  We give a new more explicit proof of a result by Kalton & Lancien stating
that on each Banach space with an unconditional basis not isomorphic to a
Hilbert space there exists a generator of a holomorphic semigroup which does
not have maximal regularity. In particular, we show that there always exists a
Schauder basis (f_m) such that the generator is a Schauder multiplier
associated to the sequence (2^m). Moreover, we show that maximal regularity
does not extrapolate: we construct consistent holomorphic semigroups (T_p(t))
on L^p for p in (1, \infty) which have maximal regularity if and only if p = 2.
These assertions were both open problems. Our approach is completely different
than the one of Kalton & Lancien. We use the characterization of maximal
regularity by R-sectoriality for our construction.
