Fourier algebra of a compact Lie group

Optimal extension of the Fourier transform and convolution operator on compact groups

Let $G$ be a compact group (not necessarily abelian) and let $Φ$ be a Young function satisfying the $Δ_2$-condition. We determine the optimal domain and the associated extended operator for both Fourier transform and the convolution operator defined on the Orlicz spaces $L^Φ(G).$

Fourier-Stieltjes transform defined by induced representation on locally compact groups

In this work we extend the Fourier-Stieltjes transform of a vector measure and a continuous function defined on compact groups to locally compact groups. To do so, we consider a representation L of a normal compact subgroup K of a locally compact group G, and we use a representation of G induced by that of L. Then, we define the Fourier-Stieltjes transform of a vector measure and that of a continuous function with compact support defined on G from the representation of G. Then, we extend the Shur orthogonality relation established for compact groups to locally compact groups by using the repre…

Fourier transform of function on locally compact Abelian groups taking value in Banach spaces

We consider Fourier transform of vector-valued functions on a locally compact group $G$, which take value in a Banach space $X$, and are square-integrable in Bochner sense. If $G$ is a finite group then Fourier transform is a bounded operator. If $G$ is an infinite group then Fourier transform $F: L_2(G,X)\to L_2(\widehat G,X)$ is a bounded operator if and only if Banach space $X$ is isomorphic to a Hilbert one.

Fourier multipliers on graded Lie groups

In this paper we study multipliers on graded nilpotent Lie groups defined via group Fourier transform. More precisely, we show that Hörmander type conditions on the Fourier multipliers imply $L^p$-boundedness. We express these conditions using difference operators and positive Rockland operators. We also obtain a more refined condition using Sobolev spaces on the dual of the group which are defined and studied in this paper.

Fourier analysis associated to a vector measure on a compact group

In this paper, we introduce and study the Fourier transform of functions which are integrable with respect to a vector measure on a compact group (not necessarily abelian). We also study the Fourier transform of vector measures. We also introduce and study the convolution of functions from $L^p$-spaces associated to a vector measure. We prove some analogues of the classical Young's inequalities. Similarly, we also study convolution of a scalar measure and a vector measure.

Characterisations of Fourier and Fourier--Stieltjes algebras on locally compact groups

Motivated by the beautiful work of M. A. Rieffel (1965) and of M. E. Walter (1974), we obtain characterisations of the Fourier algebra $A(G)$ of a locally compact group $G$ in terms of the class of $F$-algebras (i.e. a Banach algebra $A$ such that its dual $A'$ is a $W^*$-algebra whose identity is multiplicative on $A$). For example, we show that the Fourier algebras are precisely those commutative semisimple $F$-algebras that are Tauberian, contain a nonzero real element, and possess a dual semigroup that acts transitively on their spectrums. Our characterisations fall into three flavours, wh…

Beurling-Fourier algebras on Lie groups and their spectra

We investigate Beurling-Fourier algebras, a weighted version of Fourier algebras, on various Lie groups focusing on their spectral analysis. We will introduce a refined general definition of weights on the dual of locally compact groups and their associated Beurling-Fourier algebras. Constructions of nontrivial weights will be presented focusing on the cases of representative examples of Lie groups, namely $SU(n)$, the Heisenberg group $\mathbb{H}$, the reduced Heisenberg group $\mathbb{H}_r$, the Euclidean motion group $E(2)$ and its simply connected cover $\widetilde{E}(2)$. We will determin…

紧群与 Assiamoua 空间上的向量 Fourier 分析

Vector Fourier analysis on compact groups and Assiamoua spaces

研究阐明 Assiamoua 空间在紧群向量值函数 Fourier 分析中的基础作用，并说明其支撑经典结果迁移及向量值 Sobolev 空间构造。

Fourier transforms of spherical distributions on compact symmetric spaces

In our previous articles "A local Paley-Wiener theorem for compact symmetric spaces", Adv. Math. 218 (2008), 202--215, and "Fourier series on compact symmetric spaces" (submitted) we studied Fourier series on a compact symmetric space M=U/K. In particular, we proved a Paley-Wiener type theorem for the smooth functions on M, which have sufficiently small support and are K-invariant, respectively K-finite. In this article we extend those results to K-invariant distributions on M. We show that the Fourier transform of a distribution, which is supported in a sufficiently small ball around the base…

