Here is a change of variable that we did in the class and I wonder what is the justification behind the change:
$$ \int_{\mathbf{R}^N} \int_{\mathbf{R}^N} \frac{|f(x + h) - f(x)|^p}{\| h \|^{N + sp}} \,dx \,dh \leq \int_{\mathbf{R}^N} \frac{C}{\| h \|^{N + sp}} (K(f, \| h \|))^p \,dh= C \int_0 ^\infty \frac{t^{N - 1}}{t^{N+sp}} (K(f, t))^p\,dt, $$ where we claimed that the spherical coordinate $t = \| h \|$ was used. Here $K:(L^p(\mathbf{R}^N), W^{1, p}(\mathbf{R}^N))_{s, p} \times (0, +\infty)\to \mathbf{R}$ is a nonnegative measurable function. In particular, what I am confused about is $h \in \mathbf{R}^N$, but it seems like we have changed $\,dh = \,dt$, where $t \in \mathbf{R}$? I know this is just a notation and probably involves just how spherical coordinate works, but I haven't seen a rigorous justification of spherical coordinate, so this doesn't make totally sense to me yet. Can someone state the formal theorem that we are using here and explain how it is applied to this case?
The formal theorem is:
The proof is just a computation (and implicit use of Tonelli's theorem):
These images are from page 88-89 of "Measure Theory and Integration" by Taylor.