  We consider the maximal regularity problem for non-autonomous evolution
equations \begin{equation}
  \left\{
  \begin{array}{rcl} u'(t) + A(t)\,u(t) &=& f(t), \ t \in (0, \tau]
  u(0)&=&u_0.
  \end{array} \right. \end{equation} Each operator $A(t)$ is associated with a
sesquilinear form $\mathfrak{a}(t)$ on a Hilbert space $H$. We assume that
these forms all have the same domain $V$. It is proved in \cite{HO14} that if
the forms have some regularity with respect to $t$ (e.g., piecewise
$\alpha$-H\"older 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 $\mathfrak{a}(t;\cdot,\cdot) -
\mathfrak{a}(s; \cdot,\cdot)$ is continuous on a larger space than the common
domain $V$. We give three examples which illustrate our results.
