The initial value problem for the linear evolution equation \(u'(t) + A u(t) = f(t), \) \(t>0\), and a discrete counterpart that can be interpreted as the forward Euler discretization are considered. Here, \(A\) denotes a generator of a bounded analytic semigroup in a Banach space \(X\) (especially \(X=L_q(\Omega)\) for some domain \(\Omega\) and \(q\in (1,\infty)\), \(q\neq 2\)). The authors prove sufficient conditions for maximal \(L_p\)-regularity of \(A\) so that \(u'\) and \(Au\) fulfills a-priori estimates in the \(L_p(\mathbb{R}_+;X)\)-norm. The criterion relies upon weighted \(L_p\)-\(L_q\)-estimates on the semigroup \(e^{tA}\). The results are applied to the Schrödinger equation with singular potentials and some general elliptic operators.