The geometry of binary collisions and generalized Radon transforms (Q1374720)

The geometry of elastic collisions is essential for the regularizing property of the gain term in the Boltzmann equation, which was proved by P. L. Lions. In this paper, the author shows that such result can be deduced from a regularity theorem for generalized Radon transforms by \textit{C. D. Sogge} and \textit{E. M. Stein} [J. Anal. Math. 54, 165-188 (1990; Zbl 0695.42012)], when the energy of one particle can be expressed in terms of the moments as \(\phi(| v|^2)\), with \(\phi(| v|^2)=| v|^2\) and \(\phi(| v|^2)=\sqrt{1+| v|^2}\), he also shows that the same technique cannot be used with other choices of \(\phi\).