AN ASYMPTOTIC ORDER OFFOURIER TRANSFORM ON SL（2,R）

In this paper, a better asymptotic order of Fourier transform on SL(2,R) is obtained by using classical analysis and Lie analysis comparing with that of [5]、[6], and the Plancherel theorem on Cc^2(SL(2,R) ) is also obtained as an application.     

SL(2,R)上Fourier变换的绝对值的渐近阶

本文研究了SL(2，R)上的Fourier变换的绝对值的渐近阶的问题．利用Lie代数SL(2，R)的表示在表示空间的作用和经典调和分析的方法，得到了SL(2，R)上的Fourier变换的绝对值的一个渐近阶．而且作为应用给出了Gc^2(2，R))上的函数的反演公式的一种新证明．     

A CONSTRUCTIVE PROOF OF THE INVERSION FORMULA FOR ZONAL FUNCTIONS ON SL(2,R）

1.IntroductionLeSL(2,R)denotethemultiplicativegroupofall2x2realm8triceswithdet-nat1.InthispaPer,weuseGtodenotebothSL(2,R)andthellnearLiegroupbecausetheyareisomorphictoeachother.Forj={o,1/2},s=1 iA(whereAER,andRisthesetofallrealnumbers),letVi,8betheprincipalcolltinuousseriesofunitaryrepresentationsofG(cf.[4]).SetBytheIwasawadecomposition,anygEGcanbeuniquelywhttenasg=usatnr,u8ESK,afESA,nrESN.Al8oanyginGhasaCartandecompositionasfollows:Afunctionf0nGissaidtobeazonalfunctionifitsatisf…     

高维Klein群的一个不等式及其应用

本文首先得到了SL（2，Гn）中Klein群的一个不等式，并给出了它的两个应用；然后证明了对SL（2，Гn）中的非初等群G，若G中的任意斜驶元素f满足tr^2（f）〉4且当∞ 不属于fix（f）时tr（f）=tr（f），则存在h∈SL（2，Гn）使得hGh^-1属于SL（2，R），此结果是Maskit相关结果的推广。     

SL(2,R)上的Jackson型正定理

本文在SL(2,R)上引入距离、光滑模、导数等概念,给出了SL(2,R)上的连续函数用Tn和Bσ,n逼近的Jackson型正定理,得到了SL(2,R)上函数的光滑性和最佳逼近阶之间的关系.     

SL(2,R)上的Hardy-Littlewood极大函数

给出了SL(2 ,R)上的Hardy Littlewood极大函数mf 和局部Hardy Littlewood极大函数mRf 的定义 ,对f∈L1(G) ,我们得到了 | {g∈SL(2 ,R) |mf(g) >λ} |的估计 ,且证明了局部Hardy Littlewood极大函数的弱(1.1)型和强 (p ,p)型 ,p >1.     

The Moduli Space of Boundary Compactifications of SL (2, R)

In an earlier paper, the authors introduced the notion of a boundary compactification of SL(2, R) and SL(2, C), a normal projective embedding of PSL2 arising as the Zariski closure of an orbit in (P1)n under the diagonal action of SL2. Here the moduli space of such boundary compactifications of SL(2, R) is shown to be a contractible hyperbolic orbifold, by using the Schwarz–Christoffel transformation to identify it with a quotient of the moduli space of equi-angular planar polygons.     

关于SL(2,R)上速降函数的反演公式

在局部紧可分群的一般理论中，分解正则表示以及获得反演公式（或 Ｐｌａｎ－ｃｈｅｒｅｌ定理的明确表示）是调和分析的基本目标之一．ＳＬ（２，　）是最简单的非交换局部紧么模半单Ｌｉｅ群．Ｈａｒｉｓｈ－Ｃｈａｎｄｒａ在 Ｃ∞ｃ（ＳＬ（２，　））上获得了反演公式，Ｘｉａｏ和ｈｅｎｇ在文［１］中证明了Ｃ３ｃ（ＳＬ（２， ）上的反演公式．在文［２］中Ｚｈｅｎｇ引入了Ｌｉｅ群Ｇ上函数的广义微分（Ａ导数）概念．在本文中，我们利用文［２］中的微分概念来研究ＳＬ（２，　）上可微函数的Ｆｏｕｒｉｅｒ变换的阶，并获得了ＳＬ（２，　）上速降函数的反演公式．     

群在von Neumann代数上作用的自由和遍历性注记

本文研究了群在von Neumann代数上作用的自由性和遍历性问题.利用投影和群SL2(R)的Iwasawa分解,得到了可数离散群在交换von Neumann代数上作用的自由性的等价刻画,证明了SL2(R)在上半平面H上有理作用导出的SL2(R)在极大交换von Neumann代数A={Mf:f∈L2(H,dxdy/y2)}上的作用α是遍历的,但不是自由的.     

