Local existence of polynomial decay solutions to the Boltzmann equation for soft potentials

The existence of classical solutions to the Cauchy problem for the Boltzmann equation without angular cutoff has been extensively studied in the framework when the solution has Maxwellian decay in the velocity variable, cf. [6, 8] and the references therein. In this paper, we prove local existence of solutions with polynomial decay in the velocity variable for the Boltzmann equation with soft potential. In the proof, the singular change of variables between post-and pre-collision velocities plays an important role, as well as the regular one introduced in the celebrated cancelation lemma by Alexandre-Desvillettes-Villani-Wennberg [Entropy dissipation and long-range interactions, Arch. Ration. Mech. Anal. 152 (2000) 327-355].