SPHERICAL FUNCTIONS AND CONFORMAL DENSITIES ON SPHERICALLY SYMMETRIC CAT( 1)-SPACES (original) (raw)

Spherical means and the restriction phenomenon

The Journal of Fourier Analysis and Applications, 2001

LetP beasmoothcompactconvexplanarcurvewitharclengthdmandletdcr = ap dm where ap is a cutoff function. For | ~ SO(2) set cr| = (r(OE) for any measurable planar set E. Then, for suitable functions f in ~2, the inequality represents an average over rotations, of the Stein-Tomas restriction phenomenon. We obtain best possible indices for the above inequality when F is any convex curve and under various geometric assumptions.

On the characterization of spherical distributions

Journal of Information and Optimization Sciences, 1996

A necessary and sufficient condition has been derived for a random variable to have a spherical distribution. This result has been specialized to find characteristic functions of uniform distributions on or inside unit hyperspheres.

The Spherical Boundary and Volume Growth

ISRN Geometry, 2012

We consider thespherical boundary, a conformal boundary using a special class of conformal distortions. We prove that certain bounds on volume growth of suitable metric measure spaces imply that the spherical boundary is “small” (in cardinality or dimension) and give examples to show that the reverse implications fail. We also show that the spherical boundary of an annular convex proper length space consists of a single point. This result applies tol2-products of length spaces, since we prove that a natural metric, generalizing such “norm-like” product metrics on a (possibly infinite) product of unbounded length spaces, is annular convex.

2 0 Se p 20 05 Spherical Functions on Euclidean Space

2005

We study special functions on euclidean spaces from the viewpoint of riemannian symmetric spaces. Here the euclidean space En = G/K where G is the semidirect product Rn · K of the translation group with a closed subgroup K of the orthogonal group O(n). We give exact parameterizations of the space of (G,K)–spherical functions by a certain affine algebraic variety, and of the positive definite ones by a real form of that variety. We give exact formulae for the spherical functions in the case where K is transitive on the unit sphere in En

Carleson and Vanishing Carleson Measures on Radial Trees

Mediterranean Journal of Mathematics, 2012

We extend a discrete version of an extension of Carleson's theorem proved in [5] to a large class of trees T that have certain radial properties. We introduce the geometric notion of s-vanishing Carleson measure on such a tree T (with s ≥ 1) and give several characterizations of such measures. Given a measure σ on T and p ≥ 1, let L p (σ) denote the space of functions g defined on T such that |g| p is integrable with respect to σ and let L p (∂T) be the space of functions f defined on the boundary of T such that |f | p is integrable with respect to the representing measure of the harmonic function 1. We prove the following extension of the discrete version of a classical theorem in the unit disk proved by Power. A finite measure σ on T is an s-vanishing Carleson measure if and only if for any real number p > 1, the Poisson operator P : L p (∂T) → L sp (σ) is compact. Characterizations of weak type for the case p = 1 and in terms of the tree analogue of the extended Poisson kernel are also given. Finally, we show that our results hold for homogeneous trees whose forward probabilities are radial and whose backward probabilities are constant, as well as for semihomogeneous trees.

Regularity of spherical means and localization of spherical harmonic expansions

Journal of the Australian Mathematical Society, 1986

Let G/K be a compact symmetric space, and let G = KAK be a Cartan decomposition of G. For f in L1(G) we define the spherical means f(g, t) = ∫k∫k ∫(gktk′) dk dk′, g ∈ G, t ∈ A. We prove that if f is in Lp(G), 1 ≤ p ≤ 2, then for almost every g ∈ G the functions t → f(g, t) belong to certain Soblev spaces on A. From these regularity results for the spherical means we deduce, if G/K is a compact rank one symmetric space, a theorem on the almost everywhere localization of spherical harmonic expansions of functions in L2 (G/K).

Spherical means in annular regions in the n -dimensional real hyperbolic spaces

Proceedings of The Indian Academy of Sciences-mathematical Sciences

Let Z r,R be the class of all continuous functions f on the annulus Ann(r, R) in the real hyperbolic space mathbbBn\mathbb B^nmathbbBn with spherical means M s f(x) = 0, whenever s > 0 and xinmathbbBnx\in\mathbb B^nxinmathbbBn are such that the sphere S s (x) ⊂ Ann(r, R) and Br(o)subseteqBs(x).B_r(o)\subseteq B_s(x).Br(o)subseteqBs(x). In this article, we give a characterization for functions in Z r,R . In the case R = ∞, this result gives a new proof of Helgason’s support theorem for spherical means in the real hyperbolic spaces.

Uniform Distributions on Spheres in Finite DimensionalLαand Their Generalizations

Journal of Multivariate Analysis, 1998

We characterize uniform distributions on spheres in n-dimensional spaces L : by certain Cauchy-like (n&1)-dimensional distributions of the quotients and derive some properties of mixtures of uniform distributions on such spheres, i.e., :-spherical distributions. We feel that a simple Cauchy-like distribution is much simpler to deal with than the usual description of a uniform distribution on the sphere.