Counterexamples on $L_p$ -maximal regularity

Abstract

We establish a transference result for $L^p$ -maximal regularity for the abstract Cauchy problem on Banach space. From this result we deduce counterexamples to $L^p$ -maximal regularity $(1 < p < \infty).$ In particular we obtain an operator B without any $L^p$ -maximal regularity although it admits bounded imaginary powers with $\Vert B^{is}\Vert = 1$ for all $s \in \mathbb{R}$ . We also derive an operator which satisfies $L^p$ -maximal regularity on bounded intervals [0, T[ but not on the half line $\mathbb{R}_{+}.$