Pseudodifferential Operators, Rellich-Kondrachov Theorem and Hardy-Sobolev Spaces

TL;DR

This paper extends the Rellich-Kondrachov theorem to pseudodifferential operators on Hardy spaces, providing new boundedness and compact embedding results for Hardy-Sobolev spaces.

Contribution

It introduces versions of the Rellich-Kondrachov theorem for pseudodifferential operators on Hardy spaces, with new boundedness properties and applications to Hardy-Sobolev space embeddings.

Findings

Boundedness properties for pseudodifferential operators with Hörmander class symbols.

Compact embedding results for Hardy-Sobolev spaces.

Extension of classical theorems to nonhomogeneous Hardy spaces.

Abstract

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.