Diagonalization of the Linearized Non-Cutoff Radially Symmetric Boltzmann Operator

We provide some new explicit expressions for the linearized non-cutoff radially symmetric Boltzmann operator with Maxwellian molecules, proving that this operator is a simple function of the standard harmonic oscillator. In fact, that operator is shown to be diagonal in the Hermite basis and to be essentially a fractional power of the harmonic oscillator. We prove as well that this linearized operator is a pseudodifferential operator and we give a complete asymptotic expansion for its symbol whose leading part is a fractional harmonic oscillator. This provides sharp coercive estimates for the linearized non-cutoff radially symmetric Boltzmann operator.