In this paper we study Markov semigroups generated by Hörmander-Dunkl type operators on Heisenberg group.

On a Lie group NA that is a split extension of a nilpotent Lie group N by a one-parameter group of automorphisms A, the heat semigroup ${\mu }_t$ generated by a second order subelliptic left-invariant operator ${\sum }_{j=0}^m{Y}_j+Y$ is considered. Under natural conditions there is a $\mu {̌}_t$-invariant measure m on N, i.e. $\mu {̌}_t*m=m$. Precise asymptotics of m at infinity is given for a large class of operators with Y₀,...,Yₘ generating the Lie algebra of S.

The asymptotics of spherical functions for large dimensions are related to spherical functions for Olshanski spherical pairs. In this paper we consider inductive limits of Gelfand pairs associated to the Heisenberg group. The group K = U(n) × U(p) acts multiplicity free on 𝓟(V), the space of polynomials on V = M(n,p;ℂ), the space of n × p complex matrices. The group K acts also on the Heisenberg group H = V × ℝ. By a result of Carcano, the pair (G,K) with G = K ⋉ H is a Gelfand pair. The main results...

We examine the asymptotic, or large-time, behaviour of the semigroup kernel associated with a finite sum of homogeneous subcoercive operators acting on a connected Lie group of polynomial growth. If the group is nilpotent we prove that the kernel is bounded by a convolution of two Gaussians whose orders correspond to the highest and lowest orders of the homogeneous subcoercive components of the generator. Moreover we establish precise asymptotic estimates on the difference of the kernel and the...

For any connected Lie group G and any Laplacian Λ = X²₁ + ⋯ + X²ₙ ∈ 𝔘𝔤 (X₁,...,Xₙ being a basis of 𝔤) one can define the commutant 𝔅 = 𝔅(Λ) of Λ in the convolution algebra ℒ¹(G) as well as the commutant ℭ(Λ) in the group C*-algebra C*(G). Both are involutive Banach algebras. We study these algebras in the case of a "distinguished Laplacian" on the "Iwasawa part AN" of a semisimple Lie group. One obtains a fairly good description of these algebras by objects derived from the semisimple group....

In the paper we investigate the absolute convergence in the sup-norm of Harish-Chandra's Fourier series of functions belonging to Besov spaces defined on non-compact connected Lie groups.

Let $X$ be an arbitrary hyperbolic geodesic metric space and let $\Gamma$ be a countable subgroup of the isometry group $\mathrm{Iso}\left(X\right)$ of $X$. We show that if $\Gamma$ is non-elementary and weakly acylindrical (this is a weak properness condition) then the second bounded cohomology groups $H_b^2(\Gamma ,ℝ)$, $H_b^2(\Gamma ,{\ell }^p\left(\Gamma \right))$$(1<...$

Let $G$ be the Lie group ${ℝ}^2⋉{ℝ}^{+}$ endowed with the Riemannian symmetric space structure. Let $X_0,X_1,X_2$ be a distinguished basis of left-invariant vector fields of the Lie algebra of $G$ and define the Laplacian $\Delta =-(X_0^2+X_1^2+X_2^2)$. In this paper we consider the first order Riesz transforms $R_i=X_i{\Delta }^{-1/2}$ and $S_i={\Delta }^{-1/2}{X}_i$, for $i=0,1,2$. We prove that the operators $R_i$, but not the $S_i$, are bounded from the Hardy space $H^1$ to $L^1$. We also show that the second-order Riesz transforms $T_{ij}=X_i{\Delta }^{-1}{X}_j$ are bounded from $H^1$ to $L^1$, while the transforms $S_{ij}={\Delta }^{-1}{X}_i{X}_j$ and $R_{ij}=X_i{X}_j{\Delta }^{-1}$, for $i,j=0,1,2$, are not.