Harmonic Measure and the Analyst's Traveling Salesman Theorem

Abstract

We study how generalized Jones $\beta$-numbers relate to harmonic measure. Firstly, we generalize a result of Garnett, Mourgoglou and Tolsa by showing that domains in $\mathbb{R}^{d+1}$ whose boundaries are lower $d$-content regular admit Corona decompositions for harmonic measure if and only if the square sum $\beta_{\partial\Omega}$ of the generalized Jones $\beta$-numbers is finite. Secondly, for semi-uniform domains with Ahlfors regular boundaries, it is known that uniform rectifiability implies harmonic measure is $A_{\infty}$ for semi-uniform domains, but now we give more explicit dependencies on the $A_{\infty}$-constant in terms of the uniform rectifiability constant. This follows from a more general estimate that does not assume the boundary to be uniformly rectifiable. For general semi-uniform domains, we also show how to bound the harmonic measure of a subset in terms of that sets Hausdorff measure and the square sum of $\beta$-numbers on that set. Using this, we give estimates on the fluctuation of Green's function in a uniform domain in terms of the $\beta$-numbers. As a corollary, for bounded NTA domains , if $B_{\Omega}=B(x_{\Omega},c\mathrm{diam} \Omega)$ is so that $2B_{\Omega}\subseteq \Omega$, we obtain that \[ (\mathrm{diam} \partial\Omega)^{d} + \int_{\Omega\backslash B_{\Omega}} \ |\frac{\nabla^2 G_{\Omega}(x_{\Omega},x)}{G_{\Omega}(x_{\Omega},x)}\ |^{2} \mathrm{dist}(x,\Omega^c)^{3} dx \sim \mathscr{H}^{d}(\partial\Omega). \] Secondly, we also use $\beta$-numbers to estimate how much harmonic measure fails to be $A_{\infty}$-weight for semi-uniform domains with Ahlfors regular boundaries.