Do we have the continuous embedding $\|u\|_{\dot{B}^{s,1}_1}\leq C\|u\|_{\dot{W}^{s,1}}$? Here $$\|u\|_{\dot{B}^{s,1}_1} = \sum_{k=1}^{\infty}2^{ks}\|P_ku\|_{L^1}\\ \|u\|_{\dot{W}^{s,1}} = \||\nabla|^su\|_{L^1},$$
where $|\nabla|^su$ and the Littlewood-Paley projector $P_ku$ are defined in frequency space as $$\widehat{|\nabla|^s u}=|\xi|^s\hat{u}\\ \widehat{P_k u} = \phi_k(\xi)\hat{u}(\xi)$$ with $\phi_k$ a smooth cutoff function supported on the annulus $|\xi|\sim 2^k$ satisfying $\sum_{k\in\mathbb{Z}}\phi_k = 1$.
I would like to be able to say that
- $2^{ks}\|P_k u\|_{L^1} \sim \|P_k|\nabla|^su\|_{L^1}$, but as far as I can tell, the inequality only goes in the $\gtrsim$ direction.
- $\sum_{k\geq 1}\|P_k|\nabla|^s u\|_{L^1}\lesssim \||\nabla|^su\|_{L^1}$, but this feels reminiscent of the Littlewood-Paley inequality (but with $\ell^2$ replaced by $\ell^1$) which fails for $p=1$.
I feel like the embedding should be false, but I'm having trouble coming up with a counterexample. If it is indeed false, a follow up question: can one get the embedding $\|u\|_{\dot{B}^{s,1}_1}\leq C\|u\|_{\dot{W}^{r,1}}$ for $r>s$?