Muna Naik, Rudra P. Sarkar
Let $S$ be a Damek-Ricci space equipped with the Laplace-Beltrami operator $Δ$. In this paper we characterize all eigenfunctions of $Δ$ through sphere, ball and shell averages as the radius (of sphere, ball or shell) tends to infinity.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We give optimal bounds for the radial, space and time derivatives of arbitrary order of the heat kernel of the Laplace--Beltrami operator on Damek--Ricci spaces. In the case of symmetric spaces of rank one, these complete and actually improve conjectured estimates by Anker and Ji. We also provide asymptotics at infinity of all the radial and time derivates of the kernel. Along the way, we provide sharp bounds for all the derivatives of the Riemannian distance and obtain analogous bounds for those of the heat kernel of the distinguished Laplacian.
Spectral projections and resolvent estimates on Damek-Ricci spaces and their applications
该研究在Damek-Ricci空间上证明了Laplace-Beltrami算子谱投影与预解式的Lp-Lp'有界性，通过建立球函数逐点精确界完成证明，并将结果应用于复值势Schrödinger算子的特征值界研究。
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…
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 $λ\in\R\setminus\{0\}$.
We explore the possibilities of reaching the characterization of eigenfunction of Laplacian as a degenerate case of the inverse Paley-Wiener theorem (characterizing functions whose Fourier transform is supported on a compact annulus) for the Riemannian symmetric spaces of noncompact type. Most distinguished prototypes of these spaces are the hyperbolic spaces. The statement and the proof of the main result work mutatis-mutandis for a number of spaces including Euclidean spaces and Damek-Ricci spaces.
Eigenfunction equivalence for the fractional Laplace-Beltrami operator and the classical Helmholtz equation
本文建立了分数次Laplace-Beltrami算子与经典Helmholtz方程的本征函数等价性，基于Seeley的椭圆算子复幂构造与伪微分符号演算完成证明，并将其应用于固定频率逆散射问题。
We prove an uncertainty principle for certain eigenfunction expansions on $ L^2(\mathbb{R}^+,w(r)dr) $ and use it to prove analogues of theorems of Chernoff and Ingham for Laplace-Beltrami operators on compact symmetric spaces, special Hermite operator on $ \mathbb{C}^n $ and Hermite operator on $ \mathbb{R}^n.$
Mapping properties of the Schrödinger maximal function on Damek--Ricci spaces
本文研究了 Damek--Ricci 空间上 Schrödinger 极大函数的映射性质，完整刻画了其在测地球上的局部 $L^q$ 估计对，并证明了相关的全局弱型估计，结果与欧氏情形一致且具有尖锐性。
In \cite{Roe} Roe proved that if a doubly-infinite sequence $\{f_k\}$ of functions on $\R$ satisfies $f_{k+1}=(df_{k}/dx)$ and $|f_{k}(x)|\leq M$ for all $k=0,\pm 1,\pm 2,...$ and $x\in \R$, then $f_0(x)=a\sin(x+φ)$ where $a$ and $φ$ are real constants. This result was extended to $\R^n$ by Strichartz \cite{Str} where $d/dx$ is substituted by the Laplacian on $\R^n$. While it is plausible to extend this theorem for other Riemannian manifolds or Lie groups, Strichartz showed that the result holds true for Heisenberg groups, but fails for hyperbolic 3-space. This negative result can be indeed ex…
Nodal sets and continuity of eigenfunctions of Krein-Feller operators on Riemannian manifolds
该研究针对黎曼流形上的Kreĭn-Feller算子，在特征函数连续假设下证明了Courant节点域定理，并在不同流形区域条件下证明了其特征函数的连续性性质。
In this paper we study the almost everywhere convergence of the expansions related to the self-adjoint extension of the Laplace-Beltrami operator on the unit sphere. The sufficient conditions for summability is obtained. The more general properties and representation by the eigenfunctions of the Laplace-Beltrami operator of the Liouville space $L_1^\a$ is used. For the orders of Riesz means, which greater than critical index $\frac{N-1}{2}$ we proved the positive results on summability of Fourier-Laplace series. Note that when order $α$ of Riesz means is less than critical index then for estab…
In this paper we obtain some results of harmonic analysis on quantum complex hyperbolic spaces. We introduce a quantum analog for the Laplace-Beltrami operator and its radial part. The latter appear to be second order $q$-difference operator, whose eigenfunctions are related to the Al-Salam-Chihara polynomials. We prove a Plancherel type theorem for it.
Inclusions and noninclusions of Hardy type spaces on certain nondoubling manifolds
本文研究具有Ricci曲率下界、正单射半径和谱隙的非紧Riemann流形上各类Hardy型空间的包含与非包含关系，证明Riesz-Hardy空间与Goldberg型空间的同构关系及Goldberg型空间无原子刻画，并在Damek-Ricci空间上证明三类Hardy空间互不相同。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。