# Product of Besov and Lorentz functions

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 $$r_2. This may follow by some (generalized) Moser-Trudinger inequality for $$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? In case, does there exist some reference for this kind of bounds?

• 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. Commented Feb 23, 2021 at 10:29
• 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. Commented Feb 24, 2021 at 3:17