$\|u\|_{L^{3}(\mathbb R)} \leq C \|Du\|_{L^{2}(\mathbb R)}^{\alpha} \|u\|_{L^{2}(\mathbb R)}^{1-\alpha}$?

Question

Let $u\in H^{1}(\mathbb R).$

Is Gagliardo–Nirenberg interpolation inequality valid for the $p=3, q=r=2, m=1, 0< \alpha < 1$ ; and $j=0$ ? That is, is it true that,$\|u\|_{L^{3}(\mathbb R)} \leq C \|Du\|_{L^{2}(\mathbb R)}^{\alpha} \|u\|_{L^{2}(\mathbb R)}^{1-\alpha}$ ?

[My confusion is in the above link if they take $j=0,$ then they are assuming that $q=\infty$, is it necessarry]

Answer 1

Well, I don't think that $j=0$ implies $q=\infty$. In any case, you can do interpolation in $L^p$ spaces and then Sobolev embedding

$$ \|u\|_{L^3}^3\leq \|u\|_{L^2}^2\|u\|_{L^\infty}\leq C\|u\|_{L^2}^{2.5}\|\partial_x u\|_{L^2}^{0.5}. $$

You can also do Sobolev embedding and then interpolation between Sobolev spaces:

$$ \|u\|_{L^3}\leq C\|u\|_{H^{1/6}}\leq C\|u\|_{L^2}^{5/6}\|u\|_{H^1}^{1/6} $$