论文标题
从光谱簇到紧凑型歧管的均匀分辨估计值
From spectral cluster to uniform resolvent estimates on compact manifolds
论文作者
论文摘要
众所周知，均匀的分解估计意味着光谱群集估计值。我们表明，在某些情况下，相反的情况也是如此。特别是，在紧凑型Riemannian歧管上，Sogge \ cite {MR930395}的通用光谱簇估计值{MR930395}，无边界上的complatian歧管上的运算符，直接直接暗示着均匀的Sobolev不平等，暗示了Santos Ferreira，kenig和salo，kenig and kenig and kenig and kenig and salo \ cite \ cite \ cite to f to Mr3200351}的参考。该观察结果还基于Smith--Sogge \ Cite {MR2316270}的光谱群集边界以及Smith，Koch和Tataru \ cite {MR2444443996}的频谱群集{MR231270}。我们还将Canzani和Galkowski \ cite {Canzani-Galkowski}的最新光谱群集界限转换为改进的分解界限。此外，我们表明，在扰动下，解决方案估计值稳定，并使用它来为具有单数电位的Schrödinger运营商建立统一的Sobolev和光谱群集不平等。
It is well known that uniform resolvent estimates imply spectral cluster estimates. We show that the converse is also true in some cases. In particular, the universal spectral cluster estimates of Sogge \cite{MR930395} for the Laplace--Beltrami operator on compact Riemannian manifolds without boundary directly imply the uniform Sobolev inequality of Dos Santos Ferreira, Kenig and Salo \cite{MR3200351}, without any reference to parametrices. This observation also yields new resolvent estimates for manifolds with boundary or with nonsmooth metrics, based on spectral cluster bounds of Smith--Sogge \cite{MR2316270} and Smith, Koch and Tataru \cite{MR2443996}, respectively. We also convert the recent spectral cluster bounds of Canzani and Galkowski \cite{Canzani--Galkowski} to improved resolvent bounds. Moreover, we show that the resolvent estimates are stable under perturbations and use this to establish uniform Sobolev and spectral cluster inequalities for Schrödinger operators with singular potentials.
