# reference request – Product of Besov and Lorentz functions

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 $$r_2. This may follow by some (generalized) Moser-Trudinger inequality for $$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?