Refined decay rates of $$C_0$$ -semigroups on Banach spaces

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