Common boundary values of holomorphic functions for two-sided complex structures

Abstract

L et� 1, � 2 be two disjoint open sets in R 2n whose bound- aries share a smooth real hypersurface M as a relatively open sub- set. Assume thati is equipped with a complex structure J i that is smooth up to M. Suppose that at each point x ∈ M there is a vector v ∈ Tx M such that J 1 x v and J 2 x v are in the same connected component of TxR 2n Tx M.I ff is holomorphic with respect to both structures in the open sets and continuous on � 1 ∪ M ∪ � 2 ,t henf must be smooth on the union � 1 ∪ M. Although the result, as stated, is far more meaningful for integrable structures, our methods make it much more natural to deal with the general almost complex structures with- out the integrability condition. The result is therefore proved in the framework of almost complex structures.