Beurling-Fourier algebras on compact groups: spectral theory

For a compact group $G$ we define the Beurling-Fourier algebra $A_ω(G)$ on $G$ for weights $ω$ defined on the dual $\what G$ and taking positive values. The classical Fourier algebra corresponds to the case $ω$ is the constant weight 1. We study the Gelfand spectrum of the algebra realizing it as a subset of the complexification $G_{\mathbb C}$ defined by McKennon and Cartwright and McMullen. In many cases, such as for polynomial weights, the spectrum is simply $G$. We discuss the questions when the algebra $A_ω(G)$ is symmetric and regular. We also obtain various results concerning spectral s…

局部紧阿贝尔群上的Brascamp--Lieb不等式

Brascamp--Lieb inequalities on locally compact abelian groups

本文在局部紧阿贝尔群上建立Brascamp--Lieb常数的结构定理，统一并推广了欧氏空间与有限生成群情形的有限性刻画，并突出其Fourier不变性。

紧群的多维 Fourier 代数与 Varopolous 代数中的谱综合

Spectral Synthesis in the Multidimensional Fourier Algebra and the Varopolous Algebra for Compact Groups

本文定义了紧群上的多维 Varopolous 代数，证明了多维 Fourier 代数可嵌入其中，并利用该嵌入证明了涵盖前人成果的并行综合结果。

重访分次李群上的振荡奇异积分算子

Oscillating singular integral operators on graded Lie groups revisited

本文将 Fefferman–Stein 的振荡奇异积分理论推广至任意分次李群，借助 Rockland 算子的无穷小表示刻画核及其群 Fourier 变换的判据。

Weak amenability for Fourier algebras of 1-connected nilpotent Lie groups

A special case of a conjecture raised by Forrest and Runde (Math. Zeit., 2005) asserts that the Fourier algebra of every non-abelian connected Lie group fails to be weakly amenable; this was aleady known to hold in the non-abelian compact cases, by earlier work of Johnson (JLMS, 1994) and Plymen (unpublished note). In recent work (JFA, 2014) the present authors verified this conjecture for the real ax+b group and hence, by structure theory, for any semisimple Lie group. In this paper we verify the conjecture for all 1-connected, non-abelian nilpotent Lie groups, by reducing the problem to the …

Weak amenability of Fourier algebras on compact groups

We give for a compact group G, a full characterisation of when its Fourier algebra A(G) is weakly amenable: when the connected component of the identity G_e is abelian. This condition is also equivalent to the hyper-Tauberian property for A(G), and to having the anti-diagonal D^v={(s,s^{-1}):s is in G} being a set of spectral synthesis for A(GXG). We show the relationship between amenability and weak amenability of A(G), and (operator) amenability and (operator) weak amenability of A_D(G), an algebra defined by the authors in arXiv:0705.4277. We close by extending our results to some classes o…

Fourier multipliers for Triebel-Lizorkin spaces on compact Lie groups

We investigate the boundedness of Fourier multipliers on a compact Lie group when acting on Triebel-Lizorkin spaces. Criteria are given in terms of the Hörmander-Mihlin-Marcinkiewicz condition. In our analysis, we use the difference structure of the unitary dual of a compact Lie group. Our results cover the sharp Hörmander-Mihlin theorem on Lebesgue spaces and also other historical results on the subject.

Mapping properties of Fourier transforms

The paper deals with continuous and compact mappings generated by the Fourier transform between distinguished function spaces on $\mathbb{R}^n$. The degree of compactness will be measured in terms of related entropy numbers. We are more interested in the interplay of already available ingredients than in generality.

Fourier transform acting on the functions defined in the infinite LCA groups. A Grand Lebesgue Spaces approach

We derive in this article the exact norm in the Grand Lebesgue Spaces (GLS) estimates for Fourier transform acting on the functions defined in the infinite local compact Abelian (LCA) group, compact or discrete.

