Propagation of smallness near codimension two for gradients of harmonic functions

Abstract

Let $u$ be a harmonic function in the unit ball $B_1 \subset \mathbb R^n$, normalized so that its gradient has magnitude at most 1 on the unit ball. We show that if the gradient of $u$ is $\epsilon$-small in size on a set $E\subset B_{1/2}$ with positive $(n-2+\delta)$-dimensional Hausdorff content for some $\delta>0$, then $\sup_{B_{1/2}} |\nabla u| \leq C \epsilon^\alpha$ with $C,\alpha>0$ depending only on $n,\delta$ and the $(n-2+\delta)$-Hausdorff content of $E$. This is an improvement over a similar result of Logunov and Malinnikova that required $\delta>1-c_n$ for a small dimensional constant $c_n$ and reaches the sharp threshold for the dimension of the smallness sets from which propagation of smallness can occur.