Maximal L p regularity for nonautonomous parabolic equations in extrapolation spaces1

This paper deals with a parabolic problem u′ (t)=A (t) u (t)+f (t), u (0)=x, on a Banach space X. We establish maximal L^p regularity, p ∈1,+∞, on a suitable extrapolation space of X. As a consequence we obtain weak solutions that have the same regularity of the mild solutions in the autonomous case. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday