Given $u(t) \in \mathcal{L}_2$ and $y(t) \in \mathcal{L}_2$, let the induced gain is given by $\gamma$ as shown below
$$ \lVert z \rVert_2 \leq \gamma \lVert u \rVert_2 $$
similarly, for $u(t) \in \mathcal{L}_\infty$ and $y(t) \in \mathcal{L}_\infty$, let the induced gain is given by $\beta$ as shown below
$$ \lVert z \rVert_\infty \leq \beta \lVert u \rVert_\infty $$
The question that I want to ask is can we give upper bound on $\beta$ in terms of $\gamma$?