Here is my problem:
Lebesgue's dominated convergence theorem implies that $$\begin{align}\left|\int_{\Omega_+} n\,\eta'(n\,x_N)w\, u\, dx\right|&=\left|\int_{U\times]0,1/n[}n\,\eta'(n\,x_N)h\,x_N\,u\, dx\right|\\ &\leq \lVert \eta'\rVert_\infty\int_{U\times]0,1/n[}|hu|dx\to 0,\ \ \ \ n\to\infty.\end{align}$$
Some info:
- $1\leq p<\infty$;
- $u\in W^{1,p}(\Omega)$;
- $\Omega=U\times(-r,r)$, where $r>0$ and $U\subseteq\mathbb{R}^{N-1}$ is open;
- $\Omega^+=\Omega\cap\{x_N\geq0\}$, where points in $\mathbb{R}^N$ are denoted $(x',x_N)$, so $x'\in\mathbb{R}^{N-1},x_N\in\mathbb{R}$;
- $\eta$ is a smooth function which is 1 from 1 on and 0 up till $\frac12$, smooth function from $\mathbb{R}$ to itself;
- $v\in\mathcal{D}(\Omega)$ and $w(x',x_N)=v(x',x_N)-v(x',-x_N)$;
- $h(x',x_N)=\int_0^1\partial_Nw(x',tx_N)dt$.
I think that is all that is needed for this point. So how does Dominated Convergence come in here? One can try, as my teacher suggested in class, to prove $hu$ is $L^1$. $u\in W^{1,p}\subseteq L^p$. So I could try Hölder, but how is $h$ an $L^{p'}$ function, with $p'=\frac{p}{p-1}$ is conjugate to $p$?
OK, I was just missing $v$ has compact support, so it is summable at any order, and in fact bounded, being smooth on a compact support. So I could even take $\partial_Nw$, split it into two added terms (as $w$ is the difference of two terms), then substitute each of those with its sup norm, and finally have an estimate for the $p'$-norm of $h$ in terms of $\sup|\partial_Nw|$ and the measure of the support of $v$.