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It has long been suspected that the non-cutoff Boltzmann operator has similarcoercivity properties as a fractional Laplacian. This has led to the hope thatthe homogenous Boltzmann equation enjoys similar regularity properties as theheat equation with a fractional Laplacian. In particular, the weak solution ofthe fully nonlinear non-cutoff homogenous Boltzmann equation with initial datumin $L^1_2(\mathbb{R}^d)\cap L\log L(\mathbb{R}^d)$, i.e., finite mass, energyand entropy, should immediately become Gevrey regular for strictly positivetimes. We prove this conjecture for Maxwellian molecules.

Gevrey smoothing for weak solutions of the fully nonlinear homogeneous Boltzmann and Kac equations without cutoff for Maxwellian molecules