Harmonic analysis of Banach space valued functions in the study of parabolic evolution equations.

摘要

This work is motivated by the study of parabolic evolution equations and, in particular, of their regularity in a Lp sense. Such questions lead to the investigation of singular integral operators with operator valued kernels acting on a Banach space. We are interested in boundedness results for such operators and their applications to evolution equations. Our focus is on the relationship between these results and the geometry of the underlying Banach space. We study various problems, both in discrete and continuous time, and relate their behavior to the R-boundedness of certain sets of bounded linear operators acting on a UMD space (for Lp regularity-with 1 p infinity) or to the existence of a complemented copy of c 0 or ℓ1 (for ℓinfinity and ℓ1 regularity).