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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 $r_2<r_3$. 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 $r_2<r_3$? In illustration, does there animate some reference for this kindly of bounds?

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