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?
