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In the following, we assume that $u,v\in L^{\infty}\cap H^{s}$, in which $s>0$ and $s\in\mathbb{N}$, $|\alpha|+|\beta|=s$.$\alpha,\beta\in\mathbb{N}^{n}$,that is to say, they are both multi-index and we suppose futhur that $|\alpha|\geq1,|\beta|\geq1$.
When I study the book "Pseudodifferential operators and nash moser theorem" by Alinhac and Gerard, I encounter a problem. Suppose that , then the authors give$$\partial^{\alpha}u\partial^{\beta}v=\sum_{j}*\partial_{i_{j}}(\partial^{\alpha_{j}}u\partial^{\beta_{j}}v)+*u\partial^{\alpha+\beta}v.$$ $|\alpha_{j}|+|\beta_{j}|=s-1$ *denotes constant that is harmless. How can we get this identity? ----------I get it, for example, we just consider spatial dimension=1, $\partial^{2}u\partial^{3}v=\partial(\partial u\partial^{2}v)-\partial u\partial^{3}v=\partial(\partial u\partial^{2}v)-\partial(u\partial^{3}v)-u\partial^{4}v$, where $\partial$ denotes $\partial_{x}$ and I think this will not introduce any vague.
The author also says that, if we can prove that $$\color{red}{|\partial^{\alpha_{j}}u\partial^{\beta_{j}}v|_{L^{2}}\leq const\cdot\Big(|u|_{L^{\infty}}|v|_{H^{s}}+|v|_{L^{\infty}}|u|_{H^{s}}\Big)},$$ then we can get $$\color{purple}{|\partial^{\alpha}u\partial^{\beta}v|_{L^{2}}\leq const\cdot\Big(|u|_{L^{\infty}}|v|_{H^{s}}+|v|_{L^{\infty}}|u|_{H^{s}}\Big)},$$ Why?
I have understand this problem, the authors get the estimates for $$|\partial^{\alpha_{j}}u\partial^{\beta_{j}}v|_{H^{1}}\lesssim\Big(|u|_{L^{\infty}}|v|_{H^{s}}+|v|_{L^{\infty}}|u|_{H^{s}}\Big)$$ by dyadic analysis. Then according to $$|\partial_{i_{j}}(\partial^{\alpha_{j}}u\partial^{\beta_{j}}v)|_{L^{2}}\leq |\partial^{\alpha_{j}}u\partial^{\beta_{j}}v|_{H^{1}}$$ the authors get the main result. If we estimate $|\partial^{\alpha}u\partial^{\beta}v|_{L^{2}}$ directly by dyadic analysis, then we can just get $$|S_{q}(\partial^{\alpha}u)(\partial^{\beta}v)_{q}|_{L^{2}}\leq const\cdot|u|_{L^{\infty}}|v|_{H^{s}}c_{q}$$ We can not get anything about $|\partial^{\alpha}u\partial^{\beta}v|_{L^{2}}$.
In addition, where can I find the proof of Gagliardo-Nirenberg inequality: If $u\in L^{\infty}\cap H^{s}(s>0,s\in\mathbb{N})$, then for all $\alpha\in\mathbb{N}^{n}$ satisfying $0\leq|\alpha|\leq s$ and $p=\frac{2s}{|\alpha|}$, we have $$\color{orange}{|\partial^{\alpha}u|_{L^{p}}\leq const\cdot|u|_{L^{\infty}}^{1-\frac{|\alpha|}{s}}|u|_{H^{s}}^{\frac{|\alpha|}{s}}}.$$ The author offer us a reference but I can't find in the internet: T. Aubin. inégalités d'interpolation. Bull.Sci.École Norm. Sup.(4) 14(1981),229-234. But unfortunately, I can not find this and I can not understand French, any English reference?