Cauchy independent measures and almost-additivity of analytic capacity

Abstract

We show that, given a family of discs centered at a chord-arc curve, the analytic capacity of a union of subsets of these discs (one subset in each disc) is comparable with the sum of their analytic capacities. However, we need the discs in question to be separated, and it is not clear whether the separation condition is essential. We apply this result to find families {μj} of measures in ℂ with the following property. If the Cauchy integral operators $${C_{{\mu _j}}}$$ from L2(μj) to itself are bounded uniformly in j, then Cμ, μ = Σμj, is also bounded from L2(μ) to itself.