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In the original paper of Nirenberg, the following inequality is proven : if $\frac{1}{p} = \frac{j}{n} + \left( \frac{1}{r} - \frac{m}{n} \right) \alpha + \frac{1 - \alpha}{q}$ and $\frac{j}{m} \leq \alpha < 1$ (I put $< 1$ to avoid the "second exceptional case"), then

$$\| \mathrm{D}^{j} u \|_{L^{p}} \leq C_{1} \| \mathrm{D}^{m} u \|_{L^{r}}^{\alpha} \| u \|_{L^{q}}^{1 - \alpha} + C_{2} \| u \|_{L^{s}} $$

where $s > 0$ is arbitrary and the norms are on $L^p$ spaces over a bounded Lipschitz domain $\Omega$ in $\mathbb R^n$. If $\Omega = \mathbb R^n$ instead, there is no need for the second term in the right-hand side.

Nonetheless, in Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations several examples of this inequality are mentioned in a bounded domain without the second term.

Do someone know if this is wrong or if the original inequality is not optimal ?

Thanks

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Oh, sorry I got it : Brézis mentions the inequality with $W^{m,r}$ instead of only the $L^r$ norm of $D^m u$, like this

$$\| \mathrm{D}^{j} u \|_{L^{p}} \leq C_{1} \| u \|_{W^{m,r}}^{\alpha} \| u \|_{L^{q}}^{1 - \alpha} $$

So in fact I guess he just uses $s = q$, and writes $\|u\|_{L^q} = \|u\|_{L^q}^\alpha \|u\|_{L^q}^{1-\alpha}$ and controls one term with the Rellich embedding and factorizes.

I leave the question here if someone has the same problem one day, hopefully I'll save him some time...

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