  Let $(E, \| \cdot\|)$ be a Banach space such that, for some $q\geq 2$, the
function $x\mapsto \|x\|^q$ is of $C^2$ class and its first and second
Fr\'{e}chet derivatives are bounded by some constant multiples of $(q-1)$-th
power of the norm and $(q-2)$-th power of the norm and let $S$ be a
$C_0$-semigroup of contraction type on $(E, \| \cdot\|)$. We consider the
following stochastic convolution process \begin{align*}
u(t)=\int_0^t\int_ZS(t-s)\xi(s,z)\,\tilde{N}(\mathrm{d} s,\mathrm{d} z), \;\;\;
t\geq 0, \end{align*} where $\tilde{N}$ is a compensated Poisson random measure
on a measurable space $(Z,\mathcal{Z})$ and $\xi:[0,\infty)\times\Omega\times
Z\rightarrow E$ is an $\mathbb{F}\otimes \mathcal{Z}$-predictable function. We
prove that there exists a c\`{a}dl\`{a}g modification a $\tilde{u}$ of the
process $u$ which satisfies the following maximal inequality \begin{align*}
\mathbb{E} \sup_{0\leq s\leq t} \|\tilde{u}(s)\|^{q^\prime}\leq C\ \mathbb{E}
\left(\int_0^t\int_Z \|\xi(s,z) \|^{p}\,N(\mathrm{d} s,\mathrm{d}
z)\right)^{\frac{q^\prime}{p}}, \end{align*} for all $ q^\prime \geq q$ and
$1<p\leq 2$ with $C=C(q,p)$.
