A direct proof of Sobolev embeddings for Triebel-Lizorkin spaces, including mixed norms and quasi-homogeneity

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