Amenability properties of the central Fourier algebra of a compact group

We let the central Fourier algebra, ZA(G), be the subalgebra of functions u in the Fourier algebra A(G) of a compact group, for which u(xyx^{-1})=u(y) for all x,y in G. We show that this algebra admits bounded point derivations whenever G contains a non-abelian closed connected subgroup. Conversely when G is virtually abelian, then ZA(G) is amenable. Furthermore, for virtually abelian G, we establish which closed ideals admit bounded approximate identities. We also show that if ZA(G) is weakly amenable, even hyper-Tauberian, exactly when G admits no non-abelian connected subgroup. We also stud…

紧半单李群的 $q$-形变及其复化的 Beurling-Fourier 代数

Beurling-Fourier algebras of $ q $-deformations of compact semisimple Lie groups and complexification

本文研究紧半单李群 $q$-形变的 Beurling-Fourier 代数，证明其 Drinfeld double 函数代数的不可约表示可由特定中心权族对应的加权 Fourier 代数不可约表示穷尽。

Uncertainty Principles of Ingham and Paley-Wiener on Semisimple Lie Groups

Classical results due to Ingham and Paley-Wiener characterize the existence of nonzero functions supported on certain subsets of the real line in terms of the pointwise decay of the Fourier transforms. Viewing these results as uncertainty principles for Fourier transforms, we prove certain analogues of these results on connected, noncompact, semisimple Lie groups with finite center. We also use these results to show unique continuation property of solutions to the initial value problem for time-dependent Schrödinger equations on Riemmanian symmetric spaces of noncompact type.

Littlewood-Paley theorem, Nikolskii inequality, Besov spaces, Fourier and spectral multipliers on graded Lie groups

In this paper we investigate Besov spaces on graded Lie groups. We prove a Nikolskii type inequality (or the Reverse Hölder inequality) on graded Lie groups and as consequence we obtain embeddings of Besov spaces. We prove a version of the Littlewood-Paley theorem on graded Lie groups. The results are applied to obtain embedding properties of Besov spaces and multiplier theorems for both spectral and Fourier multipliers in Besov spaces on graded Lie groups. In particular, we give a number of sufficient conditions for the boundedness of Fourier multipliers in Besov spaces.

Beurling-Fourier代数的解析子代数与李群的复化

Analytic subalgebras of Beurling-Fourier algebras and complexification of Lie groups

本文通过引入解析子代数，将加权Fourier代数的Gelfand谱元素解释为李群复化中特定点上的取值，并分别给出一般连通李群的局部解、连通单连通幂零李群的全局解，以及$ax+b$-群的全局解实例。

某些紧致轨道超群的傅里叶代数

The Fourier Algebra of Certain Compact Orbit Hypergroups

本文证明了紧致离散赋值环上的中心傅里叶代数不可amenable，通过研究交换轨道超群的傅里叶代数，证实了Alaghmandan-Spronk猜想，并建立了与经典情形的对偶同构。

关于局部紧广群上 Fourier 与 Fourier-Stieltjes 代数的限制映射

On the restriction maps of the Fourier and Fourier-Stieltjes algebras over locally compact groupoids

本文将群上 Fourier 与 Fourier-Stieltjes 代数限制映射的满射性结果推广到局部紧广群，刻画 étale、传递及群丛等情形下的满射条件，并借助 FD 性质给出非满射例子及 Banach 代数性质缺失的条件。

Geometric Properties of some Banach Algebras related to the Fourier algebra on Locally Compact Groups

Let $A_p(G)$ denote the Figa-Talamanca-Herz Banach Algebra of the locally compact group}} $G$, thus $A_2(G)$ {\it{is the Fourier Algebra of $G$. If $G$ is commutative then $A_2(G)=L^1(\hat{G})^\wedge$. Let $A_p^r(G)=A_p\cap L^r(G)$ with norm $\|u\|_{A_p^r}=\| u\|_{A_p}+\| u\|_{L^r}$.We investigate for which $p$, $r$, and $G$ do the Banach algebras $A_p^r(G)$ {\it{have the Banach space geometric properties: The Radon-Nikodym Property (RNP), the Schur Property (SP) or the Dunford-Pettis Property (DPP). The results are new even if $G=R$ (the real line) or $G=Z$ (the additive integers).

