= Solution
For the globally invertible <Hamiltonian flow> from (a), the general characteristic formula is
$$
f(t,z)=f_0(\Phi(0,t,z))+\int_0^t h(\Phi(s,t,z))\,ds.
$$
Every $\Phi(s,t)$ preserves <Lebesgue measure>, by (b). Thus composition with it is an <isometry> of each <Lp space>. The <Minkowski integral inequality> gives
$$
\boxed{\|f(t)\|_p\leq\|f_0\|_p+t\|h\|_p.}
$$
Applying the <Minkowski inequality> also in time yields \b[the <finite-time Lp bound for Hamiltonian transport>]:
$$
\boxed{\|f\|_{L^p([0,T]\times\mathbb R^{2d})}
\leq T^{1/p}\|f_0\|_p+
\left(\frac{T^{p+1}}{p+1}\right)^{1/p}\|h\|_p<\infty.}
$$
The smoothness assumptions allow the characteristic construction; the norm estimate itself only uses the <Lp space> data and volume preservation.
For a concrete failure on infinite time, choose $\omega=1$, $f_0=0$, and
$$
h(x,v)=e^{-(|x|^2+|v|^2)/2}=e^{-H_1(x,v)}.
$$
This is smooth, time independent and in every finite <Lp space>; it is invariant under the <isotropic harmonic oscillator flow>. Therefore \b[the solution grows linearly]:
$$
\boxed{f(t,x,v)=t\,h(x,v),\qquad
\int_0^\infty\|f(t)\|_p^p\,dt
=\|h\|_p^p\int_0^\infty t^p\,dt=\infty.}
$$
This <invariant-source secular growth in Hamiltonian transport> supplies the counterexample even with zero initial data.
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