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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.

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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.

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  • $\begingroup$ 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 $||u||_{C^{0,\beta}(\Omega)} \leq 3||Du||_{p}$ $\endgroup$
    – Alex
    Commented Dec 10, 2013 at 15:03
  • $\begingroup$ I am right then in assuming that the Nirenberg-Gagliardo-Sobolev inequality applies to $n \geq 2$ $\endgroup$
    – Alex
    Commented Dec 10, 2013 at 15:05
  • $\begingroup$ Sorry, but I can't understand your comment. Are you asking me something? or are you showing something? $\endgroup$
    – Tomás
    Commented Dec 10, 2013 at 15:29
  • $\begingroup$ I was completing the proof. My original question was if it is implied that the N-G-S inequality applies for $n \geq 2$? $\endgroup$
    – Alex
    Commented Dec 10, 2013 at 15:52
  • $\begingroup$ 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 $n=1$. $\endgroup$
    – Tomás
    Commented Dec 10, 2013 at 16:25

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