Lp Fourier multipliers on compact Lie groups

In this paper we prove Lp multiplier theorems for invariant and non-invariant operators on compact Lie groups in the spirit of the well-known Hormander-Mikhlin theorem on Rn and its variants on tori Tn. We also give applications to a-priori estimates for non-hypoelliptic operators. Already in the case of tori we get an interesting refinement of the classical multiplier theorem.

On the structure of invertible elements in certain Fourier-Stieltjes algebras

For a locally compact abelian group $G$, J. L. Taylor (1971) gave a complete characterization of invertible elements in the measure algebra $M(G)$. Using the Fourier-Stieltjes transform, this characterization can be carried out in the context of Fourier-Stieltjes algebras $B(G)$. We obtain this latter characterization for the Fourier-Stieltjes algebra $B(G)$ of certain classes of locally compact groups, in particular, many totally minimal groups and the $ax+b$-group.

The Fourier algebra for locally compact groupoids

We introduce and investigate using Hilbert modules the properties of the Fourier algebra A(G) for a locally compact groupoid G. We establish a duality theorem for such groupoids in terms of multiplicative module maps. This includes as a special case the classical duality theorem for locally compact groups proved by P. Eymard.

Around Uncertainty Principles of Ingham-type on $\R^n$, $\T^n$ and Two Step Nilpotent Lie Groups

Classical results due to Ingham and Paley-Wiener characterize the existence of nonzero functions supported on certain subsets of the real line in terms of the pointwise decay of the Fourier transforms. We view these results as uncertainty principles for Fourier transforms. We prove certain analogues of these uncertainty principles on the $n$-dimensional Euclidean space, the $n$-dimensional torus and connected, simply connected two step nilpotent Lie groups. We also use these results to show a unique continuation property of solutions to the initial value problem for time-dependent Schrödinger …

Hardy's theorem for compact Lie groups

We show that Hardy's uncertainty principle can be reformulated in such a way that it has an analogue even for compact Lie groups and symmetric spaces of compact type.

The Fourier algebra for locally compact groupoids

We introduce and investigate using Hilbert modules the properties of the {\em Fourier algebra} $A(G)$ for a locally compact groupoid $G$. We establish a duality theorem for such groupoids in terms of multiplicative module maps. This includes as a special case the classical duality theorem for locally compact groups proved by P. Eymard.

$L^p$-海森堡-泡利-韦伊尔不确定原理在某些二步 nilpotent 利格群上的应用

$L^p$- Heisenberg--Pauli--Weyl uncertainty inequalities on certain two-step nilpotent Lie groups

本文在 Métivier 群上建立了新的 $L^p$-海森堡-泡利-韦伊尔不确定原理，通过群傅里叶变换证明非零函数与变换无法同时精确局部化，显著提升了原有不等式的精度。

A theorem for random Fourier series on compact quantum groups

Helgason showed that a given measure $f\in M(G)$ on a compact group $G$ should be in $L^2(G)$ automatically if all random Fourier series of $f$ are in $M(G)$. We explore a natural analogue of the theorem in the framework of compact quantum groups and apply the obtained results to study complete representability problem for convolution algebras of compact quantum groups as an operator algebra.

A Characterisation of the Fourier transform on the Schwartz-Bruhat space of locally compact Abelian groups

We obtain a characterisation of the Fourier transform on the space of Schwartz-Bruhat functions on locally compact Abelian groups. The result states that any appropriately additive bijection of the Schwartz space onto itself, which interchanges convolution and pointwise products is essentially the Fourier transform. The proof of this result is very similar to that obtained by the author recently for the Euclidean Fourier transform.

The Fourier-Stieltjes and Fourier algebras for locally compact groupoids

The Fourier-Stieltjes and Fourier algebras B(G), A(G) for a general locally compact group G, first studied by P. Eymard, have played an important role in harmonic analysis and in the study of operator algebras generated by G. Recently, there has been interest in developing versions of these algebras for locally compact groupoids, justification being that, just as in the group case, the algebras should play a useful role in the study of groupoid operator algebras. Versions of these algebras for the locally compact groupoid case appear in three related theories: (1) a measured groupoid theory (J…

On the isomorphisms of Fourier algebras of finite abelian groups

We consider gerenalizated permutation equivalence of Vielenkin system on commutative compact torsion finite exponent groups.

