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 $H_p^s$ 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 spaces of the quasi-homogeneous type.