, Volume 62, Issue 1, pp 77-113

Harmonic measure,L 2-estimates and the Schwarzian derivative

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We consider several results, each of which uses some type of “L 2” 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 anL 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 considerL 2 estimates for Schwarzian derivatives and the question of when a Riemann mapping ϕ has log ϕ′ in BMO.

Supported in part by NSF Grant DMS-91-00671.
Supported in part by NSF Grant DMS-86-025000.