Damek Ricci Spaces Research Articles

Titchmarsh theorems on Damek–Ricci spaces via moduli of continuity of higher order

A classical theorem of Titchmarsh relates the [Formula: see text]-Lipschitz functions and decay of the Fourier transform of the functions. In this note, we prove the Titchmarsh theorem for Damek–Ricci space (also known as harmonic [Formula: see text] groups) via moduli of continuity of higher orders. We also prove an analogue of another Titchmarsh theorem which provides integrability properties of the Fourier transform for functions in the Hölder Lipschitz spaces.

Sharp Hardy-type inequalities for non-compact harmonic manifolds and Damek–Ricci spaces

Sharp Hardy-type inequalities for non-compact harmonic manifolds and Damek–Ricci spaces

Direct and Inverse Approximation Theorems for Functions Defined in Damek–Ricci Spaces

Direct and Inverse Approximation Theorems for Functions Defined in Damek–Ricci Spaces

Regularity and pointwise convergence of solutions of the Schrödinger operator with radial initial data on Damek-Ricci spaces

Regularity and pointwise convergence of solutions of the Schrödinger operator with radial initial data on Damek-Ricci spaces

Direct and inverse approximation theorems for functions defined in Damek–Ricci spaces

UDC 517.5 We introduce the notion of k th modulus of smoothness and establish the direct and inverse theorems in terms of the quantities E s ( f ) and the moduli of smoothness generated by the spherical mean operator defined on the L 2 -space for the Damek–Ricci spaces. These theorems are analogous to the well-known theorems of Jackson and Bernstein. We also consider some problems related to the constructive characteristics of functional classes defined by the majorants of the moduli of smoothness of their elements.

Hartman-Watson distribution and hyperbolic-like heat kernels

We relate Gruet's formula for the heat kernel on real hyperbolic spaces to the commonly used one derived from Millson induction and distinguishing the parity of the dimensions. The bridge between both formulas is settled by Yor's result on the joint distribution of a Brownian motion and of its exponential functional at fixed time. This result allows further to relate Gruet's formula with real parameter to the heat kernel of the hyperbolic Jacobi operator and to derive a new integral representation for the heat kernel of the Maass Laplacian. When applied to harmonic AN groups (known also as Damek-Ricci spaces), Yor's result yields also a new integral representation of their corresponding heat kernels through the modified Bessel function of the second kind and the Hartman-Watson distribution. This newly obtained formula has the merit to unify both existing formulas in the same way Gruet's formula does for real hyperbolic spaces.

Dynamics of $ L^p $ multipliers on harmonic manifolds

<abstract><p>Let $ X $ be a complete, simply connected harmonic manifold of purely exponential volume growth. This class contains all non-flat harmonic manifolds of nonpositive curvature, and in particular all known examples of non-compact harmonic manifolds except for the flat spaces. We use the Fourier transform from <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup> to investigate the dynamics on $ L^p(X) $ for $ p > 2 $ of certain bounded linear operators $ T : L^p(X) \to L^p(X) $ which we call "$ L^p $-multipliers" in accordance with standard terminology. Examples of $ L^p $-multipliers are given by the operator of convolution with an $ L^1 $ radial function, or more generally convolution with a finite radial measure. In particular elements of the heat semigroup $ e^{t\Delta} $ act as multipliers. Given $ 2 < p < \infty $, we show that for any $ L^p $-multiplier $ T $ which is not a scalar multiple of the identity, there is an open set of values of $ \nu \in {\mathbb C} $ for which the operator $ \frac{1}{\nu} T $ is chaotic on $ L^p(X) $ in the sense of Devaney, i.e., topologically transitive and with periodic points dense. Moreover such operators are topologically mixing. We also show that there is a constant $ c_p > 0 $ such that for any $ c \in {\mathbb C} $ with $ \operatorname{Re} c > c_p $, the action of the shifted heat semigroup $ e^{ct} e^{t\Delta} $ on $ L^p(X) $ is chaotic. These results generalize the corresponding results for rank one symmetric spaces of noncompact type and harmonic $ NA $ groups (or Damek-Ricci spaces).</p></abstract>

Asymptotic mean value property for eigenfunctions of the Laplace–Beltrami operator on Damek–Ricci spaces

Let S be a Damek–Ricci space equipped with the Laplace–Beltrami operator \(\Delta \). In this paper, we characterize all eigenfunctions of \(\Delta \) through sphere, ball and shell averages as the radius (of sphere, ball or shell) tends to infinity.

