Non-autonomous maximal regularity in weighted space

We consider the problem of maximal regularity for non-autonomous Cauchy problems u ′ (t) + A(t)u(t) = f (t) (t ∈ [0, τ ]), u(0) = u 0. The time dependent operators A(t) are associated with sesquilinear forms on a Hilbert space H. We prove the maximal regularity in the weighted space L 2 (0, τ, t β dt; H), with β ∈] − 1, 1[ and we prove also other regularity properties for the solution of the previous problem. Our result is motivated by boundary value problems