Lower bounds on Bourgain’s constant for harmonic measure

Abstract

Abstract For every $n\geq 2$ , Bourgain’s constant $b_n$ is the largest number such that the (upper) Hausdorff dimension of harmonic measure is at most $n-b_n$ for every domain in $\mathbb {R}^n$ on which harmonic measure is defined. Jones and Wolff (1988, Acta Mathematica 161, 131–144) proved that $b_2=1$ . When $n\geq 3$ , Bourgain (1987, Inventiones Mathematicae 87, 477–483) proved that $b_n>0$ and Wolff (1995, Essays on Fourier analysis in honor of Elias M. Stein (Princeton, NJ, 1991), Princeton University Press, Princeton, NJ, 321–384) produced examples showing $b_n<1$ . Refining Bourgain’s original outline, we prove that $$\begin{align*}b_n\geq c\,n^{-2n(n-1)}/\ln(n),\end{align*}$$ for all $n\geq 3$ , where $c>0$ is a constant that is independent of n. We further estimate $b_3\geq 1\times 10^{-15}$ and $b_4\geq 2\times 10^{-26}$ .