A NEW WAY TO EXPRESS BOUNDARY VALUES IN TERMS OF HOLOMORPHIC FUNCTIONS ON PLANAR LIPSCHITZ DOMAINS

We decompose p - integrable functions on the boundary of a bounded simply connected Lipschitz domain U, subset of the complex plane, into the sum of the boundary values of two, uniquely determined holomorphic functions, where one is in the holomorphic Hardy space for U while the other is in the holomorphic Hardy space for the (interior of) the complement of U. Various refinements are presented showing the dependence of the decomposition on the regularity of the domain U, and/or of the boundary function. We discuss a few applications. The main tool is a regularity result for the holomorphic Hardy space of U which can be traced back to Privalov for smooth U but appears to be new for U Lipschitz. This is joint work with S. Bell and N. Wagner.