Maximal regularity and kernel bounds: observations on a theorem by Hieber and Prüss (Q5954415)

Here, the authors study the maximal \(L^p\) regularity for the following abstract Cauchy problem NEWLINE\[NEWLINE u_{t}(t)=Au(t)+f(t),\quad t\in I,\quad u(0)=0,NEWLINE\]NEWLINE in a Banach space \(X\), where \(A\) is an infinitesimal generator of an analytic semigroup of operators on \(X\) whose kernel satisfies suitable upper bounds, \(f(t)\) is a given function, and \(I\) is a bounded or unbounded interval and \(p\in [1,\infty]\) . Maximal \(L^p\) regularity means that for every \(f\in L^p(I,X)\) the considered problem has one and only one solution \(u\) belonging to \(L^p(I,X)\) and such that \(u_{t}\) and \(Au\) belong to the same spaces and depend continuously on \(f\). The authors derive results on the maximal \(L^p\) regularity for the considered problem.