Let $G $ be a noncompact semisimple Lie group with finite centre. Let $X=G/K$ be the associated Riemannian symmetric space and assume that $X$ is of rank one. The spectral projections associated to the Laplace-Beltrami operator are given by $P_λf =f\ast Φ_λ$, where $Φ_λ$ are the elementary spherical functions on $X$. In this paper, we prove an Ingham type uncertainty principle for $P_λf$. Moreover, similar results are obtained in the case of spectral projections associated to Dunkl Laplacian.
Let $X=G/K$ be a symmetric space of the non-compact type. We prove that the mean value operator over translated $K$-orbits of a fixed point is surjective on the space of smooth functions on $X$ if $X$ is either complex or of rank one. For higher rank spaces it is shown that the same statement is true for points in an appropriate Weyl subchamber.
In this note we prove various sharp boundedness results on suitable Hardy type spaces for Riesz transforms of arbitrary order on noncompact symmetric spaces of arbitrary rank.
Let X be a noncompact symmetric space of rank one and let h^1(X) be a local atomic Hardy space. We prove the boundedness from h^1(X) to L^1(X) and on h^1(X) of some classes of Fourier integral operators related to the wave equation associated with the Laplacian on X and we estimate the growth of their norms depending on time.
We establish sharp Adams type inequalities on Sobolev spaces $W^{α, n/α}(X)$ of any fractional order $α< n$ on Riemannian symmetric space $X$ of noncompact type with dimension $n$ and of arbitrary rank. We also establish sharp Hardy-Adams inequalities on the Sobolev spaces $W^{n/2, 2}(X)$. For the real hyperbolic spaces, such results were recently obtained by J. Li et al. (Trans. AMS, 2020). We use Fourier analysis on the symmetric spaces to obtain these results.
A result of Chernoff gives sufficient condition for an $L^2$-function on $\R^n$ to be quasi-analytic. This is a generalization of the classical Denjoy-Carleman theorem on $\R$ and of the subsequent work on $\R^n$ by Bochner and Taylor. In this note we endeavour to obtain an exact analogue of the result of Chernoff for $L^p, p\in [1,2]$ functions on the Riemannian symmetric spaces of noncompact type. No restriction on the rank of the symmetric spaces and no condition on the symmetry of the functions is assumed.
Let X=G/K be a symmetric space of noncompact type and let L be the Laplacian associated with a G-invariant metric on X. We show that the resolvent kernel of L admits a holomorphic extension to a Riemann surface depending on the rank of the symmetric space. This Riemann surface is a branched cover of the complex plane with a certain part of the real axis removed. It has a branching point at the bottom of the spectrum of L. It is further shown that this branching point is quadratic if the rank of X is odd, and is logarithmic otherwise. In case G has only one conjugacy class of Cartan subalgebras…
In 1975, P.R. Chernoff used iterates of the Laplacian on $\mathbb{R}^n$ to prove an $L^2$ version of the Denjoy-Carleman theorem which provides a sufficient condition for a smooth function on $\mathbb{R}^n$ to be quasi-analytic. In this paper, we prove an exact analogue of Chernoff's theorem for all rank one Riemannian symmetric spaces (of noncompact and compact types) using iterates of the associated Laplace-Beltrami operators.
In this paper we prove the Fourier restriction theorem for $p=2$ on Riemannian symmetric spaces of noncompact type with real rank one which extends the earlier result proved in \cite[Theorem 1.1]{KRS}. This result depends on the weak $L^2$ estimates of the Poisson transform of $L^2$ function. By using this estimate of the Poisson transform we also characterizes all weak $L^2$ eigenfunction of the Laplace--Beltrami operator of Riemannian symmetric spaces of noncompact type with real rank one and eigenvalue $-(λ^2+ρ^2)$ for $λ\in\R\setminus\{0\}$.
