Analytic semigroup ($S(t)=e^{tL}$)

An analytic semigroup is a strongly continuous semigroup whose bounded-operator-valued extension is holomorphic in a sector about the positive time axis. Its infinitesimal generator of a semigroup $L$ satisfies $S(t)H\subseteq D(L)$ for $t>0$ and, on each bounded time interval, $\Vert LS(t)\Vert \le C/t$. This estimate gives instantaneous spatial smoothing but creates an integrable-singularity question in an inhomogeneous variation-of-constants formula.