Restriction of a compactly supported function on a bounded domain in a surface

Let $$\Omega \subset \mathbb{R}^N$$ be a bounded domain and let $$P_k$$ a plane of dimension $$1\leq k in $$\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:

And

• Do you mean the partial derivative of $\varphi$ in the integrand? Mar 22 at 0:47
• Yes, you are right. Mar 22 at 0:50
• 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. Mar 22 at 1:39
• 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. Mar 22 at 1:54
• Is your function still a counterexample for the Edit part? Mar 22 at 1:57

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".