Bobylev identity ($\hat{Q}(\xi )=\frac{1}{4\pi }\int \hat{f}({\xi }^{+})\hat{f}({\xi }^{-})d\sigma -\hat{f}(\xi )\hat{f}(0)$)

For the Maxwell molecule collision operator and the Fourier transform with exponent $-i\xi \cdot v$, put ${\xi }^{\pm }=(\xi \pm |\xi |\sigma )/2$. The angular exchange for elastic collisions, rotation invariance of spherical surface area and factorization of the two velocity integrals give the displayed formula. This identity converts the collision integral into a spherical average of products at two related frequencies.

For finite nonnegative measures with equal mass and first moment and finite second moments, the constant and linear terms of their Fourier transforms cancel. Taylor's remainder bounds their difference by a constant times $|\xi {|}^2$, making this distance finite. Fourier uniqueness gives definiteness. The origin is excluded from the supremum; the quotient need not have a direction-independent limit there.

Unit-mass solutions of the isotropic Maxwell molecule collision operator with matching first moments obey this comparison. The Bobylev identity, the Fourier bound by one, and $|{\xi }^{+}{|}^2+|{\xi }^{-}{|}^2=|\xi {|}^2$ give a scalar damped differential inequality. Its integral form and the Gronwall inequality yield nonexpansion in the Fourier distance of order two. This does not alone establish strict contraction or equilibrium convergence.