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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?

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    $\begingroup$ Spherical harmonics are traces of homogeneous polynomials, so you can easily go into the full space and use standard estimates for harmonic functions. The higher dimensional analogues can be obtained the same way. $\endgroup$ – fedja May 28 '18 at 2:05
  • $\begingroup$ Do you mean the mean value property for harmonic functions? I thought about it, but can't seem to apply them successfully to my case. And what about the first and second derivatives of $f$? $\endgroup$ – un umile appassionato Jun 2 '18 at 11:08

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