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