Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Almost sharp local Bernstein estimates for Laplace eigenfunctions on compact Riemannian manifolds

Kévin Le Balc'h

TL;DR

This work establishes almost sharp local Bernstein inequalities for Laplace eigenfunctions on compact Riemannian manifolds, showing that locally these eigenfunctions behave like polynomials of degree ~√λ up to an arbitrarily small λ^{ε} loss in L^p norms. The authors develop refined L^2 Carleman estimates, prove uniform L^2 doubling bounds on annuli of wavelength width, and bootstrap to L^p estimates for all p∈[1,∞], including the gradient bound. An analogous theory is developed for A-harmonic functions using the doubling index N, with a similar almost sharp L^p control on annuli; the eigenfunction case is treated by extending these results to the λ-dependent operator -Δ_g - λ. The results advance the Donnelly–Fefferman paradigm by providing nearly sharp, locally valid Bernstein-type inequalities, with extensions to manifolds with boundary, elliptic inequalities, and questions about linear combinations of eigenfunctions.

Abstract

We study local growth properties of Laplace eigenfunctions on compact Riemannian manifolds. Following the paradigm introduced by Donnelly and Fefferman in the late 1980s, an eigenfunction is expected to behave locally like a polynomial of degree comparable to the square root of the eigenvalue. In this direction we establish almost sharp local $L^{p}$--Bernstein inequalities, $p\in[1,\infty]$, conjectured by Donnelly--Fefferman in 1990. We also derive analogous estimates for $A$-harmonic functions, with the square root of the eigenvalue replaced by the doubling index. Our argument refines the original Donnelly--Fefferman method based on $L^{2}$--Carleman estimates. At the $L^{2}$--level, we first prove a uniform bound for the doubling index on annuli of width comparable to the wavelength. This implies, with an arbitrarily small polynomial loss, the corresponding property at the $L^{p}$--level for all $p\in[1,\infty]$. The latter step relies on a bootstrap scheme combining elliptic regularity with a patching of local Carleman estimates on small balls.

Almost sharp local Bernstein estimates for Laplace eigenfunctions on compact Riemannian manifolds

TL;DR

This work establishes almost sharp local Bernstein inequalities for Laplace eigenfunctions on compact Riemannian manifolds, showing that locally these eigenfunctions behave like polynomials of degree ~√λ up to an arbitrarily small λ^{ε} loss in L^p norms. The authors develop refined L^2 Carleman estimates, prove uniform L^2 doubling bounds on annuli of wavelength width, and bootstrap to L^p estimates for all p∈[1,∞], including the gradient bound. An analogous theory is developed for A-harmonic functions using the doubling index N, with a similar almost sharp L^p control on annuli; the eigenfunction case is treated by extending these results to the λ-dependent operator -Δ_g - λ. The results advance the Donnelly–Fefferman paradigm by providing nearly sharp, locally valid Bernstein-type inequalities, with extensions to manifolds with boundary, elliptic inequalities, and questions about linear combinations of eigenfunctions.

Abstract

We study local growth properties of Laplace eigenfunctions on compact Riemannian manifolds. Following the paradigm introduced by Donnelly and Fefferman in the late 1980s, an eigenfunction is expected to behave locally like a polynomial of degree comparable to the square root of the eigenvalue. In this direction we establish almost sharp local --Bernstein inequalities, , conjectured by Donnelly--Fefferman in 1990. We also derive analogous estimates for -harmonic functions, with the square root of the eigenvalue replaced by the doubling index. Our argument refines the original Donnelly--Fefferman method based on --Carleman estimates. At the --level, we first prove a uniform bound for the doubling index on annuli of width comparable to the wavelength. This implies, with an arbitrarily small polynomial loss, the corresponding property at the --level for all . The latter step relies on a bootstrap scheme combining elliptic regularity with a patching of local Carleman estimates on small balls.
Paper Structure (20 sections, 20 theorems, 195 equations, 2 figures)

This paper contains 20 sections, 20 theorems, 195 equations, 2 figures.

Table of Contents

  1. Introduction
  2. On local growth and local Bernstein estimates of Laplace eigenfunctions
  3. Main results
  4. Strategy of the proof
  5. Proof of the growth estimates
  6. $L^2$-Carleman estimates
  7. Vanishing order estimate
  8. Sharp local $L^2$ estimates on annuli and consequences at $L^p$-level
  9. Almost sharp local $L^p$ estimates on annuli
  10. Proof of the growth estimates for $A$-harmonic functions
  11. Modification of the proofs for Laplace eigenfunctions
  12. Extensions
  13. Manifolds with boundaries
  14. On elliptic differential inequalities
  15. Uniform bound on the doubling index on spheres
  16. ...and 5 more sections

Key Result

Theorem 1.6

For every $p \in [1,\infty]$, $\varepsilon>0$, there exist $r_0, C >0$ depending only on $M$, $p$ and $\varepsilon$, such that for every Laplace eigenfunction $\varphi_{\lambda} \in C^{\infty}(M)$, i.e. satisfying eq:Laplaceeingenfunction, for every $x \in M$, $r \in (0,r_0)$,

Figures (2)

  • Figure 1: Covering of the annulus $A_r^{(0,2 \varepsilon)}$ by balls $B(x_i,4 r \varepsilon)$
  • Figure 2: Support of $\chi_i = \chi_\alpha \tilde{\chi}_i$

Theorems & Definitions (43)

  • Remark 1.1
  • Remark 1.2
  • Conjecture 1.3: DF90
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 33 more