Let $s\in (0, 1)$ and $u:\mathbb R^n\to \mathbb R$ be a measurable function. The Gagliardo seminorm of $u$ is defined as $$[u]^2_s =\int_{\mathbb R^{2n}} \frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}} dx dy.$$
I am reading some notes on fractional Sobolev spaces and I ran into the following statement which is not clear to me: Let $\widetilde{u}(x) = u(-x)$. Clearly $[\widetilde u]^2_s = [u]^2_s$.
First of all: what does it mean $[\widetilde u]^2_s$? I would say that $$[\widetilde u]^2_s = \int_{\mathbb R^{2n}} \frac{|u(-x)-u(-y)|^2}{|-x+y|^{n+2s}} dx dy$$ but I am not sure about that. If this is the case, hence $[\widetilde u]^2_s = [u]^2_s$ should come from a double change of variables $t=-x$ and $z=-y$. Am I right? Is that the right interpretation?
Thank you in advance.
Bear in mind that the lone $x$ and $y$ in the integral are not inputs from your function, they are variables of integration that vary across their prescribed ranges. The seminorm essentially sends a function to an integral, first of all, so $$ [\tilde{u}]_s^2 = \int_{\mathbb{R}^{2n}} \frac{ \left| \tilde{u}(x) - \tilde{u}(y) \right|^2 }{ |x-y|^{n+2s} } \, \mathrm{d} x \, \mathrm{d} y $$ Perhaps more evocatively, so you see that it is a function (not any $x$ or $y$) being plugged in: $$ [\;\cdot\;]_s^2 = \int_{\mathbb{R}^{2n}} \frac{ \left| \;\cdot\;(x) - \;\cdot\;(y) \right|^2 }{ |x-y|^{n+2s} } \, \mathrm{d} x \, \mathrm{d} y $$ From here we can use the definition of $\tilde{u}$ to get $$ [\tilde{u}]_s^2 = \int_{\mathbb{R}^{2n}} \frac{ \left| u(-x) - u(-y) \right|^2 }{ |x-y|^{n+2s} } \, \mathrm{d} x \, \mathrm{d} y $$ Technically you arrive at the same answer anyways since $|x-y|=|y-x|$, but for different reasons.
Naturally, yes, you would arrive at the same norm. $\tilde{u}$ is just $u$, with its graph having done a mirror reflection, in essence.