Using polar coordinates with radial functions in two variables

Question

Consider $N \geq 4$ and write $N = N_1 + N_2$ with $2 \leq N_2 \leq N_1$. Let $$ \Omega = \left\{ (x,y) \in \mathbb{R}^{N_1} \times \mathbb{R}^{N_2} : \frac{1}{2} < |(x,y)| < 1\right\} $$ and $$ D = \left\{ (s,t) \in \mathbb{R}^2 : s,t \geq 0, \frac{1}{2} < |(s,t)| < 1 \right\}. $$ Let $u \in C^1(\Omega)$ a symmetric function in the sense that $u(x,y) = u(w,z)$ if $|x| = |w|$ and $|y| = |z|$. In this case we can consider a function $v$ defined on some subset of $\mathbb{R}^2$ such that $u(x,y) = v(|x|, |y|)$. In a paper I am reading, the author says there exists a constant such that, up to a change of variable $$ (1) \quad \quad \quad \int_{\Omega} [u(z)]^2 d_{z} = C \int_{D} [v(s,t)]^2 s^{N_1 - 1} t^{N_2 - 1}ds dt. $$ where $z = (x,y)$. I tried to decompose the set $\Omega$ in order to use the polar coordinates formula $$ \int_{\mathbb{R}^N} f(z) dz = \int_0^\infty \int_{S^{N-1}} f(r z') r^{N-1} d_{\sigma z'}, dr $$ but I failed. How to use a change of variable to obtain (1) ? Is there any integration formula for "radial" functions like my $u$ ? Any idea or suggestion is welcome.