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.

翻译：在具有光滑边界的紧致光滑流形上，我们首先结合Almgren与Garofalo-Lin的频率函数方法及微局部分析，建立了调和函数限制的尖锐下界估计，该下界由频率函数刻画。该下界可被欧几里得球面及一类对称扭曲积流形上的Steklov本征函数达到饱和。此外，借鉴Sogge与Taylor的方法，我们通过构造泊松积分算子的参数化核并计算其与谱簇的复合，分析了调和函数的内部行为。借助微局部分析技术，我们获得了边界拟模迹函数对应的调和函数的若干尖锐估计。作为应用，我们建立了Steklov本征函数的近似正交性、双线性估计及横截限制估计，并讨论了调和函数的数值逼近问题。