Symmetries of four-dimensional Lorentzian Damek-Ricci spaces

We consider the four-dimensional Damek-Ricci spaces, equipped with theleft-invariant Lorentzian metric. We obtain a full classification of Killing andaffine vector fields as well as Ricci, curvature and matter collineations. Inparticular, we prove the non-existence of proper (that is non Killing) affinevector field

Lipschitz Conditions in Damek–Ricci Spaces

Lipschitz Conditions in Damek–Ricci Spaces

Codimension one Ricci soliton subgroups of solvable Iwasawa groups

Recently, Jablonski proved that, to a large extent, a simply connected solvable Lie group endowed with a left-invariant Ricci soliton metric can be isometrically embedded into the solvable Iwasawa group of a non-compact symmetric space. Motivated by this result, we classify codimension one subgroups of the solvable Iwasawa groups of irreducible symmetric spaces of non-compact type whose induced metrics are Ricci solitons. We also obtain the classifications of codimension one Ricci soliton subgroups of Damek-Ricci spaces and generalized Heisenberg groups.

Einstein hypersurfaces of Damek-Ricci spaces

Einstein hypersurfaces are “very rare” in rank-one symmetric spaces. Damek-Ricci spaces may be viewed as the closest and the most natural generalisations of noncompact rank-one symmetric spaces. We prove that no Damek-Ricci space admits an Einstein hypersurface.

Polar actions on Damek-Ricci spaces

A proper isometric Lie group action on a Riemannian manifold is called polar if there exists a closed connected submanifold which meets all orbits orthogonally. In this article we study polar actions on Damek-Ricci spaces. We prove criteria for isometric actions on Damek-Ricci spaces to be polar, find examples and give some partial classifications of polar actions on Damek-Ricci spaces. In particular, we show that non-trivial polar actions exist on all Damek-Ricci spaces.

Explicit [formula omitted]-harmonic functions on rank-one Lie groups of Iwasawa type

We construct explicit proper p-harmonic functions on rank-one Lie groups of Iwasawa type. This class of Lie groups includes many classical Riemannian manifolds such as the rank-one symmetric spaces of non-compact type, Damek–Ricci spaces, rank-one Einstein solvmanifolds and Carnot spaces.

A note on [formula omitted]-functional, Modulus of smoothness, Jackson theorem and Bernstein–Nikolskii–Stechkin inequality on Damek–Ricci spaces

In this paper we study approximation theorems for L2-space on Damek–Ricci spaces. We prove direct Jackson theorem of approximations for the modulus of smoothness defined using spherical mean operator on Damek–Ricci spaces. We also prove Bernstein–Nikolskii–Stechkin inequality. To prove these inequalities we use functions of bounded spectrum as a tool of approximation. Finally, as an application we prove equivalence of the K-functional and modulus of smoothness for Damek–Ricci spaces.

Gabor Transform and Donoho–Stark’s U.P. in NA-Groups

Using the basic properties of Damek–Ricci spaces and its Fourier transform, we introduce the Gabor transform and study some of its properties. We will generalize the uncertainty principle of Donoho–Stark to this solvable extension of H-type group.

Totally geodesic submanifolds of Damek–Ricci spaces

We classify totally geodesic submanifolds of Damek–Ricci spaces and show that they are either subgroups (“smaller” Damek–Ricci spaces) or isometric to rank-one symmetric spaces of negative curvature.

Radon Transform on a Harmonic Manifold

We extend to a large class of noncompact harmonic manifolds the inversion formulas for the Radon transform on horospheres in hyperbolic spaces or Damek–Ricci spaces. Horospheres are defined here as level hypersurfaces of Busemann functions. The proof uses harmonic analysis on the manifolds considered, developed in a recent paper by Biswas, Knieper and Peyerimhoff; we also give a concise proof of their Fourier inversion theorem for harmonic manifolds.

Littlewood–Paley–Stein operators on Damek–Ricci spaces

AbstractWe obtain pointwise upper bounds on the derivatives of the heat kernel on Damek–Ricci spaces and we study the ‐boundedness of Littlewood–Paley–Stein operators.

Characterization of eigenfunctions of the Laplace–Beltrami operator through heat propagation in small time

On rank one Riemannian symmetric spaces of noncompact type, which accommodates all hyperbolic spaces, we show that characterization of the eigenfunctions/eigendistributions of the Laplace–Beltrami operator, through the action of the heat operator is possible only when we confine in a small time. The results are different from their counter parts in the Euclidean spaces. All results and their proofs extend to Damek–Ricci spaces.