On Differentiable Compactifications of the Hyperbolic Plane and Algebraic Actions of SL2(\mathbb R) on Surfaces

Real-analytic actions of SL(2;R) on surfaces have been classified, up to analytic change of coordinates. In particular it is known that there exists countably many analytic equivariant compactification of the isometric action on the hyperbolic plane. In this paper we study the algebraicity of these actions. We get a classification of the algebraic actions of SL(2,R) on surfaces. In particular, we classify the algebraic equivariant compactifications of the hyperbolic plane. An erratum to this article can be found at     

On differentiable compactifications of the hyperbolic plane and algebraic actions of $${\rm SL}_2(\mathbb{R})$$ on surfaces

Real-analytic actions of SL(2;R) on surfaces have been classified, up to analytic change of coordinates. In particular it is known that there exists countably many analytic equivariant compactification of the isometric action on the hyperbolic plane. In this paper we study the algebraicity of these actions. We get a classification of the algebraic actions of SL(2;R) on surfaces. In particular, we classify the algebraic equivariant compactifications of the hyperbolic plane. The online version of the original article can be found under doi: .     

Sharp inequalities and geometric manifolds

Sharp constants for function-space inequalities over a manifold encode information about the geometric structure of the manifold. An important example is the Moser-Trudinger inequality where limiting Sobolev behavior for critical exponents provides significant understanding of geometric analysis for conformal deformation on a Riemannian manifold [5, 6]. From the overall perspective of the conformal group acting on the classical spaces, it is natural to consider the extension of these methods and questions in the context of SL(2, R), the Heisenberg group, and other Lie groups. Among the principal tools used in this analysis are the linear and multilinear operators mapping L^p(M) to L^q(M) defined by the Stein-Weiss integral kernels which extend the Hardy-Littlewood-Sobolev fractional integrals\(\mathcal{H}^1 (\mathbb{R}^d )\) conformal geometry, and the notion of equimeasurable geodesic radial decreasing rearrangement. To illustrate these ideas, four model problems will be examined here: (1) logarithmic Sobolev inequality and the uncertainty principle, (2) SL(2,R) and axial symmetry in fluid dynamics, (3) Stein-Weiss integrals on the Heisenberg group, and (4) Morpurgo’s work on zeta functions and trace inequalities of conformally invariant operators.     

The Kunze-Stein Phenomenon

Si dà una nuova dimostrazione di un, teorema di Kunze e Stein, che dice che, se 1≤p<2,L ^p(SL(2, R))*L ²(SL(2, R)) è contenuto inL ²(SL(2, R)). Questa nuova dimostrazione può essere generalizzata per provare lo stesso teorema per ogni gruppo di Lie connesso, semisemplice, col centro finito.     

Poincaré sections for the horocycle flow in covers of $$\mathrm {SL}(2,\mathbb {R})/\mathrm {SL}(2,\mathbb {Z})$$ and applications to Farey fraction statistics

For a given finite index subgroup \(H\subseteq \mathrm {SL}(2,\mathbb {Z})\), we use a process developed by Fisher and Schmidt to lift a Poincaré section of the horocycle flow on \(\mathrm {SL}(2,\mathbb {R})/\mathrm {SL}(2,\mathbb {Z})\) found by Athreya and Cheung to the finite cover \(\mathrm {SL}(2,\mathbb {R})/H\) of \(\mathrm {SL}(2,\mathbb {R})/\mathrm {SL}(2,\mathbb {Z})\). We then use the properties of this section to prove the existence of the limiting gap distribution of various subsets of Farey fractions. Additionally, to each of these subsets of fractions, we extend solutions by Xiong and Zaharescu, and independently Boca, to a Diophantine approximation problem of Erd?s, Szüsz, and Turán.     

