  We aim at understanding how the non-commutation phenomena between a linear
transport operator and a fractional diffusion allow the transport operator to
satisfy hypoelliptic estimates on the whole space. Such hypoelliptic estimates
are obtained for a large class of linear transport operators by using the
classical multiplier method. The main motivation of this work arises from the
study of the hypoelliptic regularity of the solutions of kinetic equations
associated with a free transport operator.
