Abstrac t . We consider several results, each o f which uses some type o f " L 2' ' es t imate to provide information about harmonic measure on planar domains. The first gives an a.e. characterization of tangent points o f a curve in terms of a certain geometric square function. Our next result is an LP est imate relating the derivative of a conformal mapping to its Schwarzian derivative. One c...

A holomorphic quadratic differential on a hyperbolic Riemann surface has an associated measured foliation, which can be straightened to yield a measured geodesic lamination. On the other hand, a quadratic differential can be considered as the Schwarzian derivative of a CP1 structure, to which one can naturally associate another measured geodesic lamination (using grafting). We compare these two...