Let $\Omega \subset \mathbb{R}^N$ a bounded smooth domain. Also consider $u \in C^1_{c}(\Omega)$, that is, a function which support is in $\Omega$. I would like to know if the function $\varphi : \mathbb{R} \rightarrow \mathbb{R}$ given by $$ \varphi(s) = \int_{\mathbb{R}^N} \frac{|u(x + se_j) - u(x)|^p}{s^{1 + \alpha_j p}} dx $$ also has compact support, where $1 < p < +\infty, 0 < \alpha_j < 1$ and $e_j = (0,...0,1,0...0)$, $j=1,...,N$.
The context: Is it true that $$ \int_0^1 \int_{\mathbb{R}^N} \frac{|u(x + se_j) - u(x)|^p}{s^{1 + \alpha_j p}} dx ds = \int_{\mathbb{R}} \int_{\mathbb{R}^N} \frac{|u(x + se_j) - u(x)|^p}{s^{1 + \alpha_j p}} dx ds \quad? $$ These integrals are closelly related to anisotropic Sobolev space of order $(\alpha_1,...,\alpha_N)$.
Suppose $\phi(s)=0$ for some $s\ne 0$. Then $$ u(x+se_j) = u(x) $$ for almost all $x$.
Let me assume $s>0$. Since $u$ has compact support, there is $M>0$ such that $u(x)=0$ if $x_j>M$. Using the above relation, we find $u(x)=0$ for all $x$ with $x_j > M-s$. Inductively it follows $u=0$. The same argument works for $s<0$.