Sharp lower bounds for the volumes of nodal and positivity sets of harmonic functions

Abstract

Let $B=B(p, 1)\subset\mathbb{R}^n$ be a unit ball and $n\ge 3$. We prove that there are positive constants $c$ and $C$, depending only on $n$, such that every non-zero real-valued harmonic function $u: 4B\to \mathbb{R}$ with $u(p)=0$ satisfies $$ \mathcal{H}^{n-1}\bigl(\{u=0\}\cap 2B\bigr)\ge C\mathcal{N}, $$ and $$ \mathcal{H}^n\bigl(\{u>0\}\cap {\frac 12}B\bigr) \ge c\bigl(\log(1+\mathcal{N})\bigr)^{1-n}, $$ where $\mathcal{N}$ is the doubling index defined by $$\mathcal{N}=\log_2\frac{\sup_{B}|u|}{\sup_{\frac{1}{2}B}|u|}.$$ The first estimate confirms a folklore conjecture on the nodal volume of harmonic functions, and its linear dependence on $\mathcal{N}$ is optimal. The second estimate extends the planar result of Nazarov, Polterovich, and Sodin to higher dimensions, and the logarithmic order is optimal. As a further consequence of the ideas developed in the proof, we obtain an alternative proof of Nadirashvili's conjecture that does not rely on the multiscale analysis.