Define for $p\in [1,\infty)$ and $\Omega=(0,1)^N\subset\mathbb{R}^N$ $$W^{1-1/p,p}(\Omega)=\left\{u\in L^p(\Omega): \ \int_\Omega\int_\Omega\frac{|u(x)-u(y)|^p}{|x-y|^{N-1+p}}dxdy<\infty\right\}$$
$W^{1-1/p,p}(\Omega)$ is a Banach space equipped with the norm $$\|u\|=\|u\|_p+\left(\int_\Omega\int_\Omega\frac{|u(x)-u(y)|^p}{|x-y|^{N-1+p}}dxdy\right)^{1/p}$$
It is also a Banach space equipped with the norm $$|u|=\|u\|_p+\sum_{j=1}^{N}\left(\int_0^1\int_{S_j^h}\frac{|u(x_1,...,x_j+h,...,x_N)-u(x_1,...,x_j,...,x_N)|^p}{h^p}dxdh\right)^{1/p}$$
where $S_j^h=\{x=(x_1,...,x_N)\in \Omega:\ 0\leq x_j\leq 1-h\}$.
I could prove that there exist a positive constant $c>0$ such that $\|u\|\leq c|u|$ for all $u\in W^{1-1/p,p}(\Omega)$, which implies by the Open Mapping Theorem (OMT) that $\|\cdot\|$ and $|\cdot|$ are equivalents, however, I would like to prove it without using (OMT), i.e. I would like to prove the reverse inequality only by means of calculus. Until now, I could not porve it, any help is appreciated.
Thank you