In this paper, we mainly focus on the Riesz means of eigenvalues of the subelliptic Laplacian on the Heisenberg group Hn. We establish a trace formula of associated eigenvalues, then we prove differential inequalities...In this paper, we mainly focus on the Riesz means of eigenvalues of the subelliptic Laplacian on the Heisenberg group Hn. We establish a trace formula of associated eigenvalues, then we prove differential inequalities, difference inequalities and monotonicity formulas for the Riesz means of eigenvalues of the subelliptic Laplacian.展开更多
In this paper, we are concerned with the Riesz means of Dirichlet eigenvalues for the sub-Laplace operator on the Engel group and deriver different inequalities for Riesz means. The Weyl-type estimates for means of ei...In this paper, we are concerned with the Riesz means of Dirichlet eigenvalues for the sub-Laplace operator on the Engel group and deriver different inequalities for Riesz means. The Weyl-type estimates for means of eigenvalues are given.展开更多
LetΔ<sub>1</sub>（x;φ） be the error term of the first Riesz mean of the Rankin-Selberg problem. We study the higher power moments ofΔ<sub>1</sub>（x;φ） and derive an asymptotic formula fo...LetΔ<sub>1</sub>（x;φ） be the error term of the first Riesz mean of the Rankin-Selberg problem. We study the higher power moments ofΔ<sub>1</sub>（x;φ） and derive an asymptotic formula for the 3-rd, 4-th and 5-th power moments by using Ivic’s large value arguments and other techniques.展开更多
The aim of this paper is to state some conjectures and problems on Bochner-Riesz means in multiple Fourier series and integrals. The progress on these conjectures and problems are also mentioned.
The aim of this paper is to prove the a.e.convergence of sequences of the Cesaro and Riesz means of the Walsh–Fourier series of d variable integrable functions.That is,let a=（a1,...,ad）：N→Nd（d∈P）such that aj（...The aim of this paper is to prove the a.e.convergence of sequences of the Cesaro and Riesz means of the Walsh–Fourier series of d variable integrable functions.That is,let a=（a1,...,ad）：N→Nd（d∈P）such that aj（n＋1）≥δsupk≤n aj（k）（j=1,...,d,n∈N）for someδ〉0 and a1（＋∞）=···=ad（＋∞）=＋∞.Then,for each integrable function f∈L1（Id）,we have the a.e.relation for the Cesaro means limn→∞σαa（n）f=f and for the Riesz means limn→∞σα,γa（n）f=f for any 0〈αj≤1≤γj（j=1,...,d）.A straightforward consequence of our result is the so-called cone restricted a.e.convergence of the multidimensional Cesaro and Riesz means of integrable functions,which was proved earlier by Weisz.展开更多
Let G/K be the noncompact Riemannian symmetric space SL(3, H)/Sp(3). We shall prove in this paper that for f∈L^P(SL(3, H)/Sp(3)), 1≤p≤2. the Riesz means of order z of f with respect to the eigenfunctions expansion ...Let G/K be the noncompact Riemannian symmetric space SL(3, H)/Sp(3). We shall prove in this paper that for f∈L^P(SL(3, H)/Sp(3)), 1≤p≤2. the Riesz means of order z of f with respect to the eigenfunctions expansion of Laplace operator almost everywhere converge to f for Rez】б(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.展开更多
Ⅰ. INTRODUCTION Let ∑<sub>n</sub>= {u∈R<sup>n+1</sup>; |u|=1} be the unit sphere in （n+1）dimensional Euclidean space. For any functionf on ∑<sub>n</sub>, we associate its expa...Ⅰ. INTRODUCTION Let ∑<sub>n</sub>= {u∈R<sup>n+1</sup>; |u|=1} be the unit sphere in （n+1）dimensional Euclidean space. For any functionf on ∑<sub>n</sub>, we associate its expansion in spherical harmonics:展开更多
In this article,we investigate the bilinear Riesz means S^α associated with the sublaplacian on the Heisenberg group.We prove that the operator S^α is bounded from L^P1×L^P2 into Lp for 1≤p1,P2≤∞ and 1/p=1/p...In this article,we investigate the bilinear Riesz means S^α associated with the sublaplacian on the Heisenberg group.We prove that the operator S^α is bounded from L^P1×L^P2 into Lp for 1≤p1,P2≤∞ and 1/p=1/p1+l/p2 when a is larger than the suitable smoothness index α(p1,p2)-There are some essential differences bet ween the Euclidean space and the Heisenberg group for studying the bilinear Riesz means problem.We use some special techniques to obtain lower indices α(p1,p2).展开更多
设f∈L^p(R^n),1≤p≤2(n+1)/n+3,以及δ>n/p-(n+1)/2.本文证明了f在R^n上的Bochner-Riesz平均σR(f;x)满足关系式其中权函数w满足条件w(u)≥0以及1≤1/t integral from 0 to t(w(u)du≤C)(C为一绝对常数)。结论对周期情形也成立。
文摘In this paper, we mainly focus on the Riesz means of eigenvalues of the subelliptic Laplacian on the Heisenberg group Hn. We establish a trace formula of associated eigenvalues, then we prove differential inequalities, difference inequalities and monotonicity formulas for the Riesz means of eigenvalues of the subelliptic Laplacian.
