Holomorphic extension on product Lipschitz surfaces in two complex variables

Abstract

In this work we prove a new $L^p$ holomorphic extension result for functions defined on product Lipschitz surfaces with small Lipschitz constants in two complex variables. We define biparameter and partial Cauchy integral operators that play the role of boundary values for holomorphic functions on product Lipschitz domain. In the spirit of the application of David-Journ\'e-Semmes and Christ's $Tb$ theorem to the Cauchy integral operator, we prove a biparameter $Tb$ theorem and apply it to prove $L^p$ space bounds for the biparameter Cauchy integral operator. We also prove some new biparameter Littlewood-Paley-Stein estimates and use them to prove the biparameter $Tb$ theorem.