# Gagliardo Nirenberg Sobolev inequality for n >= 2

I have a quick question regarding the Gagliardo-Nirenberg-Sobolev inequality. It states that: Assume $1 \leq p < n$. There exists a constant $C$, depending only on $p$ and $n$, such that $||u||_{L^{p^{*}}(\mathbb{R}^{n})} \leq C ||Du||_{L^{p}(\mathbb{R}^{n})}$ for all $u \in C_{c}^{1}(\mathbb{R}^{n})$.

I just want to confirm that this Gagliardo-Nirenberg-Sobolev inequality only applies to when $n \geq 2$? I assume the case for $p = n$ is dealt with separately.

In the case $n=1$ we that the Holder norm of $u$ is comparable with the $L^p$ norm of $u'$. Indeed, assume that $n=1$. Note that for all $u\in C_c^1(\mathbb{R})$$u(x)-u(y)=\int_y^x u'(t)dt\tag{1}$$ for$x,y\in\mathbb{R}$. We conclude from$(1)$that \begin{eqnarray} |u(x)-u(y)| &\le& \int_y^x|u'(t)|dt \nonumber \\ &\le& \|u'\|_p|x-y|^{1/p'} \tag{2} \end{eqnarray} Where$p'$is the conjugate exponente of$p'$. Note that we have used Holder in$(2)$and this is possible because$u'\in L^p(\mathbb{R})$for all$p$. We obtain from$(2)$that $$\|u\|_{C^{0,1/p'}(\mathbb{R})}\leq\|u'\|_p\tag{3}$$ Now let's prove that$\|u\|_\infty\leq C\|u'\|_p$. Indeed assume in$(2)$that$y\in (x-1/2,x+1/2)$. We get $$|u(x)|\leq \|u'\|_p+|u(y)|\tag{4}$$ We integrate$(4)$from$x-1/2$to$x+1/2$to conclude that $$|u(x)|\leq \|u'\|_p+\int_{x-1/2}^{x+1/2}|u'(t)|dt\tag{5}$$ We apply Holder inequality on$(5)$to conclude that $$|u(x)|\leq 2\|u'\|_p\tag{6}$$ Now you can finish the proof. • If we define the Holder norm as$||u||_{C^{0, \beta}(\Omega)} := \text{sup}|u(x)| + [u]_{\beta}$for$0 < \beta \leq 1$where$[u]_{\beta} := \text{sup}\{\frac{|u(x)-u(y)|}{|x-y|^{\beta}} \}$from your proof we have$||u||_{C^{0,\beta}(\Omega)} \leq 3||Du||_{p}$– Alex Commented Dec 10, 2013 at 15:03 • I am right then in assuming that the Nirenberg-Gagliardo-Sobolev inequality applies to$n \geq 2$– Alex Commented Dec 10, 2013 at 15:05 • Sorry, but I can't understand your comment. Are you asking me something? or are you showing something? Commented Dec 10, 2013 at 15:29 • I was completing the proof. My original question was if it is implied that the N-G-S inequality applies for$n \geq 2$? – Alex Commented Dec 10, 2013 at 15:52 • In your question you ave asked if Gagliardo-Nirenberg-Sobolev inequality "only" applies to when$n\geq 2$. I have proved that it applies also to the case where$n=1\$. Commented Dec 10, 2013 at 16:25