Multiplication properties and the Rellich-Kondrachov theorem in Herz-type Sobolev spaces

In this paper, we prove that Herz-type Sobolev spaces form a Banach algebra and establish a Rellich–Kondrachov compactness theorem for these spaces. These results extend the corresponding classical theory and further demonstrate that Herz-type Sobolev spaces provide a natural generalization of the classical Sobolev spaces.