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 $${\mathcal{H}}^{n-1}(\{u=0\}\cap 2B)\ge C\mathcal{N},$$ and $${\mathcal{H}}^n(\{u>0\}\cap \frac{1}{2}B)\ge c(\log (1+\mathcal{N}){)}^{1-n},$$ 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.

arXiv comment43 pages, new version, added a sharp lower bound for the volume of the positivity set and an alternative proof of Nadirashvili's conjecture. Comments are welcome