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 $\vert s\vert^{\beta}\log(\vert s\vert)^b$, $\beta, b \geq 0$, as $\vert s\vert\rightarrow\infty$, and with $\vert s\vert^{-\alpha}\log(1/\vert s\vert)^a$, $\alpha, a \geq 0$, as $\vert s\vert \rightarrow 0$. 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).