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I am currently studying a proof of a General Sobolev Inequality. I have the following question:

Consider the Sobolev Space $W^{k,p}(U)$. With the added assumption that $k > \frac{n}{p}$. Let $l = \frac{n}{p} -1$ and take $u \in W^{k-l,r}$ where $r = \frac{pn}{n-pl} =n$. How does it follow using Sobolev-Nirenberg-Gagliardo inequality that $D^{\alpha}u \in L^{q}(U)$ for all $n \leq q < \infty$ and all $|\alpha| \leq k-l-1 = k -[\frac{n}{p}]$?

Thanks a lot for any assistance!

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  • $\begingroup$ Yes there is, sorry I forgot, I will update the question. $\endgroup$
    – Alex
    Commented Nov 6, 2013 at 13:02

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You give that $u \in W^{k-l,r}(U)$ for $r=n$ so you cannot apply S-N-G inequality at this stage since the Sobolev-Nirenberg-Gagliardo Inequality requires that $1 \leq r < n$. After a few observations we can find a suitable $r$ such that we can apply the S-N-G inequality. We first note that $W^{k-l,r}(U) \subset W^{k-l,r_{1}}(U)$ when $r_{1} < r$.

Note then that we have $1 \leq r_{1} < r < n$ when $n \leq q < \infty$ where $q := \frac{nr_{1}}{n-r_{1}}$($q$ is the Sobolev Conjugate of $r_{1}$).

You can then apply the 'N-G-S estimates for $W^{1,p}(U)$' for $|\alpha| \leq k-l-1$. The Theorem is stated as follows:

Assume $1 \leq p < n$ and $u \in W^{1,p}(U)$. Then $u \in L^{p^{*}}(U)$, with the estimate $||u||_{L^{p^{*}}(U)} \leq C||u||_{W^{1,p}(U)}$.

This gives you your required result.

[Don't forget that you have to have $U$ bounded with a $C^{1}$ boundary, you did not specify this, but I assumed that it is given.]

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