CAUCHY-SZEGO PROJECTIONS AND RELATED TOPICS

In this survey article, we give a comprehensive review of Calder´ on-Zygmund operators from the point of view of Cauchy-Szeg¨ o projections and the sharp estimates of the operators in Hardy spaces. Cauchy-Szeg¨ o projections is closed related to the Hilbert transform which is a typical example of the so called “first generation” of singular integral operators and has been studied by mathematicians for many years, We started with the case in unit disk D1 in C1 and then move to the unbounded unit ball Bn+1 in Cn+1. Analysis on Rn and Hn are quite different. We try to explain the idea behind it carefully.