Tools for maximal regularity.

Let A be the generator of an analytic C0-semigroup on a Banach space X. We associate a closed operator ${\cal A}_{1}$ with A defined on Rad(X) and show that when X is a UMD-space, the Cauchy problem associated with A has maximal regularity if and only if the operator ${\cal A}_{1}{\rm g}$ generates an analytic C0-semigroup on Rad(X). This allows us to exploit known results on analytic C0-semigroups to study maximal regularity. Our results show that ${\cal R}$-boundedness is a local property for semigroups: an analytic C0-semigroup T of negative type is ${\cal R}$-bounded if and only if it is ${\cal R}$-bounded at z = 0. As applications, we give a perturbation result for positive semigroups. Finally, we show the following: when X is a UMD-space, T is an analytic C0-semigroup of negative type, then for every $f\in L^{p}(\RR_{+}; X)$, the mild solution of the corresponding inhomogeneous Cauchy problem with initial value 0 belongs to $W^{\theta,p}(\RR_{+}; X)$ for every $0<\theta < 1$. [ABSTRACT FROM PUBLISHER]