I have been reading the chapter of Sobolev Space in Partial Differential Equations by Lawrence C. Evan, and I came across the General Sobolev Inequality stated as follows:

Theorem (General Sobolev Inequality) Let $U\subset\mathbb{R}^n$ be a bounded open set, with $C^1$ boundary. Assume $u\in W^{k,p}(U)$.

(i) If $k<\frac{n}{p}$ then $\|u\|_{L^q\left(U\right)}\le C\|u\|_{W^{k,p}\left(U\right)}$ with $\frac{1}{q}=\frac{1}{p}-\frac{k}{n}$

(ii) If $k>\frac{n}{p}$ then$$\|u\|_{C^{k- \left\lfloor{\frac{n}{p}}\right\rfloor - 1,\gamma}\left(\bar{U}\right)}\le C\|u\|_{W^{k,p}\left(U\right)}\text{, with } \: \gamma=\begin{cases}\left\lfloor{\frac{n}{p}}\right\rfloor + 1 -\frac{n}{p},\;\text{if $\frac{n}{p}\notin \mathbb{Z}$}\\ \text{any positive number}<1,\;\text{if $\frac{n}{p}\in \mathbb{Z}$}\end{cases}$$

My question is about the case when $\frac{n}{p}\in\mathbb{Z}$ in (ii). In the book, Evan provided the proof for this case as follows:

Suppose $k>\frac{n}{p}$ and $\frac{n}{p}\in\mathbb{Z}$. Set $l=\left\lfloor{\frac{n}{p}}\right\rfloor-1=\frac{n}{p}-1$. Consequently, we have as above $u\in W^{k-l,r}\left(U\right)$ for $r=\frac{pn}{n-pl}=n$. Hence the Gagliardo-Nirenberg-Sobolev inequality shows $D^\alpha u\in L^q(U)$ for all $n\le q<\infty$ and all $\lvert \alpha\rvert \le k-l-1=k-\left\lfloor{\frac{n}{p}}\right\rfloor=k-\frac{n}{p}$. Therefore Morrey's inequality further implies $D^\alpha u\in C^{0,1-\frac{n}{q}}\left(\bar{U}\right)$ for all $n<q<\infty$ and all $\lvert \alpha\rvert \le k-\left\lfloor{\frac{n}{p}}\right\rfloor-1$. Consequently $u\in C^{k- \left\lfloor{\frac{n}{p}}\right\rfloor - 1,\gamma}\left(\bar{U}\right)$ for each $0<\gamma<1$. As before, the stated estimate follows as well.

I understand the way he gets $u\in W^{k-l,r}\left(U\right)$ for $r=n$ by iterating the Gagliardo-Nirenberg-Sobolev inequality. But what I don't understand is how he used the Gagliardo-Nirenberg-Sobolev inequality on $u\in W^{k-l,n}\left(U\right)$ to obtain that $D^\alpha u\in L^q(U)$ for all $n\le q<\infty$ and all $\lvert \alpha\rvert \le k-l-1=k-\frac{n}{p}$. Isn't the Gagliardo-Nirenberg-Sobolev inequality only valid when $1\le r<n$? Or am I missing some extra steps he skipped?

Any help would be very much appreciated!

  • 1
    $\begingroup$ As Evans says just before the GNS inequality, $\lVert f \rVert_q$ can only be controlled by $\lVert Df \rVert_p$ for one particular $p$. In order to bound $\lVert D^\alpha u \rVert_q$ for a range of $q$ then, we would want $DD^\alpha u$ to lie in $L^p$ for a range of $p$. Thus... $\endgroup$ – epimorphic May 20 '17 at 2:53
  • $\begingroup$ @epimorphic Thank you very much for the reply! I am just wondering if I should be using the fact that $\lvert U \rvert<\infty$, which implies the estimate $\|D^\alpha u\|_{L^p(U)}\le C\left(\lvert U\rvert\right)\|D^\alpha u\|_{L^n(U)} $ for $1\le p<n$? $\endgroup$ – HSea12345n May 20 '17 at 8:03
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    $\begingroup$ Exactly. Finite measure implies $L^r \subset L^p$ whenever $p < r$, so $u$ lies in a whole range of Sobolev spaces. $\endgroup$ – epimorphic May 20 '17 at 8:16
  • $\begingroup$ @epimorphic Okay I see, since for all $q>n$ we can find $p=\frac{nq}{n+q}<n$ such that $p^*=\frac{np}{n-p}=q$ and using G-N-S inequality and above estimate, one gets the result. Thank you again for the hint! $\endgroup$ – HSea12345n May 20 '17 at 8:42

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