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.

中文翻译：

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振荡函数在大集合上消失

令 $(M,g)$ 是一个 $n-$ 维的紧凑黎曼流形。我们将函数 $f \in C^{0}(M)$ 的频率标度 $\lambda$ 定义为最大数，使得 $\left\langle f, \phi_k \right\rangle =0$ 对于所有拉普拉斯算子具有特征值 $\lambda_k \leq \lambda$ 的特征函数。如果 $\lambda$ 很大，那么函数 $f$ 必须在一个大集合上消失 $$ \mathcal{H}^{n-1} \left\{x:f(x) =0\right\} \gtrsim_{} \left( \frac{ \|f\|_{L^1}}{\|f\|_{L^{\infty}}} \right)^{2 - \frac{1} {n}} \frac{ \sqrt{\lambda}}{(\log{\lambda})^{n/2}}.$$ 平面环面上的三角函数 $\mathbb{T}^d$ 表明如果 $\|f\|_{L^1} \sim \|f\|_{L^{\infty}}$，则结果锐化为对数。我们还以 $\left\{x:f(x) = 0\right\}$ 的几何规律性为条件获得了更强的结果。