  In this paper, we prove global-in-time $\dot{\mathrm{H}}^{\alpha,q}$-maximal
regularity for a class of injective, but not invertible, sectorial operators on
a UMD Banach space X , provided $q\in(1,+\infty) , $\alpha\in(-1+1/q,1/q)$. We
also prove the corresponding trace estimate, so that the solution to the
canonical abstract Cauchy problem is continuous with values in a not
necessarily complete trace space.In order to put our result in perspective, we
also provide a short review on L q-maximal regularity which includes some
recent advances such as the revisited homogeneous operator and interpolation
theory by Danchin, Hieber, Mucha and Tolksdorf. This theory will be used to
build the appropriate trace space, from which we want to choose the initial
data, and the solution of our abstract Cauchy problem to fall in.
