= Analytic semigroup
{title2=$S(t)=e^{tL}$}
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, $\|LS(t)\|\leq C/t$. This estimate gives instantaneous spatial smoothing but creates an integrable-singularity question in an inhomogeneous <variation-of-constants formula>.
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