# reference request – Product of Besov and Lorentz capabilities retort

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## reference request – Product of Besov and Lorentz capabilities

Let us moor $$ninmathbb{N}^+$$ and $$p,qin [1,infty)$$. Given $$r_1,r_2,r_3in[1,infty)$$, I’d affection to win whether or not we maintain the certain
$$|fg|_{L^{q,r_3}(mathbb{R}^n)}lesssim |f|_{B^{n/p}_{p,r_1}(mathbb{R}^n)}|g|_{L^{q,r_2}(mathbb{R}^n)}quad(*),$$
the place $$B$$ and $$L$$ denote respectively Besov and Lorentz areas.

For occasion, $$(*)$$ holds when $$r_1=1$$ and $$r_2leq r_3$$, because of 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 $$L^{infty}(mathbb{R}^n)$$, however it’s conceivable that (*) holds for apt decisions of $$r_2. This could succeed by some (generalized) Moser-Trudinger inequality for $$B^{n/p}_{p,r_1}$$ mixed with product estimates in Orlicz/Lorentz areas, however I maintain been unable neither to method up with a proof nor to discover a reference.

Does appraise $$(*)$$ really maintain for some $$(r_1,r_2,r_3)$$ with $$r_1>1$$ and $$r_2? In illustration, does there animate some reference for this kindly of bounds?

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