How to prove
$$ \lVert uv\rVert\_{\dot{B}^{\frac{N}{p}-1}\_{p,1}}\leqslant C \lVert u\rVert\_{\dot{B}^{\frac{N}{p}}\_{p,1}} \lVert v\rVert\_{\dot{B}^{\frac{N}{p}-1}\_{p,1}}$$


when $N\geqslant2 $and$1\leqslant p<2N$.


I know it needs Bony decomposition, but I don’t know how to use the condition $N\geqslant2 $and$1\leqslant p<2N$. I could handle the paraproduct. But when dealing with the remainder, I have to assume $p\geqslant2 $and$p<2N$ instead of $N\geqslant2 $.


Any hints would be appreciated!



Show a inequality in homogeneous Besov space
I shall write $\mathrm{P}\_k$ for the homogeneous Littlewood-Paley projectors, and the paraproduct decomposition as


$$ uv = u\prec v + u\succ v + u\diamond v $$


Assume $1\le p<2$. Then $p'>p$ and Bernstein applies to estimate the $L^{p'}\_x$ norm in terms of the $L^p\_x$ norm, at a cost of derivatives. The resonant term $u\diamond v$ can be estimated as follows.


$$\begin{aligned}
2^{(\frac Np-1)k}\left\|\mathrm{P}\_k(u\diamond v)\right\|\_{L^p\_x}
&\lesssim
2^{(N-1)k}\left\|\mathrm{P}\_k(u\diamond v)\right\|\_{L^1\_x}
\\&\lesssim
2^{(N-1)k}\sum\_{\substack{k\_\*\in\mathbb{Z}\,:\\k\_\*\ge k-O(1)}}
\left\|\mathrm{P}\_{k\_\*}u\right\|\_{L^{p'}\_x}
\left\|\mathrm{P}\_{\approx k\_\*}v\right\|\_{L^p\_x}
\\&\lesssim
2^{(N-1)k}\sum\_{\substack{k\_\*\in\mathbb{Z}\,:\\k\_\*\ge k-O(1)}}
2^{(\frac Np-\frac N{p'})k\_\*}\left\|\mathrm{P}\_{k\_\*}u\right\|\_{L^p\_x}
\left\|\mathrm{P}\_{\approx k\_\*}v\right\|\_{L^p\_x}
\\&\approx
\sum\_{\substack{k\_\*\in\mathbb{Z}\,:\\k\_\*\ge k-O(1)}}
2^{(N-1)(k-k\_\*)}
\left(2^{\frac Npk\_\*}\left\|\mathrm{P}\_{k\_\*}u\right\|\_{L^p\_x}\right)
\left(2^{(\frac Np-1)k\_\*}\left\|\mathrm{P}\_{\approx k\_\*}v\right\|\_{L^p\_x}\right)
\;.\end{aligned} $$


Because $N-1>0$, we may apply Young's convolution inequality on $\mathbb{Z}$ to obtain


$$ \begin{aligned}
\left\|u\diamond v\right\|\_{\dot{B}^{\frac Np-1}\_{p,1}}
&\lesssim
\sum\_{k\_\*\in\mathbb{Z}}
\left(2^{\frac Npk\_\*}\left\|\mathrm{P}\_{k\_\*}u\right\|\_{L^p\_x}\right)
\left(2^{(\frac Np-1)k\_\*}\left\|\mathrm{P}\_{\approx k\_\*}v\right\|\_{L^p\_x}\right)
\\&\lesssim
\left\|u\right\|\_{\dot{B}^{\frac Np}\_{p,1}}
\left\|v\right\|\_{\dot{B}^{\frac Np}\_{p,1}}
\;.\end{aligned} $$


