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 ${\sup }_{B_{1/2}}|\nabla u|\le C{\epsilon }^{\alpha }$ 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.

14 pages