Refined decay rates of \(C_0\)-semigroups on Banach spaces

We study rates of decay for \(C_0\) $C_0$ -semigroups on Banach spaces under the assumption that the norm of the resolvent of the semigroup generator grows with \(|s|^{\beta }\log (|s|)^b\) ${|s|}^{\beta }log{(|s|)}^b$ , \(\beta , b \ge 0\) $\beta ,b\ge 0$ , as \(|s|\rightarrow \infty \) $|s|\to \infty$ , and with \(|s|^{-\alpha }\log (1/|s|)^a\) ${|s|}^{-\alpha }log{(1/|s|)}^a$ , \(\alpha , a \ge 0\) $\alpha ,a\ge 0$ , as \(|s|\rightarrow 0\) $|s|\to 0$ . Our results do not suppose that the semigroup is bounded. In particular, for \(a=b=0\) $a=b=0$ , our results improve the rates involving Fourier types obtained by Rozendaal and Veraar (J Funct Anal 275(10):2845–2894, 2018).