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

As the homogeneous Triebel–Lizorkin space $\dot F^{s}_{p, q}$ and the space $BMO$ are defined modulo polynomials and constants, respectively, we prove that $BMO$ coincides with the realized space of $\dot F^{0}_{\infty, 2}$ and cannot be directly identified with $\dot F^{0}_{\infty, 2}$. In case $p<\infty$, we also prove that the realized space of $\dot F^{n/p}_{p, q}$ is strictly embedded into $BMO$. Then we deduce other results in this paper, that are extensions to homogeneous and inhomogeneous Besov spaces, $\dot B^{s}_{p, q}$ and $B^{s}_{p, q}$, respectively. We show embeddings between $BMO$ and the classical Besov space $ B^{0}_{\infty, \infty}$ in the first case and the realized spaces of $\dot B^{0}_{\infty, 2}$ and $\dot B^{0}_{\infty, \infty}$ in the second one. On the other hand, as an application, we discuss the acting of the Riesz operator $\mathcal{I}_{\beta}$ on $BMO$ space, where we obtain embeddings related to realized versions of $\dot B^{\beta}_{\infty, 2}$ and $\dot B^{\beta}_{\infty, \infty}$.