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

The article deals with a simplified proof of the Sobolev embedding theorem for Lizorkin–Triebel spaces (that contain the L_p-Sobolev spaces $$ as special cases). The method extends to a proof of the corresponding fact for general Lizorkin–Triebel spaces based on mixed L_p-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.