Maximal regularity for evolution problems on the line (Q637007)

From the introduction: We introduce various types of solutions of the abstract first-order evolution equation \[ u'(t)=Au(t)+f(t), \] and we recall the representation of the solutions in terms of the norms of the semigroups \((T^{\pm }(t))_{t>0},\) studying some of its properties. We prove the optimal regularity for the evolution problem to the above equation in various Banach spaces by the direct approach. The contents of this paper are part of the Ph.D. thesis of the author.