Harmonic measure, L²-estimates and the Schwarzian derivative

We consider several results, each of which uses some type of “L²” estimate to provide information about harmonic measure on planar domains. The first gives an a.e. characterization of tangent points of a curve in terms of a certain geometric square function. Our next result is an L^p estimate relating the derivative of a conformal mapping to its Schwarzian derivative. One consequence of this is an estimate on harmonic measure generalizing Lavrentiev’s estimate for rectifiable domains. Finally, we consider L² estimates for Schwarzian derivatives and the question of when a Riemann mapping ϕ has log ϕ′ in BMO.