1
$\begingroup$

Let us fix $n\in\mathbb{N}^+$ and $p,q\in [1,\infty)$. Given $r_1,r_2,r_3\in[1,\infty)$, I would like to understand whether we have the bound $$ \|fg\|_{L^{q,r_3}(\mathbb{R}^n)}\lesssim \|f\|_{B^{n/p}_{p,r_1}(\mathbb{R}^n)}\|g\|_{L^{q,r_2}(\mathbb{R}^n)}\quad(*),$$ where $B$ and $L$ denote respectively Besov and Lorentz spaces.

For example, $(*)$ holds when $r_1=1$ and $r_2\leq r_3$, due to the embeddings $B^{n/p}_{p,1}(\mathbb{R}^n)\hookrightarrow L^{\infty}(\mathbb{R}^n)$ and $L^{q,r_2}(\mathbb{R}^n)\hookrightarrow L^{q,r_3}(\mathbb{R}^n)$. When $r_1>1$, $B^{n/p}_{p,r_1}(\mathbb{R}^n)$ fails to embed in $L^{\infty}(\mathbb{R}^n)$, but it is conceivable that (*) holds for suitable choices of $r_2<r_3$. This may follow by some (generalized) Moser-Trudinger inequality for $B^{n/p}_{p,r_1}$ combined with product estimates in Orlicz/Lorentz spaces, but I have been unable neither to come up with a proof nor to find a reference.

Does estimate $(*)$ actually hold for some $(r_1,r_2,r_3)$ with $r_1>1$ and $r_2<r_3$? In case, does there exist some reference for this kind of bounds?

New contributor
Raffaele Scandone is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct.
$\endgroup$
2
  • 1
    $\begingroup$ Generally a decent reference for this kind of inequality is Runst, Sickel: Sobolev spaces of fractional order. You can combine their results for Lp-based spaces with embedding theorems for (Triebel-Lizorkin-) Lorentz spaces.Not sure whether this will give you the answer you are looking for, though. $\endgroup$ – user8707 Feb 23 at 10:29
  • $\begingroup$ Thank you! At a first look, it does not appear that the results in Runst-Sickel book are sufficient, but in any case it is a very good reference. $\endgroup$ – Raffaele Scandone 2 days ago

Your Answer

Raffaele Scandone is a new contributor. Be nice, and check out our Code of Conduct.

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.