Trivial and simple spectrum for SL(d, ℝ) cocycles with free base and fiber dynamics

Let AC_D(M,SL(d,R)) denote the pairs (f,A) so that f ∈ A ⊂ Diff¹(M) is a C¹-Anosov transitive diffeomorphisms and A is an SL(d,R) cocycle dominated with respect to f. We prove that open and densely in AC_D(M,SL(d,R)), in appropriate topologies, the pair (f,A) has simple spectrum with respect to the unique maximal entropy measure μ_f. Then, we prove prevalence of trivial spectrum near the dynamical cocycle of an area-preserving map and also for generic cocycles in AutLeb(M) × L^p(M,SL(d,R)).     

Induced Representations and Hypercomplex Numbers

In the search for hypercomplex analytic functions on the halfplane, we review the construction of induced representations of the group ${G = {\rm SL}_2(\mathbb{R})}$ . Firstly we note that G-action on the homogeneous space G/H, where H is any one-dimensional subgroup of ${{\rm SL}_2(\mathbb{R})}$ , is a linearfractional transformation on hypercomplex numbers. Thus, we investigate various hypercomplex characters of subgroups H. The correspondence between the structure of the group ${{\rm SL}_2(\mathbb{R})}$ and hypercomplex numbers can be illustrated in many other situations as well. We give examples of induced representations of ${{\rm SL}_2(\mathbb{R})}$ on spaces of hypercomplex valued functions, which are unitary in some sense. Raising/lowering operators for various subgroup prompt hypercomplex coefficients as well.     

Linear Systems and Quotients of Projective Space

A complete characterization of the categorical quotients of(P¹)ⁿ by the diagonal action of SL(2, C) with respect to anypolarization is given by M. Polito, in ‘SL(2, C)-quotientsde (P¹)ⁿ’, C. R. Acad. Sci. Paris Sér. I 321 (1995)1577–1582. In this paper, these categorical quotientsare obtained by certain linear systems on P^n–3, dependingon the given polarization. 2000 Mathematics Subject Classification14L24, 14L30     

Vertex-IRF transformations, dynamical quantum groups and harmonic analysis

It is shown that a dynamical quantum group arising from a vertex-IRF transformation has a second realization with untwisted dynamical multiplication but nontrivial bigrading. Applied to the SL 2; ) dynamical quantum group, the second realization is naturally described in terms of Koornwinder's twisted primitive elements. This leads to an intrinsic explanation why harmonic analysis on the “classical” SL(2; ) quantum group with respect to twisted primitive elements, as initiated by Koornwinder, is the same as harmonic analysis on the SL(2; C) dynamical quantum group.     

Wavelet Transforms Associated to Square Integrable Group Representations on L 2 ( C,H ; dz )

Let SL (2, C ) be the special linear group of 2 ‐ 2 complex matrices with determinant 1 and SU (2) its maximal compact subgroup. Then SL (2, C )/ SU (2) can be realized as the quaternionic upper half-plane $ {\cal H}^c $ . Let SL (2, C ) = NASU (2) be the Iwasawa decomposition and M the centerlizer of A in SU (2). Then P = NA and P a = NAM are the automorphism groups of $ {\cal H}^c $ . In this article, we define the unitary representations of P and P a on L 2 ( C , H ; dz ). From the viewpoint of square integrable group representations we discuss the wavelet transforms, and obtain the orthogonal direct sum decompositions for the function spaces $ L^2({\cal H}^c, \fraca {(dz\, d\rho)}{\rho ^3}) $ and $ L^2({\bf R}^2\times {\bf R}^2, \fraca {dx\, dy\, dx^{\prime }dy^{\prime }}{{({x^{\prime }}^2 + {y^{\prime }}^2)^{\fraca {3}{2}})}} $ .     

Minkowski's conjecture, well-rounded lattices and topological dimension

Let be the diagonal subgroup, and identify with the space of unimodular lattices in . In this paper we show that the closure of any bounded orbit

meets the set of well-rounded lattices. This assertion implies Minkowski's conjecture for and yields bounds for the density of algebraic integers in totally real sextic fields.

The proof is based on the theory of topological dimension, as reflected in the combinatorics of open covers of and .