This question concerns the abstract Cauchy Problem: u'(t) + B(u(t)) = f(t), for 0 < t < T, and u(0)=0, where 0 < T, -B is the infinitesimal generator of a bounded analytic semigroup on a complex Banach space X and u and f are X-valued functions.
Haim Brezis posed the question of whether B satisfies Lp-maximal regularity. In 1964 L. Luciano De Simon gave a positive answer to the question for Hilbert spaces. This left open the case of Lp, for p not equal to 2 and the class of general Banach spaces.
Pierre Grisvard later did the first systematic study of the problem in 1969. Since then, the problem has been studied extensively. Positive results were obtained in 1969 by Grisvard using interpolation spaces. In 1984 Thierry Coulhon and Damien Lamberton exhibited counterexamples to the general Banach space case using Banach spaces that are not UMD (Unconditional Monotone Decomposition) spaces.
In 1987, Giovanni Dore and Alberto Venni proved the remarkable result that Lp-maximal regularity holds if X is a UMD space and certain growth conditions were satisfied. In 1999, Christian Le Merdy found counterexamples for some other fundamental cases such as L1 of the circle.
In 1999, Nigel Kalton, MU Curators’ Professor and holder of the Distinguished Houchins Chair, and Gilles Lancien of the Universite de Franche-Comte, Besancon, France, gave a powerful positive solution to the problem by showing that the property that every negative generator of a bounded analytic semigroup has the Lp-maximal regularity characterizes Hilbert spaces (up to isomorphism) among large classes of Banach spaces including those having an unconditional basis. This property in general does not characterize Hilbert spaces among all Banach spaces. This follows from a result of Heinrich Lotz in 1985, which shows that every strongly continuous semigroup on L(•) is uniformly continuous and therefore Lp-regular.