Decompose Euclidian space as $\mathbb{R}^N = \mathbb{R}^{n} \times \mathbb{R}^{m}$. If $u \in H^1(\mathbb{R}^N)$ I know it is well defined the Schwarz rearrangement of $u$ and the Pólya–Szegő inequality is also valid if $u \geq 0$. For each $x \in \mathbb{R}^n$ define $v_x(y) = u(x,y)$. I am wondering why is it well defined the Schwarz rearrangement of $v_x$. For me it is not clear that $v_x \in H^1(\mathbb{R}^m)$ (if it was the case, I would know the Schwarz rearrangement would be well defined). I saw this technique in this paper, LEMMA 3.1. (for simpliicty, just take $\alpha = 0$ in the paper).
Any help about this topic is very welcome.