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Im wondering about theorems concerning extending Sobolev functions defined on the $d$-dimensional unit cube to all of $\mathbb{R}^d$. More precisely, given $f:[0,1]^d \to \mathbb{R}$ with $f\in H^k([0,1]^d)$, does there exist an extension $\bar{f}:\mathbb{R}^d\to \mathbb{R}$ with $\bar{f}\in H^k(\mathbb{R}^d)$ such that:

  1. $\bar{f}(x)=f(x)$ for all $x\in [0,1]^d$
  2. $\|\bar{f}\|_{H^k(\mathbb{R}^d)}\leq C\|f\|_{H^k([0,1]^d)}$

Moreover, if the answer is positive, is it possible to find such an $\bar{f}$ with compact support?

Best Regards

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The unit cube has lipschitz regularity so yes, you have an extension operator. Note that your extra condition on the compactness of the support can be ensured easily once you have an extension (just multiply the extension by some smooth bump function which equals $1$ on the unit cube).

For a reference, in the lipschitz case, I would suggest Theorem 5 in Chapter VI of this book of Stein which amazingly proves the following statement (and in fact covers even less regular domain)

Theorem: Fix $\Omega\subset\mathbf{R}^d$ an open set with lipschitz boundary. There exists a linear map $\mathscr{E}: \text{L}^1_{\text{loc}}(\Omega)\rightarrow \text{L}^1_{\text{loc}}(\mathbf{R}^d)$ such that:

  • For any $u\in L^1_{\text{loc}}(\Omega)$, there holds $\mathscr{E}(u) = u$ a.e. on $\Omega$
  • For any $m\in\mathbf{N}$ and any $p\in[1,\infty]$, $\mathscr{E}$ maps continuously $\text{W}^{m,p}(\Omega)$ into $\text{W}^{m,p}(\mathbf{R}^d)$.

Note that the definition of $\mathscr{E}$ is insensitive to $m$ or $p$: Stein manages to define this extension operator once and for all.

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  • $\begingroup$ Thanks! I will look into that reference. Regarding the compactness, after multiplying by a bump function, can we still say that we have the condition of the norms as I stated? I mean, if we have an extension fulfilling the inequality in 2, can multiplying by a bump function violate the inequality ? $\endgroup$
    – Jjj
    Commented Sep 2 at 15:06
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    $\begingroup$ Well no, because of Leibniz formula for derivatives: if $\theta$ is the smooth bump function, and $\alpha$ a multi-index of length not larger than $k$, then $\partial^\alpha(\theta \bar{f})$ is a linear combination of terms like $(\partial^\beta \theta)(\partial^{\alpha-\beta}\bar{f})$, where $\beta$ runs on all multi-indices not larger than $\alpha$. In particular, since $\theta$ is smooth you get $\|\partial^\alpha(\theta \bar{f})\|_2 \lesssim_\theta \|\bar{f}\|_{H^k}$ where $\lesssim_\theta$ depends irrelevantly on $\theta$. $\endgroup$
    – Ayman Moussa
    Commented Sep 2 at 15:12
  • $\begingroup$ Thanks again. One last question. When you say that $\lesssim_{\theta}$ depends irrevantly of $\theta$ what do you exactly mean? I guess the constant in the inequality is related to $\sup_{\beta \leq \alpha}|\frac{\partial^{\beta}\theta}{\partial x^{\beta}}|$? For instance, if I choose $\theta$ such that it is really close to the indicator function on $[0,1]^d$ then the constant will scale with that? $\endgroup$
    – Jjj
    Commented Sep 3 at 11:43
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    $\begingroup$ Your guess is correct. I mean if you only want 2. to hold for some constant $C$, the fact that this constant actually depends on the choice of $\theta$ is irrelevant. $\endgroup$
    – Ayman Moussa
    Commented Sep 3 at 12:47

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