Let $\mathcal{H}_k\subset L^2(S^n)$ be the space of sphere harmonics of degree $k$, i.e., they are the eigenfunctions of $\Delta_{S^n}$. And let $\Pi_k:L^2(S^n)\to \mathcal{H}_k$ be the orthogonal projection operator. It has a kernel defined by $$ \Pi_kf(x)=\int_{S^n}\Pi_k(x,y)f(y)dS(y) $$ A Zonal spherical harmonics $Y_0^{k}(x)$ is defined as $$ Y_0^{k}(x)=\frac{\Pi_k(x,y_0)}{\sqrt{dim \mathcal{H}_k}} $$ here, $y_0$equal to the north pole. From the definition, we can immediately get $$ \|Y_0^{k}(x)\|_\infty\leq C\sqrt{dim \mathcal{H}_k}\leq Ck^{\frac{n-1}{2}} $$ But I also want to get the $L^p$ norms, so I want to know how they look like, then I can compute.

Thanks for any reference.