Lower bounds on Bourgain's constant for harmonic measure
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 $nb_n$ for every domain in $\mathbb{R}^n$ on which harmonic measure is defined. Jones and Wolff (1988) proved that $b_2=1$. When $n\geq 3$, Bourgain (1987) proved that $b_n>0$ and Wolff (1995) produced examples showing $b_n<1$. Refining Bourgain's original outline, we prove that \[ b_n\geq c\,n^{2n(n1)}/\ln(n)\] 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 $b_4\geq 2\times 10^{26}$.
 Publication:

arXiv eprints
 Pub Date:
 May 2022
 DOI:
 10.48550/arXiv.2205.15101
 arXiv:
 arXiv:2205.15101
 Bibcode:
 2022arXiv220515101B
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 Mathematics  Analysis of PDEs;
 31B15 (Primary);
 28A75;
 31B25;
 60J65 (Secondary)
 EPrint:
 20 pages, 4 figures, 1 table