Geodesic
Some embeddings related to homogeneous Triebel--Lizorkin spaces and the $BMO$ functions
Problemy analiza, Tome 13 (2024) no. 2, pp. 25-48.

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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 $\dot B^{\beta}_{\infty, \infty}$.
Keywords: $BMO$ functions, realizations, Triebel–Lizorkin spaces.
Mots-clés : Besov spaces 
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B. Gheribi; M. Moussai. Some embeddings related to homogeneous Triebel--Lizorkin spaces and the $BMO$ functions. Problemy analiza, Tome 13 (2024) no. 2, pp. 25-48. http://geodesic.mathdoc.fr/item/PA_2024_13_2_a1/

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