On maximal regularity and semivariation of $α$-times resolvent families

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

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