A multiplier theorem for the sublaplacian on the Heisenberg group

A multiplier theorem for the sublaplacian on the Heisenberg group is proved using Littlewood-Paley-Stein theory ofg-functions.     

Bilinear Riesz means on the Heisenberg group

Science China Mathematics - In this article, we investigate the bilinear Riesz means Sα associated with the sublaplacian on the Heisenberg group. We prove that the operator Sα is bounded...     

Differential inequalities for Riesz means and Weyl-type bounds for eigenvalues

We derive differential inequalities and difference inequalities for Riesz means of eigenvalues of the Dirichlet Laplacian,

     

Frames and Riesz bases for shift invariant spaces on the abstract Heisenberg group

Let $G$ be a second countable locally compact abelian group. The aim of this paper is to characterize the left translates on the Heisenberg group $\mathbb{H}\left(G\right)$ to be frames and Riesz bases in terms of the group Fourier transform.     

Almost everywhere convergence of Riesz means on noncompact symmetric spaceSL(3,H)/Sp(3)

LetG/K be the noncompact Riemannian symmetric spaceSL(3,H)/Sp(3). We shall prove in this paper that forf∈L ^p(SL(3,H)/Sp(3)), 1≤p≤2, the Riesz means of orderz off with respect to the eigenfunctions expansion of Laplace operator almost everywhere converge tof for Rez >δ(n,p). The critical index δ(n,p) is the same as in the classical Stein's result for Euclidean space, and as in the noncompact symmetric spaces of rank one and of complex type. Partially supported by National Natural Science Foundation of China     

Poisson transform for the Heisenberg group and eigenfunctions of the sublaplacian

We define an analogue of Poisson transform on the Heisenberg group and use it to characterise joint eigenfunctions of the sublaplacian and T=i∂_t in terms of certain analytic functionals.     

流形上的Riesz变换

设 Ｍ为一完备 Ｒｉｅｍａｎｎ流形, Ｓｔｒｉｃｈａｒｔｚ Ｒ． Ｓ, Ｌｏｈｏｕｅ Ｎ．, Ｂａｋｒｙ Ｄ．及作者等建立了 Ｍ上 Ｒｉｅｓｚ变换Ｒ的 Ｌ~ｐ（１＜ Ｐ＜ ∞）与弱型（１,１）有界性．本文将用分析的方法对曲率非负的流形建立Ｒ的Ｌ*－有界性．     

Shannon multiresolution analysis on the Heisenberg group

We present a notion of frame multiresolution analysis on the Heisenberg group, abbreviated by FMRA, and study its properties. Using the irreducible representations of this group, we shall define a sinc-type function which is our starting point for obtaining the scaling function. Further, we shall give a concrete example of a wavelet FMRA on the Heisenberg group which is analogous to the Shannon MRA on R.     

紧对称空间上Riesz平均的强逼近

本文在紧对称空间中给出了全测度集上Ｒｉｅｓｚ平均强逼近的阶     

On fundamental solution for powers of the sub-Laplacian on the Heisenberg group

We discuss the fundamental solution for m-th powers of the sub-Laplacian on the Heisenberg group. We use the representation theory of the Heisenberg group to analyze the associated m-th powers of the sub-Laplacian and to construct its fundamental solution. Besides, the series representation of the fundamental solution for square of the sub-Laplacian on the Heisenberg group is given and we also get the closed form of the fundamental solution for square of the sub-Laplacian on the Heisenberg group with dimension n=2,3,4.     

由schr(o)dinger算子定义的Riesz变换的Lp估计

本文主要讨论了当非负位势V(x)属于某逆Holder类时,由一致椭圆算子L=-div(A(x)(△))+V(x)所定义的Riesz变换在Lp空间的有界性.     

Fourier-Jacobi展开的Riesz平均

对ｆ∈Ｌｐ（Ｒ＋，Δ（ｔ）ｄｔ），Δ（ｔ）＝（２ｓｉｎｈｔ）２α＋１（２ｃｏｓｈｔ）２β＋１，１ｐ２，本文证明了当Ｒｅｚ＞（２／ｐ－１）（α＋１／２）时，ｆ的Ｆｏｕｒｉｅｒ－Ｊａｃｏｂｉ展开的ｚ－阶Ｒｉｅｓｚ平均几乎处处收敛于ｆ．该结果推广了Ｇｉｕｌｉｎｉ和Ｍａｕｃｅｒｉ在实秩为１的对称空间上的相应结果     

齐次群上的限制算子与Riesz平均

主要讨论一类齐次群上限制算子的映照性质,并且应用所得结果证明这类齐次群上Riesz平均的混合范数有界性。     

The Fundamental Solutions of the Space, Space-Time Riesz Fractional Partial Differential Equations with Periodic Conditions

In this paper, the space-time Riesz fractional partial differential equations with periodic conditions are considered. The equations are obtained from the integral partial differential equation by replacing the time derivative with a Caputo fractional derivative and the space derivative with Riesz potential. The fundamental solutions of the space Riesz fractional partial differential equation (SRFPDE) and the space-time Riesz fractional partial differential equation (STRFPDE) are discussed, respectively. Using methods of Fourier series expansion and Laplace transform, we derive the explicit expressions of the fundamental solutions for the SRFPDE and the STRFPDE, respectively.     

On Weighted Norms of Riesz Transforms Equal to One

We consider the vector Riesz transform ^t-(t+s)/2 div^s of even order s + t in the weighted space L ₂(ⁿ;|x|^a). We establish that for t s, n >3 its norm is equal to one on some interval of values of a, while inside the interval a stronger estimate for a subordinate norm is valid.     

局部紧Abel群上的平移Riesz基和Riesz谱集

设G是局部紧Abel群,Ω■G是Haar可测集,L~2(Ω)是Ω上Haar平方可积函数构成的Hilbert空间,PW_Ω(G):={f∈L~2(G):supp f(ξ)■Ω}是G上的Paley-Wiener空间.本文研究Paley-Wiener空间PW_Ω(G)上平移Riesz基和Riesz谱集Ω之间的关系.     

由Schr?dinger算子定义的Riesz变换的Lp估计

本文主要讨论了当非负位势 V(x)属于某逆Holder类时,由一致椭圆算子L=-div(A(x))+V(x)所定义的 Riesz变换在 L^p空间的有界性。