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

We consider the maximal regularity problem for non-autonomous evolution equations 0.1u′(t)+A(t)u(t)=f(t),t∈(0,τ]u(0)=u0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{l}{u'(t) + A(t)\,u(t) = f(t), \quad t \in (0, \tau]}\\ {u(0) = u_0.}\end{array}$$\end{document}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 α-Hölder continuous for some α > ½) then the above problem has maximal Lp-regularity for all u0 in the real-interpolation space (H,D(A(0)))1-1/p,p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(H, \fancyscript{D}(A(0)))_{1-{1}/{p},p}$$\end{document}. 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.