Journal d’Analyse Mathematique

, Volume 62, Issue 1, pp 77–113

Harmonic measure,L2-estimates and the Schwarzian derivative

  • Christoper J. Bishop
  • Peter W. Jones

DOI: 10.1007/BF02835949

Cite this article as:
Bishop, C.J. & Jones, P.W. J. Anal. Math. (1994) 62: 77. doi:10.1007/BF02835949


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 anLp 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 considerL2 estimates for Schwarzian derivatives and the question of when a Riemann mapping ϕ has log ϕ′ in BMO.

Copyright information

© Hebrew University 1994

Authors and Affiliations

  • Christoper J. Bishop
    • 1
  • Peter W. Jones
    • 2
  1. 1.Department of MathematicsSUNY at Stony BrookStony BrookUSA
  2. 2.Department of MathematicsYale UniversityNew HavenUSA