I am reading the following paper (preprint here) and the author starts by stating the Sobolev inequality on the Sphere $\mathbb{S}^d$ $$\frac{p-2}{d}\int |\nabla u|^2 + \int |u|^2 \geq \left(\int |u|^p\right)^{2/p} \tag 1$$ where the integral is taken over the Sphere and $2<p\leq \frac{2d}{d-2}.$ The author mentions that this inequality can be derived by considering the Sobolev inequality on $\mathbb{R}^d$ with optimal constant, which is \begin{align} \int |\nabla u|^2 \geq S \int |u|^p \tag2 \end{align}

where $S$ is the best constant. I am not sure how to deduce $(1)$ from $(2)$ using Stereographic projection. Any comments/remarks will be much appreciated.