Stephan Fackler

J.-L. Lions' problem concerning maximal regularity of equations governed by non-autonomous forms

An old problem due to J.-L. Lions going back to the 1960s asks whether the abstract Cauchy problem associated to non-autonomous forms has maximal regularity if the time dependence is merely assumed to be continuous or even measurable. We give a negative answer to this question and discuss the minimal regularity needed for positive results.

The Kalton–Lancien theorem revisited: Maximal regularity does not extrapolate

Abstract We give a new more explicit proof of a result by Kalton and Lancien stating that on each Banach space with an unconditional basis not isomorphic to a Hilbert space there exists a generator A 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 A can be chosen of the form A ( ∑ m = 1 ∞ a m f m ) = ∑ m = 1 ∞ 2 m a m f m . Moreover, we show that maximal regularity does not extrapolate: we construct consistent holomorphic semigroups ( T p ( t ) ) t ⩾ 0 on L p ( R ) for p ∈ ( 1 , ∞ ) 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 and Lancien. We use the characterization of maximal regularity by R -sectoriality for our construction.

Regularity Properties of Sectorial Operators: Counterexamples and Open Problems

We give a survey on the different regularity properties of sectorial operators on Banach spaces. We present the main results and open questions in the theory and then concentrate on the known methods to construct various counterexamples.

Maximal regularity: Positive counterexamples on UMD-Banach lattices and exact intervals for the negative solution of the extrapolation problem

Using methods from Banach space theory, we prove two new structural results on maximal regularity. The first says that there exist positive analytic semigroups on UMD-Banach lattices, namely

A toolkit for constructing dilations on Banach spaces: A TOOLKIT FOR CONSTRUCTING DILATIONS ON BANACH SPACES

\ell_p(\ell_q)

A short counterexample to the inverse generator problem on non-Hilbertian reflexive L p -spaces

for

Regularity of semigroups via the asymptotic behaviour at zero

p \neq q \in (1, \infty)

Isometric dilations and ^{∞} calculus for bounded analytic semigroups and Ritt operators

, without maximal regularity. In the second result we show that the extrapolation problem for maximal regularity behaves in the worst possible way: for every interval

Nonautonomous maximal Lp-regularity under fractional Sobolev regularity in time

I \subset (1, \infty)

J. L. Lions’ problem on maximal regularity

with