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By Baillon's result, it is known that maximal regularity with respect to thespace of continuous functions is rare; it implies that either the involvedsemigroup generator is a bounded operator or the considered space contains$c_{0}$. We show that the latter alternative can be excluded under a refinedcondition resembling maximal regularity with respect to $\mathrm{L}^{\infty}$.

A refinement of Baillon's theorem on maximal regularity