  We will present versions of the Rellich-Kondrachov theorem for
pseudo-differential operators acting on localizable Hardy spaces. One of the
techniques includes boundedness properties for pseudodifferential operators
with symbols in the H\"ormander class, that might be of interest in his own
right, extending results previously obtained by Goldberg, Alvarez and Hounie,
Taylor - among others. As application, we obtain compact embedding results for
distributions in nonhomogeneous localizable Hardy-Sobolev spaces.

Analysis of PDEs