Hardy-Littlewood inequality and $L^p$-$L^q$ Fourier multipliers on compact hypergroups

This paper deals with the inequalities devoted to the comparison between the norm of a function on a compact hypergroup and the norm of its Fourier coefficients. We prove the classical Paley inequality in the setting of compact hypergroups which further gives the Hardy-Littlewood and Hausdorff-Young-Paley (Pitt) inequalities in the noncommutative context. We establish Hörmander's $L^p$-$L^q$ Fourier multiplier theorem on compact hypergroups for $1<p \leq 2 \leq q<\infty$ as an application of Hausdorff-Young-Paley inequality. We examine our results for the hypergroups constructed from the conju…

Titchmarsh theorems for Fourier transforms of Hölder-Lipschitz functions on compact homogeneous manifolds

In this paper we extend classical Titchmarsh theorems on the Fourier transform of Hölder-Lipschitz functions to the setting of compact homogeneous manifolds. As an application, we derive a Fourier multiplier theorem for $L^2$-Hölder-Lipschitz spaces on compact Lie groups. We also derive conditions and a characterisation for Dini-Lipschitz classes on compact homogeneous manifolds in terms of the behaviour of their Fourier coefficients.

Smooth and weak synthesis of the anti-diagonal in Fourier algebras of Lie groups

Let $G$ be a Lie group of dimension $n$, and let $A(G)$ be the Fourier algebra of $G$. We show that the anti-diagonal $\checkΔ_G=\{(g,g^{-1})\in G\times G \mid g\in G\}$ is both a set of local smooth synthesis and a set of local weak synthesis of degree at most $[\frac{n}{2}]+1$ for $A(G\times G)$. We achieve this by using the concept of the cone property in \cite{ludwig-turowska}. For compact $G$, we give an alternative approach to demonstrate the preceding results by applying the ideas developed in \cite{forrest-samei-spronk}. We also present similar results for sets of the form $HK$, where …

控制混合光滑空间中傅里叶变换的映射性质

Mapping properties of the Fourier transform in spaces with dominating mixed smoothness

该论文聚焦控制混合光滑函数空间，探究傅里叶变换的连续及紧映射特性，为相关函数空间理论提供映射性质层面的研究支撑。

Fourier algebras of hypergroups and central algebras on compact (quantum) groups

This paper concerns the study of regular Fourier hypergroups through multipliers of their associated Fourier algebras. We establish hypergroup analogues of well-known characterizations of group amenability, introduce a notion of weak amenability for hypergroups, and show that every discrete commutative hypergroup is weakly amenable with constant 1. Using similar techniques, we provide a sufficient condition for amenability of hypergroup Fourier algebras, which, as an immediate application, answers one direction of a conjecture of Azimifard--Samei--Spronk [J. Funct. Anal. 256(5) 1544-1564, 2009…

On Fourier algebra of a hypergroup constructed from a conditional expectation on a locally compact group

We prove that the Fourier space of a hypergroup constructed from a conditional expectation on a locally compact group has a Banach algebra structure.

$L^p$-Fourier and Fourier-Stieltjes algebras for locally compact groups

Let $G$ be a locally compact group and $1\leq p<\infty$. A continuous unitary representation $π\!: G\to B(\mathcal{H})$ of $G$ is an $L^p$-representation if the matrix coefficient functions $s\mapsto \langle π(s)x,x\rangle$ lie in $L^p(G)$ for sufficiently many $x\in \mathcal{H}$. Brannan and Ruan defined the $L^p$-Fourier algebra $A_{L^p}(G)$ to be the set of matrix coefficient functions of $L^p$-representations. Similarly, the $L^p$-Fourier-Stieltjes algebra $B_{L^p}(G)$ is defined to be the weak*-closure of $A_{L^p}(G)$ in the Fourier-Stieltjes algebra $B(G)$. These are always ideals in the…

On operator amenability of Fourier-Stieltjes algebras

We consider the Fourier-Stietljes algebra B(G) of a locally compact group G. We show that operator amenablility of B(G) implies that a certain semitolpological compactification of G admits only finitely many idempotents. In the case that G is connected, we show that operator amenability of B(G) entails that $G$ is compact.

