= Solution
Taking the <essential supremum> in the characteristic solution gives the sharper estimate
$$
\boxed{\|f_t\|_\infty\leq\|f_0\|_\infty+\int_0^t\|h(s,\cdot,\cdot)\|_\infty\,ds.}
$$
Indeed, the bijective <characteristic flow map> preserves each spatial-velocity <essential supremum>. If $H_t=\operatorname*{ess\,sup}_{0\leq s\leq t,\ x,v\in\mathbb R}|h(s,x,v)|$, this proves $\|f_t\|_\infty\leq\|f_0\|_\infty+tH_t$. A time-independent source has $H_t=\|h\|_{L^\infty(\mathbb R^2)}$, which is the displayed form. For a time-dependent source, the same symbol must mean a bound uniform over the elapsed time interval; its <norm> at the final time alone need not bound the accumulated forcing.
The bound is \b[sharp]. Take $f_0=1$ and $h=1$, giving $f(t,x,v)=1+t$ and equality for every $t\geq0$. Both functions are smooth and bounded, although their finite-$p$ <integrals> over the whole plane are infinite.
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