Can term (I) improve the Fisher-information monotonicity criterion?

Determine whether exploiting the term (I) in the decomposition of the Fisher-information dissipation −I′(f)·Q(f,f) for the spatially homogeneous Boltzmann equation (the momentum-conservation part in equations (9.1)–(9.3)) yields a more general or stronger sufficient condition for the monotonicity of Fisher information than the one obtained by controlling the unsigned term (III) only via term (II) (the energy-conservation part).

Background

In Section 9, the dissipation −I′(f)·Q(f,f) is decomposed into several contributions: (I) (linked to momentum conservation), (II) (linked to tangential variations on the collision sphere and energy conservation, further split into (II)1–(II)3), and an unsigned term (III). The proof of Proposition 10.1 uses only term (II) to control (III) and deduce the key inequality underpinning the Fisher-information monotonicity criterion.

The author remarks that the argument might conceivably be strengthened by also exploiting term (I), but the extent to which this yields a more general result is not established.