Beurling-Fourier algebras of compact quantum groups: characters and finite dimensional representations

In this paper we study weighted versions of Fourier algebras of compact quantum groups. We focus on the spectral aspects of these Banach algebras in two different ways. We first investigate their Gelfand spectrum, which shows a connection to the maximal classical closed subgroup and its complexification. Secondly, we study specific finite dimensional representations coming from the complexification of the underlying quantum group. We demonstrate that the weighted Fourier algebras can detect the complexification structure in the special case of $SU_q(2)$, whose complexification is the quantum L…

Groups whose Fourier algebra and Rajchman algebra coincide

We study locally compact groups for which the Fourier algebra coincides with the Rajchman algebra. In particular, we show that there exist uncountably many non-compact groups with this property. Generalizing a result of Hewitt and Zuckerman, we show that no non-compact nilpotent group has this property, whereas non-compact solvable groups with this property are known to exist. We provide several structural results on groups whose Fourier and Rajchman algebras coincide as well as new criteria for establishing this property. Finally, we study the relation between groups with completely reducible…

Cubature formulas and discrete fourier transform on compact manifolds

The goal of the paper is to describe essentially optimal cubature formulas on compact Riemannian manifolds which are exact on spaces of band- limited functions.

非 Kac 紧量子群上中心 Fourier 级数的 Khintchine 不等式

A Khintchine inequality for central Fourier series on non-Kac compact quantum groups

本文在非Kac紧量子群框架内，对中心Fourier级数建立了带算子系数的Khintchine不等式，填补了紧连通半单Lie群上相关不等式缺失的空白。

Hardy-Littlewood, Hausdorff-Young-Paley inequalities, and Lp-Lq Fourier multipliers on compact homogeneous manifolds

In this paper we prove new inequalities describing the relationship between the "size" of a function on a compact homogeneous manifold and the "size" of its Fourier coefficients. These inequalities can be viewed as noncommutative versions of the Hardy-Littlewood inequalities obtained by Hardy and Littlewood on the circle. For the example case of the group SU(2) we show that the obtained Hardy-Littlewood inequalities are sharp, yielding a criterion for a function to be in $L^p$ on SU(2) in terms of its Fourier coefficients. We also establish Paley and Hausdorff-Young-Paley inequalities on gener…

通过局部化的紧致伪微分与傅里叶积分算子

Compact pseudodifferential and Fourier integral operators via localization

本文提出局部化算子的一般框架，证明其在调制空间之间的有界性与紧致性，并用于统一推广伪微分和傅里叶积分算子理论，得到三参数伪微分算子的新结果。

Fourier multipliers and group von Neumann algebras

In this paper we establish the $L^p$-$L^q$ boundedness of Fourier multipliers on locally compact separable unimodular groups for the range of indices $1<p\leq 2 \leq q<\infty$. Our approach is based on the operator algebras techniques. The result depends on a version of the Hausdorff-Young-Paley inequality that we establish on general locally compact separable unimodular groups. In particular, the obtained result implies the corresponding Hörmander's Fourier multiplier theorem on $\mathbb{R}^{n}$ and the corresponding known results for Fourier multipliers on compact Lie groups.

每个无限紧阿贝尔群上的递减重排傅里叶级数的发散性

Divergence of decreasing rearranged Fourier series on every infinite compact abelian group

该研究证明每个无限紧豪斯多夫阿贝尔群上存在复值连续函数，其递减重排傅里叶级数几乎处处发散，这类函数构成稠密Gδ集，通过结合相关引理与构造完成证明。

$p$-Fourier algebras on compact groups

Let $G$ be a compact group. For $1\leq p\leq\infty$ we introduce a class of Banach function algebras $\mathrm{A}^p(G)$ on $G$ which are the Fourier algebras in the case $p=1$, and for $p=2$ are certain algebras discovered in \cite{forrestss1}. In the case $p\not=2$ we find that $\mathrm{A}^p(G)\cong \mathrm{A}^p(H)$ if and only if $G$ and $H$ are isomorphic compact groups. These algebras admit natural operator space structures, and also weighted versions, which we call $p$-Beurling-Fourier algebras. We study various amenability and operator amenability properties, Arens regularity and represen…