Baillon's Theorem on Maximal Regularity

The aim of this note is to give a proof of Baillon's Theorem on Maximal Regularity. Though it is in some sense a negative result (it states that for abstract Cauchy problems maximal regularity can occur only in very special cases), it is commonly accepted that it is important. Many people believe that its proof is very complicated. This might be due to the fact that Baillon's note in the Comptes Rendus is rather short and sometimes difficult to understand. The proof outlined here follows basically Baillon's lines. However it is simplified and (hopefully) easier to understand.