I first asked this question on math.stackexchange here but it seems it is more a research level question ...
At the $0$ order of derivatives of Sobolev spaces and for a fixed integrability order $p$, we find Besov spaces $\dot{B}^0_{p,q}$, Triebel Lizorkin spaces $\dot{F}^0_{p,q}$ and Lorentz spaces $L^{p,q}$, with in particular if $p≥ 2$ $$ \begin{align*} \dot{B}^0_{p,1} ⊂ \dot{B}^0_{p,2} ⊂ L^p &= \dot{F}^0_{p,2} \subset \dot{F}^0_{p,p} =\dot{B}^0_{p,p} ⊂ \dot{B}^0_{p,\infty} \\ L^{p,1} \subset L^{p,2} ⊂ L^{p} &= L^{p,p} \subset L^{p,\infty}. \end{align*} $$ So, the ordering is well understood when remaining either in the $B,F$ setting, either in the Lorentz setting. so my question is Is there any embedding from one setting into the other one? (except the trivial embeddings on/from $L^p$).
The way of building these spaces is different (in one case one cuts in frequency and in the other one cuts in height). However, Sobolev embeddings tells us that we can trade a bit of local regularity for a bit of local integrability. Moreover the function $|x|^{-a}$ is in both $\dot{B}^{0}_{d/a,\infty}$ and $L^{d/a,\infty}$, but not in $L^{d/a}$.
