A direct proof of Sobolev embeddings for quasi-homogeneous Lizorkin–Triebel spaces with mixed norms

Abstract

The article deals with a simplified proof of the Sobolev embedding theorem for Lizorkin–Triebel spaces (that contain the Lp-Sobolev spaces Hps as special cases). The method extends to a proof of the corresponding fact for general Lizorkin–Triebel spaces based on mixed Lp-norms. In this context a Nikol' skij–Plancherel–Polya inequality for sequences of functions satisfying a geometric rectangle condition is proved. The results extend also to anisotropic spaces of the quasi-homogeneous type.