Asian Journal of Mathematics

Volume 24 (2020)

Oscillatory functions vanish on a large set

Pages: 177 – 190

DOI: https://dx.doi.org/10.4310/AJM.2020.v24.n1.a8

Author

Stefan Steinerberger (Department of Mathematics, Yale University, New Haven, Connecticut, U.S.A.)

Abstract

Let $(M, g)$ be an $n$-dimensional, compact Riemannian manifold. We will show that functions that are orthogonal to the first few Laplacian eigenfunctions have to have a large zero set. Let us assume $f \in C^0 (M)$ is orthogonal $\langle f, \phi_k \rangle = 0$ to all eigenfunctions $\phi_k$ with eigenvalue $\lambda_k \leq \lambda$. If $\lambda$ is large, then the function $f$ has to vanish on a large set$\mathcal{H}^{n-1} {\lbrace x : f(x) = 0 \rbrace} \gtrsim\frac{\sqrt{\lambda}}{(\operatorname{log} \lambda)^{n/2}}{\left (\frac{{\lVert f \rVert}_{L^1}}{{\lVert f \rVert}_{L^\infty}}\right )}^{2-\frac{1}{n}} \quad \textrm{.}$Trigonometric functions on the flat torus $\mathbb{T}^d$ show that the result is sharp up to a logarithm if ${\lVert f \rVert}_{L^1} \sim {\lVert f \rVert}_{L^\infty}$. We also obtain a stronger result conditioned on the geometric regularity of $\lbrace x : f(x) = 0 \rbrace$. This may be understood as a very general higher-dimensional extension of the Sturm oscillation theorem.

Keywords

Sturm oscillation theorem, nodal set, Laplacian eigenfunction

2010 Mathematics Subject Classification

28A75, 34B24, 34C10, 35B05, 35J05, 35K08, 46E35

This work is supported by the NSF (DMS-1763179) and by the Alfred P. Sloan Foundation.

Received 8 May 2018

Accepted 10 May 2019

Published 21 August 2020