Maximal regularity for non-autonomous evolution equations governed by forms having less regularity

We consider the maximal regularity problem for non-autonomous evolution equations u' (t) + A(t) u(t) = f(t), t is an element of (0, tau] u(0) = u(0). Each operator A(t) is associated with a sesquilinear form a(t) on a Hilbert space H. We assume that these forms all have the same domain V. It is proved in Haak and Ouhabaz (Math Ann, doi: 10.1007/s00208-015-1199-7, 2015) that if the forms have some regularity with respect to t (e.g., piecewise alpha-Holder continuous for some alpha > 1/2) then the above problem has maximal L-p-regularity for all u(0) in the real-interpolation space (H, D(A(0)))(1-1/p),(p). In this paper we prove that the regularity required there can be improved for a class of sesquilinear forms. The forms considered here are such that the difference a(t;.,.) - a(s;.,.) is continuous on a larger space than the common domain V. We give three examples which illustrate our results.