\(L_ p\)-regularity of the Cauchy problem and the geometry of Banach spaces (Q1913491)

We investigate the different notions of regularity of the vector-valued Cauchy problem: \(u'+ Au= f\), \(u(0)= 0\), that is: if \(L[0,T ]\) is one of the lattices \(L_p [0,T ]\), \(1\leq p\leq +\infty\) or \(C[0,T ]\), then \(\text{CP}_A\) is \(L\)-regular if there exists a constant \(C\) such that, for all \(f\in L([0,T ]; X): |Au |_{L(X)}= |A({d\over dt}+ A)^{-1}f|_{L(X)}\leq C|f|_{L(X)}\). We prove its \(L_2\)-regularity for some recent examples of unbounded operators \(A\) on \(L_p\), \(1< p\neq 2< +\infty\). We give a characterization of the \(L_1\)-regularity. We characterize the regularity of the problems associated with diagonal operators on classical sequence spaces \(\ell_p\) and \(c_0\).