Question about application of Gagliardo-Nirenberg inequality

Question

I have seen the following version of the Gagliardo-Nirenberg inequality for $p>2,$ $$\forall v \in H^{1}\left(\mathbb{R}^{d}\right), \quad \int|v|^{p+1} \leq C\left(\int|\nabla v|^{2}\right)^{\frac{d}{4}(p-1)}\left(\int|v|^{2}\right)^{1+\frac{1}{4}(2-d)(p-1)}.$$ I want to know if the following inequality is valid: $$|| v^p||_{L^2}\leq C||v||^{p}_{H^1}.$$ I tried to use the above inequality by replacing $p+1$ by $2p$ and this gave me, $$||v^p||_{L^2}\leq C ||\nabla v ||_{L^2}^{\frac{d}{2}(p-1)} ||v||_{L^2}^{1+\frac{1}{2}(2-d)(p-1)}$$ and then I used the fact that $||v||_{L^2}\leq ||v||_{H^1}$ and $||\nabla v||_{L^2}\leq ||v||_{H^1}$ to get that, $$||v^p||_{L^2}\leq C ||v ||_{H^1}^{\frac{d}{2}(p-1) + 1+\frac{1}{2}(2-d)(p-1)} = C ||v||_{H^1}^{p}.$$ Is this argument correct?

Answer 1

Since $\||v|^p\|_{L^2} = \|v\|_{L^{2p}}^p$, you are essentially asking whether $H^1 \hookrightarrow L^{2p}$. That is true if and only if $$ \begin{cases} 2p \leq \infty, & d = 1, \\ 2p < \infty, & d = 2, \\ 2p \leq \frac{2d}{d-2}, & d \geq 3, \end{cases}$$

see Wikipedia.

Edit: Regarding your argument, I do not think that your “version of the Gagliardo--Nirenberg inequaltiy” is correct. In particular, I do not think it holds for $p+1 > \frac{2d}{(d-2)_+}$. Your calculation then seems in general correct, although I believe that the exponents are wrong: You have not replaced $p+1$ by $2p$ in all places, do you?