Let us fix $ninmathbb{N}^+$ and $p,qin (1,infty)$. Given $r_1,r_2,r_3in(1,infty)$, 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 $r_2leq r_3$, 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?**