= Solution
The reference to part (f) within this part is a printed self-reference; the needed product estimate is part (e). Since both masses are one, subtracting the <Bobylev identities> gives, for $\xi\ne0$,
$$
\partial_t\frac{\widehat f(\xi)-\widehat g(\xi)}{|\xi|^2}
+\frac{\widehat f(\xi)-\widehat g(\xi)}{|\xi|^2}
=\frac1{|\mathbb S^2|}\int_{\mathbb S^2}
\frac{\widehat f(\xi^+)\widehat f(\xi^-)-\widehat g(\xi^+)\widehat g(\xi^-)}{|\xi|^2}\,d\sigma.
$$
The normalized angular average and part (e) give
$$
\boxed{\left|\partial_t\frac{\widehat f-\widehat g}{|\xi|^2}+\frac{\widehat f-\widehat g}{|\xi|^2}\right|\leq d(f,g)}.
$$
For completeness, the <Duhamel principle> for this scalar equation yields $d(t)\leq e^{-t}d(0)+\int_0^te^{-(t-s)}d(s)\,ds$. Apply the <Gronwall inequality> to $e^td(t)$ to obtain the <Fourier nonexpansion for Maxwell molecules>, $\boxed{d(f_t,g_t)\leq d(f_0,g_0)}$. This estimate is nonexpansion; by itself it does not prove strict decay or convergence to a specified equilibrium.
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