite abelian group. Then we define the Fourier transforms as functions on \({\widehat{G}}\), and obtain characterizations of a bent function by its Fourier transforms (as functions on \({\widehat{G}}\)). For a function f from G to another finite group, we define a dual function \({\widetilde{f}}\) on \({\widehat{G}}\), and characterize the nonlinearity of f by its dual function \({\widetilde{f}}\). Some known results are direct consequences. Constructions of bent functions and perfect nonlinear functions are also presented.     

On theorems of Beurling and Cowling-Price for certain nilpotent Lie groups

Let G be a connected simply connected nilpotent Lie group. In [A. Baklouti, N. Ben Salah, The Lp−Lq version of Hardy's Theorem on nilpotent Lie groups, Forum Math. 18 (2006) 245-262], we proved for 2?p,q?+∞ the Lp−Lq version of Hardy's Theorem known as the Cowling-Price Theorem. In the setup where 1?p,q?+∞, the problem is still unsolved and the upshot is known only for few cases. We prove in this paper such a result in the context of 2-NPC nilpotent Lie groups. A proof of the analogue of Beurling's Theorem is also provided in the same context.     

Monogenic functions and representations of nilpotent Lie groups in quantum mechanics

We describe several different representations of nilpotent step two Lie groups in spaces of monogenic Clifford‐valued functions. We are inspired by the classic representation of the Heisenberg group in the Segal–Bargmann space of holomorphic functions. Connections with quantum mechanics are described. Copyright © 1999 John Wiley & Sons, Ltd.     

Anosov actions of nilpotent Lie groups

We study Anosov actions of nilpotent Lie groups on closed manifolds. Our main result is a generalization to the nilpotent case of a classical theorem by J.F. Plante in the 70's. More precisely, we prove that, for what we call a good Anosov action of a nilpotent Lie group on a closed manifold, if the non-wandering set is the entire manifold, then the closure of stable strong leaves coincide with the closure of the strong unstable leaves. This implies the existence of an equivariant fibration of the manifold onto a homogeneous space of the Lie group, having as fibers the closures of the leaves of the strong foliation.     

Norm inequalities for certain classes of functions and their Fourier transforms

We apply tools of interpolation theory and a commutative property of the Hilbert transform to prove necessary and sufficient conditions related to trigonometric series. These results extend and improve related theorems proven by several authors, summarized by Boas. In addition, we explore inequalities and operators, both connected to Hardy's inequalities, on certain classes of functions, including quasimonotone functions.     

Harmonic functions on nilpotent groups

For a probability measure on a locally compact groupG which is not supported on any proper closed subgroup, an elementF ofL (G) is called -harmonic if F(st)d(t)=F(s), for almost alls inG. Constant functions are -harmonic and it is known that for abelianG all -harmonic functions are constant. For other groups it is known that non constant -harmonic functions exist and the question of whether such functions exist on nilpotent groups is open, though a number of partial results are known. We show that for nilpotent groups of class 2 there are no non constant -harmonic functions. Our methods also enable us to give new proofs of results similar to the known partial results.     

Holomorphic representations of nilpotent Lie groups

Let G be a connected, simply-connected complex nilpotent Lie group, and G_r ?( G a real form of G. Motivated by the problem of analytic continuation of Banach-space representations of G_R to holomorphic representations of G, we construct translation-invariant locally-convex algebras of entire functions on G (generalizing the classical spaces of entire functions of finite exponential order). The dual spaces of these algebras are naturally identified with algebras of left-invariant differential operators of infinite order on G. In connection with analytic continuation of unitary representations of G_R, we study the convex cone of entire functions on G whose restrictions to G_R are positive-definite, and determine the minimal order of growth at infinity of such functions.     

Fourier transforms and bent functions on faithful actions of finite abelian groups

Let G be a finite abelian group acting faithfully on a finite set X. The G-bentness and G-perfect nonlinearity of functions on X are studied by Poinsot and co-authors (Discret Appl Math 157:1848–1857, 2009; GESTS Int Trans Comput Sci Eng 12:1–14, 2005) via Fourier transforms of functions on G. In this paper we introduce the so-called \(G\)-dual set \(\widehat{X}\) of X, which plays the role similar to the dual group \(\widehat{G}\) of G, and develop a Fourier analysis on X, a generalization of the Fourier analysis on the group G. Then we characterize the bentness and perfect nonlinearity of functions on X by their own Fourier transforms on \(\widehat{X}\). Furthermore, we prove that the bentness of a function on X can be determined by its distance from the set of G-linear functions. As direct consequences, many known results in Logachev et al. (Discret Math Appl 7:547–564, 1997), Carlet and Ding (J Complex 20:205–244, 2004), Poinsot (2009), Poinsot et al. (2005) and some new results about bent functions on G are obtained. In order to explain the theory developed in this paper clearly, examples are also presented.