Paper detail

Almost-additivity of analytic capacity and Cauchy independent measures

We show that, given a family of discs centered at a chord-arc curve, the analytic capacity of a union of arbitrary subsets of these discs (one subset in each disc) is comparable with the sum of their analytic capacities. We show a sort of converse to this geometric statement as well. However, we need that the discs in question would be separated, and it is not clear whether the separation condition is essential or not. We apply this result to find families $\{μ_j\}$ of measures in $\mathbb{C}$ with the following property. If the Cauchy integral operators $\mathcal{C}_{μ_j}$ from $L^2(μ_j)$ to itself are bounded uniformly in $j$, then $\mathcal{C}_μ$, $μ=\sumμ_j$, is also bounded from $L^2(μ)$ to itself.