Besov or Triebel-Lizorkin spaces versus Lorentz spaces

Question

At the $0$ order of derivatives of Sobolev spaces, 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?

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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 $L^{d/a}$.

Answer 1

(I came across this problem by accident, hopefully it's not outdated in the sense of, say, looking for some research problems.)

This turns to be a difficult problem. There are Besov-Lorentz spaces $B_{(p,r),q}^s$ and Triebel-Lorentz space $F_{(p,r),q}^s$ where the $L^p$ integral is replaced by the $L^{p,r}$ norm. In this setting we have $L^{p,r}=F_{(p,r),2}^0$ when $1<p<\infty$ and $0<r\le\infty$.

According to Seeger-Trebels (2019), when $1<p<\infty$, $B_{p,q}^0\hookrightarrow L^{p,r}$ holds iff one of the following:

• $2\neq p\le r$ and $q\le\min(p,2)$;
• $2=p\le r$ and $q\le \min(p,r)$.

And $L^{p,r}\hookrightarrow B_{p,q}^0$ holds iff one of the following:

• $2\neq p\ge r$ and $q\ge\max(p,2)$;
• $2=p\ge r$ and $q\ge \max(p,r)$.

$F_{(p,r_0),q_0}^0\hookrightarrow F_{(p,r_1),q_1}^0$ holds iff $r_0\le r_1$ and $q_0\le q_1$.