Mouhot–Strain spectral gap conjecture for linearized Boltzmann operators

Let $B$ be a Boltzmann collision kernel with parameters $\gamma\in(-d,\infty)$ and $\alpha\in[0,2)$. A spectral gap means that the associated linearized Boltzmann collision operator has a positive spectral gap. Mouhot–Strain conjecture. The linearized Boltzmann collision operator associated with $B$ admits a spectral gap if and only if

$$\gamma+\alpha\geq 0.$$

The same statement should hold formally for angular cutoff, with $\alpha=0$, and for the linearized Landau collision operator as the limiting case $\alpha=2$. The sufficient condition was supported by constructive coercivity estimates, while the necessary condition was subsequently proved by Gressman and Strain using sharp constructive upper and lower bounds in terms of a geometric fractional Sobolev norm.