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Let $1< \alpha <2$ and $A$ be the generator of an $\alpha$-times resolventfamily $\{S_\alpha(t)\}_{t \ge 0}$ on a Banach space $X$. It is shown that thefractional Cauchy problem ${\bf D}_t^\alpha u(t) = Au(t)+f(t)$, $t \in [0,r]$;$u(0), u'(0) \in D(A)$ has maximal regularity on $C([0,r];X)$ if and only if$S_\alpha(\cdot)$ is of bounded semivariation on $[0,r]$.

On maximal regularity and semivariation of $\alpha$-times resolvent families