Analytic eigendistributions and applications to homogeneous trees

An analytic distribution on is an element, ν, of the dual of the space of analytic functions on K. In particular, ν defines a linear functional on the polynomial ring . In this work, we study the converse problem: given a linear functional on , try to find a minimal set K such that ν extends to an analytic distribution on K. This study was motivated by the desire to generalize a result that allows the representation of functions on a homogeneous tree as integrals of z-harmonic functions oven a certain interval. A function f on a homogeneous tree T of degree q+1 is said to be z-harmonic, if μ₁f=zf, where μ₁ is the nearest neighbor averaging operator. It was proved in [Cohen, Colonna, Adv. Appl. Math. 20 (1998) 253–274] that if |f(v)|MC^|v| for constants M>0 and , then there exist z-harmonic functions k_z such that where I is the interval with endpoints . In the present paper, we study the case when the above exponential growth condition holds with , which necessitates replacing k_z(v) dz with an analytic distribution ν_v satisfying the z-harmonicity condition μ₁ν=zν. We show that to each function on the tree satisfying the above exponential growth condition there corresponds an eigendistribution on an elliptical region containing I as the interval between its foci.     

Embedding of RCD⁎(K,N) spaces in L2 via eigenfunctions

In this paper we study the family of embeddings ${\mathrm{\Phi }}_t$ of a compact ${\mathrm{RCD}}^{?}(K,N)$ space $(X,\mathsf{d},\mathfrak{m})$ into $L^2(X,\mathfrak{m})$ via eigenmaps. Extending part of the classical results [10], [11] known for closed Riemannian manifolds, we prove convergence as $t\downarrow 0$ of the rescaled pull-back metrics ${\mathrm{\Phi }}_t^{?}{g}_{L^2}$ in $L^2(X,\mathfrak{m})$ induced by ${\mathrm{\Phi }}_t$. Moreover we discuss the behavior of ${\mathrm{\Phi }}_t^{?}{g}_{L^2}$ with respect to measured Gromov-Hausdorff convergence and t. Applications include the quantitative $L^p$-convergence in the noncollapsed setting for all $p<\infty$, a result new even for closed Riemannian manifolds and Alexandrov spaces.     

High-frequency propagation for the Schrödinger equation on the torus

The main objective of this paper is understanding the propagation laws obeyed by high-frequency limits of Wigner distributions associated to solutions to the Schrödinger equation on the standard d-dimensional torus Td. From the point of view of semiclassical analysis, our setting corresponds to performing the semiclassical limit at times of order 1/h, as the characteristic wave-length h of the initial data tends to zero. It turns out that, in spite that for fixed h every Wigner distribution satisfies a Liouville equation, their limits are no longer uniquely determined by those of the Wigner distributions of the initial data. We characterize them in terms of a new object, the resonant Wigner distribution, which describes high-frequency effects associated to the fraction of the energy of the sequence of initial data that concentrates around the set of resonant frequencies in phase-space T^*Td. This construction is related to that of the so-called two-microlocal semiclassical measures. We prove that any limit μ of the Wigner distributions corresponding to solutions to the Schrödinger equation on the torus is completely determined by the limits of both the Wigner distribution and the resonant Wigner distribution of the initial data; moreover, μ follows a propagation law described by a family of density-matrix Schrödinger equations on the periodic geodesics of Td. Finally, we present some connections with the study of the dispersive behavior of the Schrödinger flow (in particular, with Strichartz estimates). Among these, we show that the limits of sequences of position densities of solutions to the Schrödinger equation on T² are absolutely continuous with respect to the Lebesgue measure.     

Eigenvalues and eigenfunctions of one-dimensional fractal Laplacians defined by iterated function systems with overlaps

Under the assumption that a self-similar measure defined by a one-dimensional iterated function system with overlaps satisfies a family of second-order self-similar identities introduced by Strichartz et al., we obtain a method to discretize the equation defining the eigenvalues and eigenfunctions of the corresponding fractal Laplacian. This allows us to obtain numerical solutions by using the finite element method. We also prove that the numerical eigenvalues and eigenfunctions converge to the true ones, and obtain estimates for the rates of convergence. We apply this scheme to the fractal Laplacians defined by the well-known infinite Bernoulli convolution associated with the golden ratio and the 3-fold convolution of the Cantor measure. The iterated function systems defining these measures do not satisfy the open set condition or the post-critically finite condition; we use second-order self-similar identities to analyze the measures.     

A splitting theorem for manifolds of almost nonnegative Ricci curvature

Cheeger and Gromoll proved that a closed Riemannian manifold of nonnegative Ricci curvature is, up to a finite cover, diffeomorphic to a direct product of a simply connected manifold and a torus. In this paper, we extend this theorem to manifolds of almost nonnegative Ricci curvature.     

小负曲率流形上Laplace算子第一特征值的下界估计

本文首先对流形的测地球上的Sobolev常数进行讨论,并利用它进行Moser迭代,最终得到具有小负曲率的闭的黎曼流形上Laplace算子特征值的一个下界估计.     

Asymptotics of simple eigenvalues and eigenfunctions for the Laplace operator in a domain with an oscillating boundary

The asymptotic behavior of solutions to spectral problems for the Laplace operator in a domain with a rapidly oscillating boundary is analyzed. The leading terms of the asymptotic expansions for eigenelements are constructed, and the asymptotics are substantiated for simple eigenvalues. The text was submitted by the authors in English.     

