Coeffective basic cohomologies of $K$--contact and Sasakian manifolds

In this paper we define coeffective de Rham cohomology for basic forms on a $K$--contact or Sasakian manifold $M$ and we discuss its relation with usually basic cohomology of $M$. When $M$ is of finite type (for instance it is compact) several inequalities relating some basic coeffective numbers to classical basic Betti numbers of $M$ are obtained. In the case of Sasakian manifolds, we define and study coeffective Dolbeault and Bott-Chern cohomologies for basic forms. Also, in this case, we prove some Hodge decomposition theorems for coeffective basic de Rham cohomology, relating this cohomology with coeffective basic Dolbeault or Bott-Chern cohomology. The notions are introduced in a similar manner with the case of symplectic and K\"ahler manifolds.