An explicit counterexample for the Lp-maximal regularity problem

In this short Note we give a self-contained example of a consistent family of holomorphic semigroups $(T_p{(t))}_{t?0}$ such that $(T_p{(t))}_{t?0}$ does not have maximal regularity for $p>2$. This answers negatively the open question whether maximal regularity extrapolates from $L^2$ to the $L^p$-scale.