  In this paper we study the maximal regularity property for non-autonomous
evolution equations $\partial_t u(t)+A(t)u(t)=f(t), u(0)=0.$ If the equation is
considered on a Hilbert space $H$ and the operators $A(t)$ are defined by
sesquilinear forms $ a(t,.,.)$ we prove the maximal regularity under a Holder
continuity assumption of $t \to a(t,.,.)$. In the non-Hilbert space situation
we focus on Schrodinger type operators $A(t):= -\Delta + m(t, .)$ and prove
$L^p-L^q$ estimates for a wide class of time and space dependent potentials
$m$.
