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We consider the kinetic theory of a three-dimensional fluid of weaklyinteracting bosons in a non-equilibrium state which includes both normal fluidand a condensate. More precisely, we look at the previously postulatednonlinear Boltzmann-Nordheim equations for such systems, in a spatiallyhomogeneous state which has an isotropic momentum distribution, and welinearize the equation around an equilibrium state which has a condensate. Westudy the most singular part of the linearized operator coming from the threewaves collision operator for supercritical initial data. The operator has twotypes of singularities, one of which is similar to the marginally smoothingoperator defined by the symbol $\ln(1+p^2)$. Our main result in this context isthat for initial data in a certain Banach space of functions satisfying aH\"{older} type condition, at least for some finite time, evolution determinedby the linearized operator improves the H\"{o}lder regularity. The maindifficulty in this problem arises from the combination of a point singularityand a line singularity present in the linear operator, and we have to use somefine-tuned function spaces in order to carry out our analysis.

Smoothing Properties of a Linearization of the Three Waves Collision Operator in the bosonic Boltzmann-Nordheim Equation