How to see or prove $L_p(\mathbb R^{n+1})= t^{-\frac{n+1}{p}}L_p(\mathbb R_+\times S^n ,\frac{dt}{t}d\phi)$. I know that i need to use change to polar coordinate because in all book says the same, $1<p<\infty$ and $S^n$ is the sphere unitary in $\mathbb R^n$.
I want to star taking one function $u$ in $t^{-\frac{n+1}{p}}L_p(\mathbb R_+\times S^n ,\frac{dt}{t}d\phi)$ then $u(t,\phi)=t^{-\frac{n+1}{p}}\tilde{u}(t,\phi)$ with $\int_{0}^{\infty}\int_{S^n}|\tilde{u}(t,\phi)|^p\frac{dt}{t}d\phi<\infty$, how can transform this, what change of variable worth?
The idea is calculate $$\int_{\mathbb R^{n+1}}|t^{-\frac{n+1}{p}}\tilde{u}(t,\phi)|^p dx= \int_{\mathbb R^{n+1}}|t^{-(n+1)}||\tilde{u}(t,\phi)|^p dx$$ i do not how is $dx$ and how transform the variable $(t,\phi)\in \mathbb R_{+} \times S^n$ in one variable $x\in R^{n+1}$, and perhaps apply Holder's inequality.
I was thinking in perhaps exist a stereographic representation in $S^n$ as some day saw in calculus, with the pole $(0,0,1)$ and project one line between the pole and the points in the sphere unitary in $\mathbb R^3$ etc, please i want one hint, for me is important understand how to star please, thank you!!!