Let $E$ be a $\mathbb R$-Banach space, $v:E\to[1,\infty)$ be continuous and $(\kappa_t)_{t\ge0}$ be a Markov semigroup on $(E,\mathcal B(E))$.
Assume there are $(C,R,M)\in[0,\infty)\times[0,1)\times[1,\infty)$ and a nonincreasing $\xi:[0,1]\to[0,1)$ with $$\kappa_tv^r\le Cv^{r\xi(t)}\;\;\;\text{for all }t\in[0,1]\text{ and }r\in(R,2M]\tag1.$$ How can we conclude that there is a $\tilde C\ge0$ such that $$\kappa_tv^r\le\tilde Cv^{r\tilde\xi(t)}\;\;\;\text{for all }t\ge0\text{ and }r\in[0,2M)\tag2,$$ where $$\tilde\xi(t):=\xi(1)^{\lfloor t\rfloor}\xi(t-\lfloor t\rfloor)\;\;\;\text{for }t\ge0?$$
Inductively applying $(1)$ with $t=1$ easily yields $$\kappa_nv^r\le C^nv^{r\xi(1)^n}\;\;\;\text{for all }n\in\mathbb N_0\text{ and }r\in(R,2M]\tag3.$$ The problematic thing is the power $n$ of $C$ which occurs in $(3)$: If $t\ge0$, we may clearly write $t=\lfloor t\rfloor+\iota(t)$, where $\iota(t):=t-\lfloor t\rfloor\in[0,1)$. Now, by $(1)$, $$\kappa_{\iota(t)}v^r\le Cv^{r\xi(\iota(t))}\;\;\;\text{for all }r\in(R,2M]\tag4$$ and hence, by $(3)$, $$\kappa_tv^r\le C^{\lfloor t\rfloor}\kappa_{\iota(t)}v^{r\xi(1)^{\lfloor t\rfloor}}\le C^{\lfloor t\rfloor+1}v^{r\tilde\xi(t)}\tag5$$ for all $r\in(R,2M]$.
I've no idea how to extend this to $r\in(0,2M]$, but my main problem is finding the constant $\tilde C$ such that $(2)$ at least holds for $r\in(R,2M]$.
The claim can be found in this paper, in Remark 3.1 on p. 12. It is claimed that it follows from $(1)$ and Jensen's inequality, but I've no idea at which point Jensen's inequality could be of use.