Let $G/K$ be an irreducible symmetric space where $G$ is a non-compact, connected Lie group and $K$ is a compact, connected subgroup. We use decay properties of the spherical functions to show that the convolution product of any $r=r(G/K)$ continuous orbital measures has its density function in $% L^{2}(G)$ and hence is an absolutely continuous measure with respect to Haar measure. The number $r$ is approximately the rank of $G/K$. For the special case of the orbital measures, $ν_{a_{i}}$, supported on the double cosets $Ka_{i}K$ where $a_{i}$ belongs to the dense set of regular elements, we p…
Building on MacDonald's formula for the distance from a rank-one projection to the set of nilpotents in $\mathbb{M}_n(\mathbb{C})$, we prove that the distance from a rank $n-1$ projection to the set of nilpotents in $\mathbb{M}_n(\mathbb{C})$ is $\frac{1}{2}\sec\left(\fracπ{\frac{n}{n-1}+2}\right)$. For each $n\geq 2$, we construct examples of pairs $(Q,T)$ where $Q$ is a projection of rank $n-1$ and $T\in\mathbb{M}_n(\mathbb{C})$ is a nilpotent of minimal distance to $Q$. Furthermore, we prove that any two such pairs are unitarily equivalent. We end by discussing possible extensions of these …
We will see that vectors in $\CC^n$ have natural analogs as rank 2 projections in $\RR^{2n}$ and that this association transfers many vector properties into properties of rank two projections on $\RR^{2n}$. We believe that this association will answer many open problems in $\CC^n$ where the corresponding problem in $\RR^n$ has already been answered - and vice versa. As a application, we will see that phase retrieval (respectively, phase retrieval by projections) in $\CC^n$ transfers to a variation of phase retrieval by rank 2 projections (respectively, phase retrieval by projections) on $\RR^{…
The purpose of this article is to prove a generalisation of the Besicovitch-Federer projection theorem about a characterisation of rectifiable and unrectifiable sets in terms of their projections. For an $m$-unrectifiable set $Σ\subset\mathbb{R}^n$ having finite Hausdorff measure and $\varepsilon>0$, we prove that for a mapping $f\in\mathcal{C}^1(U,\mathbb{R}^n)$ having constant, equal to $m$, rank of the Jacobian matrix there exists a mapping $f_\varepsilon$ whose rank of the Jacobian matrix is also constant, equal to $m$, such that $\|f_\varepsilon-f\|_{\mathcal{C}^1}<\varepsilon$ and $\math…
The Damek-Ricci spaces are solvable Lie groups and noncompact harmonic manifolds. The rank one Riemannian symmetric spaces of noncompact type sits inside it as a thin subclass. In this note we establish that for any Damek-Ricci space $S$, the heat semigroup generated by certain perturbation of the Laplace-Beltrami operator is {\em chaotic} on the Lorentz spaces $L^{p,q}(S)$, $2<p<\infty, 1\le q<\infty$ and subspace-chaotic on the weak $L^p$-spaces. We show that both the amount of perturbation and the range of $p$ are sharp. This generalizes a result in \cite{J-W} which proves that under identi…
Given a complex Hilbert space H and the von Neumann algebra L(H) of all bounded linear operators on H, we study the Grassmann manifold M of all projections in L(H) that have a fixed finite rank r. We take the Jordan-Banach triple theory approach which allows us to define a natural Levi-Civita connection on M. We identify the geodesics, compute the Riemann distance and prove some properties of M
Let $H$ be a complex Hilbert space whose dimension is not less than $3$ and let ${\mathcal F}_{s}(H)$ be the real vector space formed by all self-adjoint operators of finite rank on $H$. For every non-zero natural $k<\dim H$ we denote by ${\mathcal P}_{k}(H)$ the set of all rank $k$ projections. Let $H'$ be other complex Hilbert space of dimension not less than $3$ and let $L:{\mathcal F}_{s}(H)\to {\mathcal F}_{s}(H')$ be a linear operator such that $L({\mathcal P}_{k}(H))\subset {\mathcal P}_{m}(H')$ for some natural $k,m$ and the restriction of $L$ to ${\mathcal P}_{k}(H)$ is injective. If …
Given a complex Hilbert space H, we study the differential geometry of the manifold M of all projections in V:=L(H). Using the algebraic structure of V, a torsionfree affine connection $\nabla$ (that is invariant under the group of automorphisms of V) is defined on every connected component of M, which in this way becomes a symmetric holomorphic manifold that consists of projections of the same rank r, (0< r < \infty). We prove that M admits a Riemann structure if and only if M consists of projections that have the same finite rank r or the same finite corank, and in that case $\nabla$ is the …
The hierarchical (multi-linear) rank of an order-$d$ tensor is key in determining the cost of representing a tensor as a (tree) Tensor Network (TN). In general, it is known that, for a fixed accuracy, a tensor with random entries cannot be expected to be efficiently approximable without the curse of dimensionality, i.e., a complexity growing exponentially with $d$. In this work, we show that the ground state projection (GSP) of a class of unbounded Hamiltonians can be approximately represented as an operator of low effective dimensionality that is independent of the (high) dimension $d$ of the…
