SOME EMBEDDINGS RELATED TO HOMOGENEOUS TRIEBEL–LIZORKIN SPACES AND THE BMO FUNCTIONS

Abstract

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

Keywords

• besov spaces
• 𝐵𝑀𝑂 functions
• realizations
• triebel– lizorkin spaces