Counterexamples on \(L_p\)-maximal regularity (Q1297926)

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 \(\| B^{is}\|= 1\) for all real numbers \(s\). We also derive an operator which satisfies \(L^p\)-maximal regularity on bounded intervals \([0,T[\) but not on the half line \([0,+\infty[\).