Poincaré inequality with a difference quotient

Question

I am working on the following problem:

For $\Omega\in \mathbb{R}^N$ an open set and $K\subset\subset\Omega$ a compact subset, define the difference quotient in $j$ direction by $$D^h_ju(x)=\frac{u(x+he_j)-u(x)}{h}$$ prove that there exists a constant $C>0$ such that for every $u\in H^1(\Omega)$ and every small $|h|$, we have $$\|D^h_ju\|_{L^2(K)}\le C\|\nabla u\|_{L^2(\Omega)}$$

I imitate the procedure in proving Poincaré inequality that $$\frac{1}{h}(u(x+he_j)-u(x))=\frac{1}{h}\int_{x_j}^{x_j+h}\partial_ju(x_1,...,x_j',...,x_N){\rm d}x_j'$$ squre both side, integrate and employ the Cauchy-Schwarz inequality to get $$\|D_j^hu\|^2_{L^2(K)}\le\int_\Omega(\frac{1}{h}\int_{x_j}^{x_j+h}\partial_ju(x_1,...,x_j',...,x_N){\rm d}x_j')^2{\rm d}x$$ $$\le\frac{1}{h}\int_\Omega(\int_{x_j}^{x_j+h}|\partial_ju|^2{\rm d}x_j'){\rm d}x$$ However, I get stuck here since though $|\partial_ju|$ is square integrable, $\frac{1}{h}\int_{x_j}^{x_j+h}|\partial_ju|^2{\rm d}x_j'$ may not be bounded. I guess there must be some property of $u\in H^1(\Omega)$ that I haven't used. By the way, I am not familiar with the Sobolev space. Literally I just know the definition of it. I know there are already tons of developed inequalities in it which might be helpful, but I want to avoid them unless there is no other way.

Also there is another problem following this one that if $u\in L^2(\Omega)$ and there is a constant $C>0$ such that $\|D^h_ju\|_{L^2(K)}\le C$ for every small $|h|$, then $u\in H^1(K)$ and $\|\nabla u\|_{L^2(K)}\le C$.

This feels like a converse proposition to the one above. I thought about Dominated Convergence Theorem and the alike but didn't make much progress.

Answer 1

Note that the integration domain for $x$ is $K$ not $\Omega$. After decoupling the integral regions, use Fubini to reverse the order of integration $$ \frac1h\int_K \int_{x_j}^{x_j+h} |\partial_j(x_j', x_i|_{i\ne j})|^2 dx_j' dx = \frac1h\int_K \int_0^h |\partial_j u(x+se_j)|^2 ds\, dx\\ = \frac1h\int_0^h \int_K |\partial_j u(x+se_j)|^2 dx\, ds\\ \le \frac1h\int_0^h \int_\Omega |\partial_j u(x)|^2 dx\, ds =\int_\Omega |\partial_j u(x)|^2 dx \\ $$