Show a inequality in homogeneous Besov space 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!
 A: 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
$$ \begin{aligned}
\left\|u\diamond v\right\|_{\dot{B}^{\frac Np-1}_{p,1}}
&\lesssim
\sum_{k_*\in\mathbb{Z}}
\left(2^{\frac Npk_*}\left\|\mathrm{P}_{k_*}u\right\|_{L^p_x}\right)
\left(2^{(\frac Np-1)k_*}\left\|\mathrm{P}_{\approx k_*}v\right\|_{L^p_x}\right)
\\&\lesssim
\left\|u\right\|_{\dot{B}^{\frac Np}_{p,1}}
\left\|v\right\|_{\dot{B}^{\frac Np}_{p,1}}
\;.\end{aligned} $$
