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Let $\Omega \subset \mathbb{R}^N$ be a bounded domain and let $P_k$ a plane of dimension $1\leq k<N$ in $\mathbb{R}^N$. Denote by $\sigma_k$ the surface measure in the surface $\Omega_k = \Omega\cap P_k$. Given $\varphi \in C^\infty_0(\Omega)$, how to prove directly that there exist a positive constant $C$ such that $$ \int_{\Omega_k} |\varphi_{x_i}(w)|^2 d \sigma_{k w} \leq C \int_{\Omega} |\varphi_{x_i}(x)|^2 d x ? $$ Any reference about this is also very welcome.

Edit: How to prove that $$ \int_{\Omega_k} |\nabla \varphi(w)|^2 d \sigma_{k w} \leq C \int_{\Omega} |\nabla \varphi(x)|^2 d x ? $$ If I am wrong, what should be the interpretation of the following result from Adams, Sobolev Space,2003: enter image description here

And

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  • $\begingroup$ Do you mean the partial derivative of $\varphi$ in the integrand? $\endgroup$
    – Liding Yao
    Mar 22 at 0:47
  • $\begingroup$ Yes, you are right. $\endgroup$ Mar 22 at 0:50
  • $\begingroup$ I don't think this is true. I believe you can consider $\varphi_{x_i}(w)=1/(1+(x_j/\epsilon)^2)$ and $j\neq k$. This makes the constant blow up. $\endgroup$
    – Liding Yao
    Mar 22 at 1:39
  • $\begingroup$ Well, I think what I asked is the 1, Theorem 4.2, p. 79 from the book Sobolev Spaces of Adams, 2003. In this theorem is said that $H^1_0(\Omega)$ is continuously embedded in $H^1_0(\Omega_k)$. I am just looking for a direct proof for the case of a compactly supported function. However I will think about your example, thanks. $\endgroup$ Mar 22 at 1:54
  • $\begingroup$ Is your function still a counterexample for the Edit part? $\endgroup$ Mar 22 at 1:57

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I am sure is no. In fact we have trace theorem $[f\mapsto f|_S]:H^1(\mathbb R^n)\to H^{1/2}(S)$ whenever $S$ is a bounded smooth surface. Here $H^1$ is $W^{1,2}$ Sobolev space, $H^{1/2}(\mathbb R^m)$ the fractional Sobolev space which is the completion of $f\in C_c^\infty(\mathbb R^m)$ with respect to $\|f\|=(\int|f\cdot\nabla f|^2)^{1/2}$.

In particular $[f\mapsto f|_S]:H^1\not\to H^1$.

Edited: For the part in Adams' book is not the theorem, but a description on "what is called a Sobolev embedding". The actual theorem on specifying what $W^{m,p}(\Omega)\to W^{j,q}(\Omega)$ is true for $j,m,p,q$ will be seen later in this chapter.

Plus I don't think the type of maps $W^{m,p}(\Omega)\to W^{j,q}(\Omega_k)$ are "embedding", as it is never injective. I prefer the usage of "trace theorem".

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  • $\begingroup$ Thank you for your reply. I put in the Edit part the Theorem from Adams's book. Could you please help me ? $\endgroup$ Mar 22 at 2:23
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    $\begingroup$ @ThiagoGM I see. I think that just misunderstanding of a theorem and a general description. $\endgroup$
    – Liding Yao
    Mar 22 at 2:55
  • $\begingroup$ Thank you! You are right. $\endgroup$ Mar 22 at 3:00

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