I believe that for $p\ge 2$, we have the continuous injection (for $p=2$, it is an equality), $$ B^0_{p,2}(\mathbb R^n)\subset L^p(\mathbb R^n), $$ where $B^0_{p,2}(\mathbb R^n)$ is the Besov space. Is there a straightforward proof?
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1$\begingroup$ Here is a ``straightforward proof." Write the dyadic Littlewood-Paley characterizations of $B^{0}_{p,2}$ and $F^{0}_{p,2} = L^p$ and use Minkowkski's inequality to take the $L^p$ norm inside the $\ell^2$ norm. In any case, this is not really a research-level question. $\endgroup$– sharpendCommented Mar 24, 2019 at 22:20
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Theorem 2.3.2 d) in Triebel says that for $- \infty < s < \infty$ we have (continuous inclusions)
$$B^s_{p,2}(\mathbb{R}^n) \subset F^s_{p,2}(\mathbb{R}^n) = H^s_p(\mathbb{R}^n) \subset B^s_{p,p}(\mathbb{R}^n) \quad \text{for}~2 \leq p < \infty, \\ B^s_{p,p}(\mathbb{R}^n) \subset F^s_{p,2}(\mathbb{R}^n) = H^s_p(\mathbb{R}^n) \subset B^s_{p,2}(\mathbb{R}^n) \quad \text{for}~1 < p \leq 2.$$
The proof seems like a direct estimate for the norms in the respective spaces, which are the ones by dyadic decomposition. Maybe that qualifies as straightforward, at least where $s=0$?
Triebel, Hans, Interpolation theory, function spaces, differential operators., Leipzig: Barth. 532 p. (1995). ZBL0830.46028.
