Generalized Harmonic Conjugation Operator And Universal Teichmüller Space

A quasisymmetric homeomorphism h of the unit circle S¹ onto itself determines a pull-back operator B_h on the real-valued harmonic Dirichlet space D(â–³). The harmonic con-jugation operator A defines a complex structure on D(â–³). A_h=B_hAB_h^-1 is called thegeneralized harmonic conjugation operator.The paper deals mainly with the pull-back operator B_h and the generalized harmonicconjugation operator A_h. We first embed the universal Teichmuller space into the space ofbounded linear operators on D(â–³) by the pull-back operator B_h. Then we prove the operatorAA_h is a negative operator by a simple and new method, and give some characterizations onh for which the operator AA_h has some specific properties.