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

TL;DR

This paper derives sharp bounds for harmonic functions on smooth manifolds, applies microlocal analysis to Steklov eigenfunctions, and explores their boundary behavior, leading to new estimates and applications in spectral theory.

Contribution

It introduces new sharp lower bounds for harmonic functions based on frequency functions and applies microlocal analysis to study Steklov eigenfunctions and their boundary restrictions.

Findings

Sharp lower bounds for harmonic functions in terms of frequency functions

New estimates for Steklov eigenfunctions and their boundary restrictions

Applications to numerical approximation and spectral analysis

Abstract

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.