  In many works, the linearized non-cutoff Boltzmann operator is considered to
behave essentially as a fractional Laplacian. In the present work, we prove
that the linearized non-cutoff Boltzmann operator with Maxwellian molecules is
exactly equal to a fractional power of the linearized Landau operator which is
the sum of the harmonic oscillator and the spherical Laplacian. This result
allows to display explicit sharp coercive estimates satisfied by the linearized
non-cutoff Boltzmann operator for both Maxwellian and non-Maxwellian molecules.
