The ordinary Sobolev embedding states that, if $\Omega\subset \mathbb R^n$ is a bounder Lipschitz domain, then for $kp>n$ we have an embedding $W^{k,p}(\Omega) \subset C^0(\Omega)$; while for $kp<n$ we have $W^{k,p}(\Omega) \subset L^q(\Omega)$ for $q< \frac{np}{n-kp}$.

Question: Is there any version of Sobolev embedding withe respect to $W^{k,p}_0$ instead of $W^{k,p}$? What if we assume $\Omega$ is unbounded?

For my interest, is there a version that states $W^{1,p}(\mathbb R) \subset C^0(\mathbb R)$?

Notation: Denote $C^\infty_0$ the set of all real-valued smooth function $f$ on $\mathbb R$ such that $\lim_{x\to \pm\infty} f(x)=0$, and $W^{1,p}_0$ is the completion of $C^\infty_0$ with respect to the Sobolev norm $||\cdot||_{W^{1,p}}$

  • $\begingroup$ For the case $k = 1$ you can refer to Chapter 9 of the book of Brezis (Functional Analysis, Sobolev Spaces and PDEs). In particular you should see Corollary 9.14 and Theorem 9.16 for the case with $W^{1,p}(\Omega)$ with $\Omega$ bounded, open and regular, Remark 14 for the case where $\Omega$ is not bounded and Remark 20 for the case with $W_0^{1,p}(\Omega)$ (in the latter you don't need $\Omega$ to be regular but just bounded and open). $\endgroup$ – Arthur11 Jun 11 '17 at 20:58
  • $\begingroup$ Thanks. How about $W^{1,p}(\mathbb R) \subset C^0 (\mathbb R)$? Is this true? $\endgroup$ – Hang Jun 11 '17 at 21:07
  • $\begingroup$ For $p>N$, $W^{1,p}(\mathbb{R}^N) \subset L^\infty (\mathbb{R}^N)$ with continuous injection. And every function in $W^{1,p}(\mathbb{R}^N)$ with $p>N$ admits a continuous representative. $\endgroup$ – Arthur11 Jun 11 '17 at 21:14
  • $\begingroup$ Any reference do you know? $\endgroup$ – Hang Jun 11 '17 at 21:24
  • $\begingroup$ In the book by Haim Brezis, "Functional Analysis, Sobolev Spaces and Partial Differential Equations" (Chapter 9), there is written exactly what I wrote here. I didn't read it but I think also the book "Sobolev Spaces" by Adams and Fournier can be very useful. Surely there is something also on "Partial Differential Equations" by Lawrence Evans. $\endgroup$ – Arthur11 Jun 11 '17 at 21:27

For $W_0^{k,p}(\Omega)$ with $\Omega \subset \mathbb R^n$ open, yes. The idea is to establish the relevant inequalities for $u \in C_c^{\infty}(\Omega)$ (which is dense in $W_0^{1,p}(\Omega)$ as you define it by mollification) and extend by density, which is probably what you learnt in the ordinary version of the theorem minus the extra step of extending it via extension operators.

This is done for example in Evans' PDE book, where he proves the Gagliardo-Nirenburg-Sobolev inequality and Morrey's inequality in $C_c^1(\mathbb R^n).$ The same proof however, can also be used to establish the result in $C_c^1(\Omega)$ for $\Omega \subset \mathbb R^n$ open.


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