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

Abstract

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 $|s|\to \infty$, and with $|s{|}^{-\alpha }\log (1/|s|{)}^a$, $\alpha ,a\ge 0$, as $|s|\to 0$. 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).