Some embeddings related to homogeneous Triebel-Lizorkin spaces and the BMO functions

As the homogeneous Triebel-Lizorkin space 9 , and the space are defined modulo polynomials and constants, respectively, we prove that coincides with the realized space of 9 0 8, 2 and cannot be directly identified with 9 0 8, 2 .In case ă 8, we also prove that the realized space of 9 { , is strictly embedded into .Then we deduce other results in this paper, that are extensions to homogeneous and inhomogeneous Besov spaces, 9 , and , , respectively.We show embeddings between and the classical Besov space 0 8, 8 in the first case and the realized spaces of 9 0 8, 2 and 9 0 8, 8 in the second one.On the other hand, as an application, we discuss the acting of the Riesz operator ℐ on space, where we obtain embeddings related to realized versions of 9 8, 2 and 9 8, 8 .