Besov spaces and analytic semigroups of linear operators (Q749823)

The author investigates the smoothness property of the function F(t) defined by \(F(t)=\int^{t}_{a}e^{-(t-s)A}f(s)ds\) where f is a strongly continuous function on an interval \(I=(a,b)\) with values in a Banach space X and where \(e^{-tA}\) is an analytic semigroup of linear operators in X. Among other results obtained he shows that if \(\sigma\) is any real number, \(1\leq p\leq \infty\), \(1\leq q\leq \infty\), and if the function f belongs to \(B^{\sigma}_{p,q}(I;X)\cap_{loc}L_ 1(I;X)\) then the function F defined above belongs to \(B^{\sigma +1}_{p,q}(I;X)\).