= Solution
Interpret $L$ as the velocity operator appearing in the equation and in the stated quadratic-form hypothesis. Let $E(t)=\int\!\!\int|f(t,x,v)|^2\,dx\,dv$. Smoothness turns the distributional equation into the usual equation. Multiply it by $f$ and integrate; for complex functions use the real part of the <inner product>. Spatial <integration by parts> gives $\int v\cdot\nabla_x(f^2)=0$, while the coercivity hypothesis at each $x$ gives
$$
\frac12E'(t)=-\int\!\!\int (Lf)f\,dv\,dx\leq-\delta E(t).
$$
Thus $e^{2\delta t}E(t)$ is nonincreasing. The <coercive energy estimate for kinetic transport> is
$$
\boxed{\|f(t_2)\|_2\leq e^{-\delta(t_2-t_1)}\|f(t_1)\|_2,\qquad t_1\leq t_2}.
$$
No division by $E$ is required, so the zero solution causes no exception. Compact spatial support, or sufficient decay to eliminate the boundary flux, is enough; <compact support> in time is unnecessary.
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