The Cauchy Integral On Chord-arc Curves And The Related Research

In this thesis,the Cauchy integral on chord-arc curves,BMO-Teichmüller space and VMO-Teichmüller space are studied.The main work is showed as follows.1.We discuss the boundedness of Faber operator on BMOA,a new subject which turns out to be closely related to the BMO theory of the universal Teichmüller space.We show that the Faber operator acts as a bounded operator on BMOA and VMOA if the symbol conformal map stays nearly to the base point in the BMO-Teichmüller space.Meanwhile,we obtain several results on quasiconformal mappings,BMO-Teichmüller space and chord-arc curves as well.As by-products,this provides a complex analysis approach to the boundedness of the Cauchy integral acting on BMO functions on chord-arc curves near to the unit circle in the BMO-Teichmüller space.2.We introduce the critical Besov space B_p¹,_p^/p（Γ）on a quasicircleΓ,and prove the solvability of Riemann-Hilbert problem on general quasicircles when p=2 and on d-regular quasicircles when d<p<2.As applications,we obtain the B_p¹,_p^/p（1<p<∞）boundedness of the Cauchy integral and Faber operator on chord-arc curves.3.We prove that the Cauchy integral on chord-arc curves is a bounded opera-tor on BMO（VMO）.Some applications are given to the Faber operator on BMOA（VMOA）and to the jump problem for BMO（VMO）,which recovers the results on chord-curve curves with small norm mentioned earlier.4.We study some problems concerning with Cauchy transform and Beurling transform acting on Carleson measure space.In particular,we prove the boundedness of the nonlinear operator Imz|S|²acting on the“Good”Carleson measure space,where S denotes the Beurling transform.5.We give some characterizations of VMO_s（R）to clarify the essential difference between the VMO-Teichmüller space on the real line and on the unit circle,where VMO_s（R）is the completion of continuous functions with compact support in the BMO（R）norm.Correspondingly,we introduce the conformally invariant VMO_s（R）-Teichmüller space and the strongly vanishing Carleson measure with respect to R.We then show that the element in this space has a quasiconformal extension to the upper half-plane so that its complex dilatation induces a strongly vanishing Carleson measure.As an immediate corollary,the well-known Douady-Earle extension also has the same property.