Almost sharp lower bound for the nodal volume of harmonic functions

This paper focuses on a relation between the growth of harmonic functions and the Hausdorff measure of their zero sets. Let $u$ be a real-valued harmonic function in $\mathbb{R}^n$ with $u(0)=0$ and $n\geq 3$. We prove $$\mathcal{H}^{n-1}(\{u=0\} \cap B(0,2)) \gtrsim_{\varepsilon} N^{1-\varepsilon},$$ where the doubling index $N$ is a notion of growth defined by $$\sup_{B(0, 1)}|u| = 2^N \sup_{B(0,\frac{1}{2})}|u|.$$ This gives an almost sharp lower bound for the Hausdorff measure of the zero set of $u$, which is conjectured to be linear in $N$. The new ingredients of the article are the notion of stable growth, and a multi-scale induction technique for a lower bound for the distribution of the doubling index of harmonic functions. It gives a significant improvement over the previous best-known bound $\mathcal{H}^{n-1}\left(\{u=0\} \cap 2B\right)\geq \exp (c \log N/\log\log N )$, which implied Nadirashvili's conjecture.