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

Xing Wang, Cheng Zhang

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.

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

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.

Paper Structure

This paper contains 17 sections, 24 theorems, 261 equations.

Table of Contents

  1. Introduction
  2. Geometric quantities
  3. Lower bounds
  4. Upper bounds
  5. Comparable interior and boundary norms
  6. Almost-orthogonality, transversal restrictions and bilinear estimates
  7. Organization of the paper
  8. Notations
  9. Preliminaries
  10. Exponential lower bounds
  11. Proof of Theorem \ref{['thm1']}
  12. Sharpness of Theorem \ref{['thm1']}
  13. Parametrix for the Poisson integral operator
  14. Kernel of the composition operator
  15. Interior decay and comparable norms
  16. ...and 2 more sections

Key Result

Theorem 1

Fix any $0<\delta_0 < \delta$. There is a constant $C>0$ dependent on $(\Omega,h)$ and $\delta_0$, such that for any $0 \le t \le \delta_0$, and any $f\in L^2(M)$, we have where the frequency $\Lambda = {\int_\Omega |\nabla u|^2}/{\int_{M}u^2}$ with $u=\mathcal{H}f$, and $K(t)=t+O(t^2)$ is defined in Kt.

Theorems & Definitions (32)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Remark 1
  • Corollary 2
  • Theorem 3
  • Corollary 3
  • Theorem 4: Almost-orthogonality
  • Corollary 4
  • Theorem 5: Bilinear estimates
  • ...and 22 more