  We describe regularizing effects in the linearization of a kinetic equation
that arises in study of a system of nonlinear waves satisfying the
Schr\"odinger equation in terms of weak turbulence and condensate. The problem
is first considered in spaces of bounded functions with weights, where
existence of solutions and some first regularity properties are proved. After a
suitable change of variables the equation is written in terms of a pseudo
differential operator. Homogeneity of the equation and classical arguments of
freezing of coefficients may then be used to prove regularizing effect in local
Sobolev type spaces.
