Decay rates given by regularly varying functions for $C_0$-semigroups on Banach spaces

We study rates of decay for (possibly unbounded) $C_0$-semigroups on Banach spaces under the assumption that the norm of the resolvent of the respective semigroup generator grows as a regularly varying function of type $\beta >0$, that is, as $\mathrm{\mid }s{\mathrm{\mid }}^{\beta }\mathrm{\ell }(1+\mathrm{\mid }s\mathrm{\mid })$ or $\mathrm{\mid }s{\mathrm{\mid }}^{\beta }\mathrm{/}\kappa (1+\mathrm{\mid }s\mathrm{\mid })$, where $\mathrm{\ell },\kappa$ are arbitrary monotone and slowly varying functions. The main result extends the estimates obtained by Deng, Rozendaal and Veraar (J. Evol. Equ. 24, 99 (2024)) to this setting of regularly varying functions and improves the estimates obtained by Santana and Carvalho (J. Evol. Equ. 24, 28 (2024)) in case $\mathrm{\mid }s{\mathrm{\mid }}^{\beta }\log (1+\mathrm{\mid }s\mathrm{\mid }{)}^b$, with $b\ge 0$.