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The Duhamel formula for Hamiltonian transport, phase space preservation and Minkowski integral inequality give the displayed bound. A time-independent source in an Lp space yields finite space-time norm on every bounded interval; it need not yield integrability on infinite time.

Transport a phase-space density along Hamilton's equations. The Hamiltonian vector field is $b=({\nabla }_{v}H,-{\nabla }_{x}H)$ and the equation is ${\partial }_{t}f+b\cdot {\nabla }_{x,v}f=h$. Its zero divergence gives volume-preserving transport. For $h=0$, the density is constant along characteristics; a source is integrated by the Duhamel formula for Hamiltonian transport. This is distinct from the elliptic Liouville equation.

If $f_0=0$ and a nonzero time-independent source is invariant along the Hamiltonian flow, then the Duhamel formula for Hamiltonian transport gives $f_t=th$. For the nonzero-frequency oscillator, $h=e^{-H_{\omega }}$ is smooth and belongs to all finite Lp spaces, yet its resulting solution has infinite space-time norm on $(0,\infty )$.

The characteristic equations for a transport equation are $\dot{X}=V$ and $\dot{V}=X$, so $\ddot{X}=X$. For an initial point $(x_0,v_0)$ their solution isThis is the hyperbolic characteristic flow for an inverted oscillator. The addition formulas give $A_s{A}_t=A_{s+t}$ and $A_t^{-1}=A_{-t}$. In particular, the backward characteristic flow map from the point $(x,v)$ at time $t$ to time $s$ isAlong this characteristic curve, the chain rule changes the transport equation into $d[f(s,S_{s,t}(x,v))]/ds=h(s,S_{s,t}(x,v))$. Integrating from zero to $t$ givesThe assumed $C^1$ regularity makes this a classical solution: on every compact set, the integrand and its needed derivatives are continuous, so differentiation under the finite-time integral is justified. At $t=0$ it has the required initial value, and the characteristic calculation verifies the equation. Conversely every classical solution must satisfy the same integrated identity, proving uniqueness. This is the Duhamel formula for Hamiltonian transport, with Hamiltonian $(v^2-x^2)/2$.

For the isotropic harmonic oscillator flow, Hamilton's equations are $\dot{X}=V$, $\dot{V}=-{\omega }^{2}X$. In dimension three these are three identical uncoupled pairs. For $\omega \ne 0$, put $c_t=\cos (\omega t)$, $s_t=\sin (\omega t)$. The solution from $(x,v)$ at time zero isThe inverse flow is obtained by replacing $t$ by $-t$:Integrating the source along the backward characteristic gives the Duhamel formula for Hamiltonian transport:Indeed $f(t,S_{t}z)=f_0(z)+{\int }_0^{t}h(s,S_{s}z)ds$, and setting $z=S_{-t}(x,v)$ gives the formula. It has the prescribed initial value and differentiation along the characteristic gives the source.

The zero-frequency limit has $s_t/\omega \to t$, $\omega s_t\to 0$, so $S_t(x,v)=(x+tv,v)$ and