Oscillatory functions vanish on a large set

TL;DR

This paper establishes that functions orthogonal to low-frequency Laplacian eigenfunctions on compact manifolds must vanish on large sets, extending oscillation principles to higher dimensions with sharpness and geometric conditions.

Contribution

It introduces a quantitative relation between frequency scale and zero set size for functions on manifolds, generalizing Sturm's oscillation theorem to higher dimensions.

Findings

Functions with high frequency scale vanish on large sets.

The result is sharp up to a logarithmic factor on flat tori.

Stronger zero set bounds are obtained under geometric regularity conditions.

Abstract

Let $(M,g)$ be a $n-$dimensional, compact Riemannian manifold. We define the frequency scale $\lambda$ of a function $f \in C^{0}(M)$ as the largest number such that $\left\langle f, \phi_k \right\rangle =0$ for all Laplacian eigenfunctions with eigenvalue $\lambda_k \leq \lambda$. If $\lambda$ is large, then the function $f$ has to vanish on a large set $$ \mathcal{H}^{n-1} \left\{x:f(x) =0\right\} \gtrsim_{} \left( \frac{ \|f\|_{L^1}}{\|f\|_{L^{\infty}}} \right)^{2 - \frac{1}{n}} \frac{ \sqrt{\lambda}}{(\log{\lambda})^{n/2}}.$$ Trigonometric functions on the flat torus $\mathbb{T}^d$ show that the result is sharp up to a logarithm if $\|f\|_{L^1} \sim \|f\|_{L^{\infty}}$. We also obtain a stronger result conditioned on the geometric regularity of $\left\{x:f(x) = 0\right\}$. This may be understood as a very general higher-dimensional extension of the Sturm oscillation theorem.