Growth rate of eventually positive Kreiss bounded $C_0$-semigroups on $L^p$ and $\mathcal{C}(K)$

In this paper, we compare several Cesàro and Kreiss type boundedness conditions for a $C_0$-semigroup on a Banach space and we show that those conditions are all equivalent for a positive semigroup on a Banach lattice. Furthermore, we give an estimate of the growth rate of a Kreiss bounded and eventually positive $C_0$-semigroup $(T_t{)}_{t\ge 0}$ on certain Banach lattices $X$. We prove that if $X$ is an $L^p$-space, $1<p<+\mathrm{\infty }$, then $\mathrm{\parallel }{T}_t\mathrm{\parallel }=\mathcal{O}(t\mathrm{/}\log (t{)}^{\max (1\mathrm{/}p,1\mathrm{/}{p}^{\mathrm{\prime }})})$ and if $X$ is an $(\text{AL})$ or $(\text{AM})$-space, then $\mathrm{\parallel }{T}_t\mathrm{\parallel }=\mathcal{O}(t^{1-\epsilon })$ for some $\epsilon \in (0,1)$, improving previous estimates.