Relation between semigroup growth and resolvent decay for immediately differentiable semigroups

We study rates of growth of $\|AT(t)\|$ as $t \downarrow 0$ for an immediately differentiable $C_0$-semigroup $(T(t))_{t \geq 0}$ with generator $A$. We assume that the resolvent of the semigroup generator decays on the imaginary axis at rates described by functions of positive increase, which enable estimates on scales finer than polynomial ones. First, in the Banach space setting, we present lower and upper bounds for the semigroup growth. Next, we improve the upper estimate for Hilbert space semigroups. Finally, for semigroups of normal operators on Hilbert spaces and multiplication $C_0$-semigroups on $L^p$-spaces, we establish an estimate that exactly captures the asymptotic behavior of $\|AT(t)\|$ as $t \downarrow 0$.