Suppose that, for all $(t,x) \in [0,T]\times \mathbb{R}$, a two-parameter stochastic process $Z(t,x)$ satisfies $$ \| Z(t,x)\|_{L^p(\Omega)} \leq C_{p,T} $$ for some constant depending on $p$ and $T$. Is it necessarily true that the following holds? $$ \Big\| \sup_{(t,x) \in [0,T]\times \mathbb{R}} Z(t,x) \Big\|_{L^p(\Omega)} < \infty $$
EDIT: I should mention that this process has some Holder regularity in both $t$ and $x$.