arXiv 0806.29212008-06-18math.CAmath.AP

Damek-Ricci空间上的Wave equation和乘子估计

Wave equation and multiplier estimates on Damek-Ricci spaces

Detlef Mueller, Maria Vallarino

摘要：

设S是Damek-Ricci空间，L是S上的一个特殊的左不变Laplacian.我们证明了与L相关的谱局域波算子的卷积核的逐点估计。这推广了D. Mueller和C. Thiele关于ax+ b-群。我们还证明了这些卷积核的梯度的逐点估计。作为推论，我们重新证明了W. Hebisch和T. Steger和M. Vallarino通过不同的方法。最后，我们得到了与L相关的波动方程解的Sobolev估计。

英文摘要：

Let S be a Damek-Ricci space and L be a distinguished left invariant Laplacian on S. We prove pointwise estimates for the convolution kernels of spectrally localized wave operators associated with L. This generalizes previous results proved by D. Mueller and C. Thiele on ax+b-groups. We also prove pointwise estimates of the gradient of these convolution kernels. As a corollary we reprove basic multiplier estimates from previous papers of W. Hebisch and T. Steger and M. Vallarino by different methods. Finally we derive Sobolev estimates for the solution to the wave equation associated with L.

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。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

摘要英文摘要相关论文

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

相关论文

The wave equation on Damek-Ricci spaces

We study the dispersive properties of the wave equation associated with the shifted Laplace-Beltrami operator on Damek-Ricci spaces, and deduce Strichartz estimates for a large family of admissible pairs. As an application, we obtain global well-posedness results for the nonlinear wave equation.

2010-12-03math.APmath.CA

Schrödinger equation on Damek-Ricci spaces

In this paper we consider the Laplace-Beltrami operator Δon Damek-Ricci spaces and derive pointwise estimates for the kernel of exp(τΔ), when τ\in C* with Re(τ) \geq 0. When τ\in iR*, we obtain in particular pointwise estimates of the Schrödinger kernel associated with Δ. We then prove Strichartz estimates for the Schrödinger equation, for a family of admissible pairs which is larger than in the Euclidean case. This extends the results obtained by Anker and Pierfelice on real hyperbolic spaces. As a further application, we study the dispersive properties of the Schrödinger equation associated …

2010-10-11math.APmath.CA

A note on $K$-functional, Modulus of smoothness, Jackson theorem and Nikolskii-Stechkin inequality on Damek-Ricci spaces

In this paper we study approximation theorems for $L^2$-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 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.

2020-05-04math.CA

Optimal heat kernel bounds and asymptotics on Damek--Ricci spaces

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.

2022-10-05math.APmath.CAmath.FA

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 $Δ$. 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.

2020-05-06math.CAmath.FA

Damek-Ricci 空间上具有径向初值的薛定谔算子解的 regularity 与逐点收敛性

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

本文研究 Damek-Ricci 空间上径向初值薛定谔方程的 Carleson 问题，证明了解逐点收敛的最优正则性指标为 $\alpha \ge 1/4$，与欧几里得情形一致。

2024-07-18math.APmath.CA

更多相关论文收起更多论文

Damek-Ricci空间上区分拉普拉斯算子的里斯变换及算子值多变量谱乘子

Riesz transforms for the distinguished Laplacian on Damek-Ricci spaces and operator-valued multivariate spectral multipliers

Northwestern Polytechnical University（西北工业大学）；Politecnico di Torino（都灵理工大学）

本文研究了Damek-Ricci空间上区分拉普拉斯算子的里斯变换的L^p有界性，并通过算子值谱乘子定理证明了其在p∈(1,∞)范围内的有界性。

2026-02-02math.CAmath.FA

关于Damek-Ricci空间上波动方程解的增长估计

Growth Estimates for Solutions to the Wave Equation on Damek--Ricci Spaces

Sun Yat-sen University（中山大学）；Harbin Institute of Technology（哈尔滨工业大学）；Universität Paderborn（帕德博恩大学）

本文研究了Damek-Ricci空间上波动方程解的增长估计，建立了精确的L^p界并解决了相关猜想。

2026-09-01math.APmath.CA

On a Class of Non-Integrable Multipliers for the Jacobi Transform

We show that a bounded function $m$ on $\R$ not necessarily integrable at infinity may still yield $L^p$-bounded convolution operators for the Jacobi transform if the nontangential boundary values of $ωm$ along the edges of a certain strip in $\C$ yield Euclidean Fourier multipliers, for $ω$ suitably defined. This partially generalizes similar results by Giulini, Mauceri, and Meda (on rank one symmetric spaces) and Astengo (on Damek--Ricci spaces).

2011-08-18math.CA

$L^p$-results for fractional integration and multipliers for the Jacobi transform

We use precise asymptotic expansions for Jacobi functions $ϕ^{(α,β)}_λ$ parameters $α$, $β$ satisfying $α>1/2$, $α>β>-1/2$, to generalizing classical Hörmander-type multiplier theorem for the spherical transform on a rank one Riemannian symmetric space (by Clerc/Stein and Stanton/Tomas) to the framework of Jacobi analysis. In particular, multiplier results for the spherical transform on Damek--Ricci spaces are subsumed by this approach, and it yields multiplier results for the hypergeometric `Heckman--Opdam transform' associated with a rank one root system. We obtain near-optimal $L^p-L^q$ est…

2011-08-17math.CA

$L^p$-$L^q$ Fourier乘子与非紧型黎曼对称空间上的Hausdorff-Young-Paley不等式

$L^p$-$L^q$ Fourier multipliers and Hausdorff-Young-Paley inequalities on Riemannian symmetric spaces of noncompact type

本文在任意秩非紧型黎曼对称空间上建立Hörmander型Lp-Lq Fourier乘子定理，推导谱乘子定理，证明Laplace-Beltrami算子函数的有界性，并给出Hausdorff-Young和Paley不等式的混合范数版本，结果适用于Damek-Ricci空间。

2024-11-06math.CAmath.FA

Flag Hardy spaces and Marcinkiewicz multipliers on the Heisenberg group: an expanded version

Marcinkiewicz multipliers are L^{p} bounded for 1<p<\infty on the Heisenberg group H^{n}\simeqC^{n}\timesR (D. Muller, F. Ricci and E. M. Stein) despite the lack of a two parameter group of automorphic dilations on H^{n}. This lack of dilations underlies the inability of classical one or two parameter Hardy space theory to handle Marcinkiewicz multipliers on H^{n} when 0<p\leq1. We address this deficiency by developing a theory of flag Hardy spaces H_{flag}^{p} on the Heisenberg group, 0<p\leq1, that is in a sense `intermediate' between the classical Hardy spaces H^{p} and the product Hardy sp…

2012-08-13math.CAmath.CV

某些非倍测度流形上Hardy型空间的包含与非包含关系

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。

2022-07-06math.CAmath.FA