Let $f$ be a continuous stochastic process bounded in $L^p(\Omega)$, i.e., $\mathbb E|f(t)|^p \leq C$ for all $t\geq 0$, $g \in L^\infty(\mathbb R) \cap C^\infty(\mathbb R)$ for all $1 \leq p \leq \infty$ with support in $[0, \infty)$. Let the convolution be defined as $$ F(t) = (f \star g)(t) =\int_0^t f(s)g(t-s)ds. $$ I would like to bound, for all $p \geq 1$, the quantity $\mathbb E|F(t)|^p$. I looked into Young's inequality for convolutions, which gives for $p,q,r \geq 1$ such that $p^{-1} + q^{-1} = 1 + r^{-1}$, $$ \| F\|_r \leq \|f\|_p \|g\|_q. $$ Now, choosing $r = \infty$, we have for all $t > 0$ and a generic constant $C$ $$ \mathbb E |F(t)|^p \leq \int_0^t \mathbb E|f(s)|^p ds \left(\int_0^t |g(s)|^q ds \right)^{p-1} \leq Ct. $$ Since the convolution should be a "smoother version" of $f$, I would have guessed to find a bound independent of time, i.e., that $\mathbb E |F(t)|^p \leq C$.
What am I missing here?
EDIT: I forgot the assumption $$ \int_0^t |g(s)|^p ds \leq C $$ for all $p \geq 1$.