Some Estimates for the Fourier Transform on Rank 1 Symmetric Spaces

Two estimates useful in applications are proved for the Fourier transform in the space L~2(X), where X a symmetric space, as applied to some classes of functions characterized by a generalized modulus of continuity.     

The Fourier Transform of Functions Satisfying the Lipschitz Condition on Rank 1 Symmetric Spaces

We prove an analog of the classical Titchmarsh theorem on the image under the Fourier transform of a set of functions satisfying the Lipschitz condition in L² for functions on noncompact rank 1 Riemannian symmetric spaces.     

Jackson Type Theorems for Compact Rank 1 Symmetric Spaces

Siberian Mathematical Journal -     

Polynomial Quantization on Rank One Para-Hermitian Symmetric Spaces

We construct the polynomial quantization on the space G/H where G=SL(n,R),H=GL(n–1,R). It is a variant of quantization in the spirit of Berezin. In our case covariant and contravariant symbols are polynomials on G/H. We introduce a multiplication of covariant symbols, establish the correspondence principle, study transformations of symbols (the Berezin transform) and of operators. We write a full asymptotic decomposition of the Berezin transform.     

Riesz Means on Compact Riemannian Symmetric Spaces

We study approximation properties of the Riesz means on compact symmetric spaces of rank one. To do so we establish equivalences between the Riesz means and Peetre K-moduli and estimate the weak type and the uniform approximation of the Riesz means at the critical index. The relations between the Riesz means and the best approximation as well as the Cesàro means are also considered.     

Patterson-Sullivan Theory in Higher Rank Symmetric Spaces

Let X = G/K be a Riemannian symmetric space of noncompact type and a discrete “generic” subgroup of G with critical exponent . Denote by the set of regular elements of the geometric boundary of X. We show that the support of all -invariant conformal densities of dimension on (e.g. Patterson-Sullivan densities) lie in a same and single regular G-orbit on . This provides information on the large-scale growth of -orbits in X. If in addition we assume to be of divergence type, then there is a unique density of the previous type. Furthermore, we explicitly determine and this G-orbit for lattices, and show that they are of divergence type. Submitted: November 1997, revised: January 1999.     

有界对称域上的VMOp与VO空间的乘子

该文利用Berezin变换,自同构群及Bergman再生核理论,对有界对称域上的VMOp 与VO空间的点态乘子进行了刻划．     

紧对称空间上Riesz平均的强逼近

本文在紧对称空间中给出了全测度集上Ｒｉｅｓｚ平均强逼近的阶     

Energy of Radial Vector Fields on Compact Rank One Symmetric Spaces

We consider the energy (or the total bending) of unit vector fields oncompact Riemannian manifolds for which the set of its singularitiesconsists of a finite number of isolated points and a finite number ofpairwise disjoint closed submanifolds. We determine lower bounds for theenergy of such vector fields on general compact Riemannian manifolds andin particular on compact rank one symmetric spaces. For this last classof spaces, we compute explicit expressions for the total bending whenthe unit vector field is the gradient field of the distance function toa point or to special totally geodesic submanifolds (i.e., for radialunit vector fields around this point or these submanifolds).     

A Characterization of Higher Rank Symmetric Spaces Via Bounded Cohomology

Let M be complete nonpositively curved Riemannian manifold of finite volume whose fundamental group Γ does not contain a finite index subgroup which is a product of infinite groups. We show that the universal cover is a higher rank symmetric space iff is injective (and otherwise the kernel is infinite dimensional). This is the converse of a theorem of Burger–Monod. The proof uses the celebrated Rank Rigidity Theorem, as well as a new construction of quasi-homomorphisms on groups that act on CAT(0) spaces and contain rank 1 elements. Received: September 2007, Accepted: March 2008     

Some Properties of Boundaries of Symmetric Spaces of Rank One

M. Cowling, A. H. Dooley, A. Korányi and F. Ricci used groups of Heisenberg type in order to study the symmetric spaces of rank one of noncompact type in a unified manner. This paper extends this work by using the same formulation to investigate the boundaries of these spaces. In particular, we prove a conjecture of Korányi concerning metrics on the boundary and demonstrate that the classical Cayley transform extends to a 1-quasiconformal map of the boundary.     

Eigenfunction Expansions on Geodesic Balls and Rank One Symmetric Spaces of Compact Type

We find sharp conditions for the pointwise convergence ofeigenfunction expansions associated with the Laplace operator and otherrotationally invariant differential operators. Specifically, we considerthis problem for expansions associated with certain radially symmetricoperators and general boundary conditions and the problem in the contextof Jacobi polynomial expansions. The latter has immediate application toFourier series on rank one symmetric spaces of compact type.     

非紧致一秩Rienann对称空间上的中心极限定理

我们在本文中研究非紧致一秩Riemann对称空间上初等球函数的渐近表示,并利用LohoueN.和RychnerTh.得到的热核表达式,建立起这类空间上的非欧中心极限定理,所得结果包含了Terras的定理作为其特例.     

非紧致一秩Riemann对称空间上的中心极限定理

我们在本文中研究非紧致一秩Ｒｉｅｍａｎｎ对称空间上初等球函数的渐近表示,并利用Ｌｏｈｏｕｅ Ｎ．和Ｒｙｃｈｎｅｒ Ｔｈ．得到的热核表达式,建立起这类空间上的非欧中心极限定理,所得结果包含了Ｔｅｒｒａｓ的定理作为其特例．     

The Problem of the Pointwise Fourier Inversion for Piecewise Smooth Functions of Several Variables

We study the pointwise convergence problem for the inverse Fourier transform of piecewise smooth functions, i.e., whether S_rD f (\bx) ? f (\bx)S_{\rho D} f (\bx) \to f (\bx) as r? ￥\rho \to \infty . r? ￥\rho \to \infty . Here for \bx,\bxi ? \Rn\bx,\bxi \in \Rn S_rDf(\bmx)=\dsf1(2p)^n/2\intli_rD [^(f)](\bxi) e^{\dst iá\bmx,\bxi?} d\bxi . S_{\rho D}f(\bm{x})=\dsf1{(2\pi)^{n/2}}\intli_{\rho D} \widehat{f}(\bxi) e^{\dst i\langle\bm{x},\bxi\rangle} d\bxi~. is the partial sum operator using a convex and open set DD containing the origin, and rD={ r\bxi:\bxi ? D }\rho D=\left\{ \rho \bxi:\bxi\in D \right\}.     

Pointwise Fourier inversion of distributions on spheres

Given a distribution T on the sphere we define, in analogy to the work of ?ojasiewicz, the value of T at a point ξ of the sphere and we show that if T has the value τ at ξ, then the Fourier-Laplace series of T at ξ is Abel-summable to τ.     

Walsh—Fourier级数在p—级数域上的Hardy空间和Herz空间中的Cesaro可和性

讨论了二维Walsh-Fourier级数在p－级数域的整环上的两个函数空间Hardy空间和Herz空间中的Cesaro可和性。