Keywords

• p-integrable function
• Lipschitz domain
• holomorphic function
• Hardy space

Abstract We decompose $p$-integrable functions on the boundary of a bounded simply connected Lipschitz domain $\Omega \subset \mathbb{C}$ into the sum of the boundary values of two, uniquely determined holomorphic functions, where one is in the holomorphic Hardy space for $\Omega$ while the other is in the holomorphic Hardy space for the (interior of) the complement of $\Omega$. Various refinements are presented showing the dependance of the decomposition on the regularity of the domain $\Omega$, and/or of the boundary function. We discuss a few applications. The main tool is a regularity result for the holomorphic Hardy space which appears to be new even for smooth $\Omega$.

This is joint work with S. Bell and N. Wagner.