Summary: Let \(X\) be a Banach space and let \(A\) be a closed linear operator on \(X\). It is shown that the abstract Cauchy problem \[ \dot u(t)+Au(t)=f(t),\;t>0,\;u(0)=0 \] enjoys maximal regularity in weighted \(L_p\)-spaces with weights \(\omega(t)=t^{p(1-\mu)}\), where \(1/p<\mu\), if and only if it has the property of maximal \(L_p\)-regularity. Moreover, it is also shown that the derivation operator \(D=d/dt\) admits an \({\mathcal H}^\infty\)-calculus in weighted \(L_p\)-spaces.