Zygmund Functions On The Real Line And Their Quasiconformal Deformation Extensions

We prove among other things that a continuous function on the real line belongs to the Zygmund class if and only if it can be extended to a normalized quasiconformal deformation to the whole plane.As applications,we obtain some results about the Zygmund functions on the real line,which corresponds to some classical results concerning the Zygmund functions on the unit circle.