Spherical maximal operator on symmetric spaces of constant curvature

We prove an endpoint weak-type maximal inequality for the spherical maximal operator applied to radial funcions on symmetric spaces of constant curvature and dimension . More explicitly, in the Lorentz space associated with the natural isometry-invariant measure, we show that, for every radial function ,

The proof uses only geometric arguments and volume estimates, and applies uniformly in every dimension.

     

Manifolds with Boundary and of Bounded Geometry

For non–compact manifolds with boundary we prove that bounded geometry defined by coordinate–free curvature bounds is equivalent to bounded geometry defined using bounds on the metric tensor in geodesic coordinates. We produce a nice atlas with subordinate partition of unity on manifolds with boundary of bounded geometry and we study the change of geodesic coordinate maps.     

Distributional radius of curvature

We show that any continuous plane path that turns to the left has a well‐defined distribution that corresponds to the radius of curvature of smooth paths. We show that the distributional radius of curvature determines the path uniquely except for a translation. We show that Dirac delta contributions in the radius of curvature correspond to facets, that is, flat sections of the path, and show how a path can be deformed into a facet by letting the radius of curvature approach a delta function. Copyright © 2005 John Wiley & Sons, Ltd.     

On the non existence of unbiased estimators of risk for spherically symmetric distributions

We study existence of unbiased estimators of risk for estimators of the location parameter of a spherically symmetric distribution, when a residual vector is available to estimate scale, under invariant quadratic loss. We show such existence often characterizes normality.     

Riemann流形上Laplace算子第一特征值的一点注记

本文研究了黎曼流形上Laplace算子的第一特征值,利用流形的测地球上的Sobolev常数进行讨论并进行Moser迭代,得到闭的黎曼流形上Laplace算子第一特征值的一个下界估计.     

Chaotic dynamics of the heat semigroup on Riemannian symmetric spaces

We show that the heat semigroup generated by certain perturbations of the Laplace–Beltrami operator on the Riemannian symmetric spaces of noncompact type is chaotic   on their L^p$L^p$-spaces when 2<p<∞$2<p<\infty$. Both the range of p and the range of chaos-inducing perturbation are sharp. This extends a result of Ji and Weber [17] where it was shown that under identical conditions the heat operator is subspace-chaotic on these spaces.     

Asymptotics of the convex hull of spherically symmetric samples

In this paper we consider the convex hull of a spherically symmetric sample in R^d. Our main contributions are some new asymptotic results for the expectation of the number of vertices, number of facets, area and the volume of the convex hull assuming that the marginal distributions are in the Gumbel max-domain of attraction. Further, we briefly discuss two other models assuming that the marginal distributions are regularly varying or O-regularly varying.     

A geometric characterization of the nonlinear Schr?dinger equation and its applications

We prove that the nonlinear Schr?dinger equation of attractive type (NLS⁺) describes just spherical surfaces (SS) and the nonlinear Schr?dinger equation of repulsive type (NLS^-) determines only pseudospherical surfaces (PSS). This implies that, though we show that given two differential PSS (resp. SS) equations there exists a local gauge transformation (despite of changing the independent variables or not) which transforms a solution of one into any solution of the other, it is impossible to have such a gauge transformation between the NLS⁺ and the NLS^-.     

Riemannian Manifold in Which the Skew-Symmetric Curvature Operator has Pointwise Constant Eigenvalues

We study manifolds where the natural skew-symmetric curvature operator has pointwise constant eigenvalues. We give a local classification (up to isometry) of such manifolds in dimension 4. In dimension 3, we describe such manifolds up to a classification of three - dimensional Riemannian manifolds with principal Ricci curvatures r₁ = r₂ = 0, r₃- arbitrary. We give examples of such manifolds in all dimensions which do not have constant sectional curvature; these manifolds are not pointwise Osserman manifolds in general.     

A geometric characterization of the nonlinear Schrodinger equation and its applications

We prove that the nonlinear Schrodinger equation of attractive type (NLS+ describes just spher-ical surfaces (SS) and the nonlinear Schrodinger equation of repulsive type (NLS-) determines only pseudo-spherical surfaces (PSS). This implies that, though we show that given two differential PSS (resp. SS) equationsthere exists a local gauge transformation (despite of changing the independent variables or not) which trans-forms a solution of one into any solution of the other, it is impossible to have such a gauge transformationbetween the NLS+ and the NLS-.     

Complete space-like submanifolds in locally symmetric semi-definite spaces

The purpose of this paper is to study complete space-like submanifolds with parallel mean curvature vector and flat normal bundle in a locally symmetric semi-defnite space satisfying some curvature conditions. We first give an optimal estimate of the Laplacian of the squared norm of the second fundamental form for such submanifold. Furthermore, the totally umbilical submanifolds are characterized     

Remarks on two theorems of Qi-Keng Lu

Two alternate arguments in the framework of intrinsic metrics and measures respectively of part of the proof of a famous theorem due to Qi-Keng Lu on Bergman metric with constant negative holomorphic sectional curvature are presented.A relationship between the Lu constant and the holo- morphic sectional curvature of the Bergman metric is given.Some recent progress of the Yau's porblem on the characterization of domain of holomorphy on which the Bergman metric is K(?)hler-Einstein is described.