标题： 一些调和函数的精确估计及其在斯捷克洛夫特征函数中的应用 显示英文标题

标题： A few sharp estimates of harmonic functions with applications to Steklov eigenfunctions

摘要： 在具有光滑边界的光滑紧致流形上，我们首先通过结合微局部分析和Almgren以及Garofalo-Lin的频率函数技术，建立了调和函数在频率函数方面的精确下界。这些下界可以通过欧几里得球上的Steklov本征函数和一族对称扭曲乘积流形来达到饱和。此外，如Sogge和Taylor所做的那样，我们通过构造泊松积分算子的参数解并计算其与谱簇的复合，分析了调和函数的内部行为。通过使用微局部分析，我们获得了其边界迹为准模式的调和函数的几个精确估计。作为应用，我们建立了Steklov本征函数的几乎正交性、双线性估计和横截限制估计，并讨论了调和函数的数值逼近。 显示英文摘要

摘要： On smooth compact manifolds with smooth boundary, we first establish the sharp lower bounds for the restrictions of harmonic functions in terms of their frequency functions, by using a combination of microlocal analysis and frequency function techniques by Almgren and Garofalo-Lin. The lower bounds can be saturated by Steklov eigenfunctions on Euclidean balls and a family of symmetric warped product manifolds. Moreover, as in Sogge and Taylor, we analyze the interior behavior of harmonic functions by constructing a parametrix for the Poisson integral operator and calculate its composition with the spectral cluster. By using microlocal analysis, we obtain several sharp estimates for the harmonic functions whose traces are quasimodes on the boundary. As applications, we establish the almost-orthogonality, bilinear estimates and transversal restriction estimates for Steklov eigenfunctions, and discuss the numerical approximation of harmonic functions.