On optimal \(L^p\) regularity in evolution equations (Q1848113)

Summary: Using interpolation techniques, we prove an optimal regularity theorem for the convolution \(u(t)=\int^t_0T(t-s)f(s) ds\), where \(T(t)\) is a strongly continuous semigroup in general Banach space. In the case of abstract parabolic problems -- that is, when \(T(t)\) is an analytic semigroup -- it lets us recover in a unified way previous regularity results. It may be applied also to some nonanalytic semigroups, such as the realization of the Ornstein-Uhlenbeck semigroup in \(L^p(\mathbb{R}^n)\), \(1<p <\infty\), in which case it yields new optimal regularity results in fractional Sobolev spaces.