In this article, it is claimed that (Claim 2.2), for any spherical harmonic $f$ of degree $n$ and any point $x \in \mathbb{S}^2$, we have
$$|f(x)|^2 \leq C_1 n^2 \int_{D(x, \frac{1}{n})}f^2, $$ $$|\nabla f(x)|^2 \leq C_2 n^4 \int_{D(x, \frac{1}{n})}f^2, $$ $$|\nabla \nabla f(x)|^2 \leq C_3 n^6 \int_{D(x, \frac{1}{n})}f^2, $$ for some positive constants $C_i$, where $D(x, \frac{1}{n})$ denotes the spherical disk of radius $\frac{1}{n}$ centered in $x$.
How do I go about proving that? Does this generalize to higher dimensional spherical harmonics?
Edit: I could indeed prove the first one going into $\mathbb{R}^3$ and employing the mean value property for harmonic functions, but the last two seem harder. I was wondering, could they perhaps follow from $L^p$ estimates or similar results about (elliptic) PDEs?