首页> 外文OA文献 >Uniform Sobolev Resolvent Estimates for the Laplace-Beltrami Operator on Compact Manifolds
【2h】

Uniform Sobolev Resolvent Estimates for the Laplace-Beltrami Operator on Compact Manifolds

机译:Laplace-Beltrami算子的均匀sobolev预估估计   紧凑型歧管

代理获取
本网站仅为用户提供外文OA文献查询和代理获取服务,本网站没有原文。下单后我们将采用程序或人工为您竭诚获取高质量的原文,但由于OA文献来源多样且变更频繁,仍可能出现获取不到、文献不完整或与标题不符等情况,如果获取不到我们将提供退款服务。请知悉。

摘要

In this paper we continue the study on the resolvent estimates of theLaplace-Beltrami operator $\Delta_g$ on a compact manifolds $M$ with dimension$n\geq3$. On the Sobolev line $1/p-1/q=2/n$ we can prove that the resolvent$(\Delta_g+\zeta)^{-1}$ is uniformly bounded from $L^p$ to $L^q$ when $(p,q)$are within the admissible range $p\leq2(n+1)/(n+3)$ and $q\geq2(n+1)/(n-1)$ and$\zeta$ is outside a parabola opening to the right and a small disk centered atthe origin. This naturally generalizes the previous results in \cite{Kenig} and\cite{bssy} which addressed only the special case when $p=2n/(n+2),q=2n/(n-2)$. Using the shrinking spectral estimates between $L^p$ and $L^q$ wealso show that when $(p,q)$ are within the interior of the admissible range,one can obtain a logarithmic improvement over the parabolic region forresolvent estimates on manifolds equipped with Riemannian metric ofnon-positive sectional curvature, and a power improvement depending on theexponent $(p,q)$ for flat torus. The latter therefore partially improves Shen'swork in \cite{Shen} on the $L^p\to L^2$ uniform resolvent estimates on thetorus. Similar to the case as proved in \cite{bssy} when$(p,q)=(2n/(n+2),2n/(n-2))$, the parabolic region is also optimal over theround sphere $S^n$ when $(p,q)$ are now in the admissible range. However, wemay ask if the admissible range is sharp in the sense that it is the onlypossible range on the Sobolev line for which a compact manifold can haveuniform resolvent estimate for $\zeta$ being ouside a parabola.
机译:在本文中,我们继续研究在维数为$ n \ geq3 $的紧流形$ M $上的Laplace-Beltrami算子$ \ Delta_g $的分解估计。在Sobolev线$ 1 / p-1 / q = 2 / n $上,我们可以证明可分解的对象$(\ Delta_g + \ zeta)^ {-1} $从$ L ^ p $均匀地绑定到$ L ^ q $当$(p,q)$在允许范围$ p \ leq2(n + 1)/(n + 3)$和$ q \ geq2(n + 1)/(n-1)$和$ \ zeta范围内时$位于抛物线的右侧开口,小圆盘位于原点中心。这自然地将\ cite {Kenig}和\ cite {bssy}中的先前结果归纳为仅解决$ p = 2n /(n + 2),q = 2n /(n-2)$的特殊情况。使用在$ L ^ p $和$ L ^ q $之间的收缩谱估计,我们还表明,当$(p,q)$在允许范围内时,对于抛物线估计,可以得到抛物线区域的对数改进。歧管配备了非正截面曲率的黎曼度量,并且功率的提高取决于平坦圆环的指数$(p,q)$。因此,后者可以部分地将$ L ^ p \上的\ cite {Shen}中的Shen的工作改进为在耳模上的L ^ 2 $统一分解估计。与\ cite {bssy}中证明的情况类似,当$ {p,q)= {2n /(n + 2),2n /(n-2))$时,抛物线区域在圆球$ S上也是最优的现在$(p,q)$在允许范围内时,^ n $。但是,我们可能会问,在Sobolev线中唯一的可能范围是紧凑的流形可以针对$ \ zeta $被抛物线抛弃的统一解析度估算的,从这个意义上说,允许范围是否较窄。

著录项

  • 作者

    Shao, Peng; Yao, Xiaohua;

  • 作者单位
  • 年度 2013
  • 总页数
  • 原文格式 PDF
  • 正文语种 {"code":"en","name":"English","id":9}
  • 中图分类

相似文献

  • 外文文献
  • 中文文献
  • 专利
代理获取

客服邮箱:kefu@zhangqiaokeyan.com

京公网安备:11010802029741号 ICP备案号:京ICP备15016152号-6 六维联合信息科技 (北京) 有限公司©版权所有
  • 客服微信

  • 服务号