文摘In this paper, we are concerned with the Riesz means of Dirichlet eigenvalues for the sub-Laplace operator on the Engel group and deriver different inequalities for Riesz means. The Weyl-type estimates for means of eigenvalues are given.
基金supported by the National Natural Science Foundation of China(Grant No.10301018)the Natural Science Foundation of Shandong Province(Grant No.2006A31)
文摘LetΔ<sub>1</sub>（x;φ） be the error term of the first Riesz mean of the Rankin-Selberg problem. We study the higher power moments ofΔ<sub>1</sub>（x;φ） and derive an asymptotic formula for the 3-rd, 4-th and 5-th power moments by using Ivic’s large value arguments and other techniques.
文摘The aim of this paper is to state some conjectures and problems on Bochner-Riesz means in multiple Fourier series and integrals. The progress on these conjectures and problems are also mentioned.
基金Supported by project TMOP-4.2.2.A-11/1/KONV-2012-0051
文摘The aim of this paper is to prove the a.e.convergence of sequences of the Cesaro and Riesz means of the Walsh–Fourier series of d variable integrable functions.That is,let a=（a1,...,ad）：N→Nd（d∈P）such that aj（n＋1）≥δsupk≤n aj（k）（j=1,...,d,n∈N）for someδ〉0 and a1（＋∞）=···=ad（＋∞）=＋∞.Then,for each integrable function f∈L1（Id）,we have the a.e.relation for the Cesaro means limn→∞σαa（n）f=f and for the Riesz means limn→∞σα,γa（n）f=f for any 0〈αj≤1≤γj（j=1,...,d）.A straightforward consequence of our result is the so-called cone restricted a.e.convergence of the multidimensional Cesaro and Riesz means of integrable functions,which was proved earlier by Weisz.
基金Partially supported by National Natural Science Foundation of China
文摘Let G/K be the noncompact Riemannian symmetric space SL(3, H)/Sp(3). We shall prove in this paper that for f∈L^P(SL(3, H)/Sp(3)), 1≤p≤2. the Riesz means of order z of f with respect to the eigenfunctions expansion of Laplace operator almost everywhere converge to f for Rez】б(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.
基金Project supported by the National Natural Science Foundation of China
文摘Ⅰ. INTRODUCTION Let ∑<sub>n</sub>= {u∈R<sup>n+1</sup>; |u|=1} be the unit sphere in （n+1）dimensional Euclidean space. For any functionf on ∑<sub>n</sub>, we associate its expansion in spherical harmonics:
基金supported by National Natural Science Foundation of China(Grant No.11371036)supported by China Scholarship Council(Grant No.201606010026)
文摘In this article,we investigate the bilinear Riesz means S^α associated with the sublaplacian on the Heisenberg group.We prove that the operator S^α is bounded from L^P1×L^P2 into Lp for 1≤p1,P2≤∞ and 1/p=1/p1+l/p2 when a is larger than the suitable smoothness index α(p1,p2)-There are some essential differences bet ween the Euclidean space and the Heisenberg group for studying the bilinear Riesz means problem.We use some special techniques to obtain lower indices α(p1,p2).
文摘设f∈L^p(R^n),1≤p≤2(n+1)/n+3,以及δ>n/p-(n+1)/2.本文证明了f在R^n上的Bochner-Riesz平均σR(f;x)满足关系式其中权函数w满足条件w(u)≥0以及1≤1/t integral from 0 to t(w(u)du≤C)(C为一绝对常数)。结论对周期情形也成立。
