ON THE REGULARITY OF SINGULAR INTEGRAL OPERATORS ON COMPLEX DOMAINS

Abstract

Given a doman $\Omega$ in $C^n$, it is a classical problem to study the boundary behavior of functions which are holomorphic on $\Omega$. The boundary values of a given function are often expressed by means of singular integral operators. In this thesis we study this problem in two different settings with different motivations. In the first part we deal with a non-smooth version of the so-called worm domain in order to understand the role played by the pathological geometry of this domain. In the second part we study the problem in the case of a product Lipschitz surface and some boundedness results for biparameter singular integral operators are proved