Let us consider the sphere $S^n$ and the Laplacian $-\Delta$ on it. Let $L^2(S^n) = \bigoplus\_k V\_k$, where $V\_k$ represents the eigenspace of the Laplacian with eigenvalue $k(k + n - 1)$. We know from the general elliptic theory that the Laplacian is elliptic of order $2$, and $-\Delta u \in H^s\_{\text{loc}}$ implies $u \in H^{s + 2}\_{\text{loc}}$. I was wondering if there is any way of deriving the last conclusion in this specific setting of the sphere without using the general elliptic theory, just from the knowledge of spherical harmonics. 


I guess, to answer this question, one needs to answer the following: if $u \in L^2(S^n)$ can be expressed as $u = \sum\_k u\_k$, such that $u\_k \in V\_k$, what is the necessary sufficient condition on $u\_k$ such that $u \in H^s(S^n)$? For good measure, if there is some condition ensuring that $u$ is smooth, I would love to know that too. I think that the answer could be some sort of decay condition on $\Vert u\_k\Vert\_{L^2}$, but I am not sure. Any hints would be appreciated.



Spherical harmonics and ellipticity of the Laplacian
Perhaps this example is not very sophisticated, but it has the virtue of being very tangible, while less cliched than the case of $S^1$ or products thereof. Specifically, on the circle, the eigenfunctions (=exponentials) for the Laplacian are (uniformly... !) bounded, which makes many things perhaps-misleadingly simple.


As @ChristianRemling and @JoonasIlmavirta pointed out, the $s$-th Sobolev norm-squared of $u=\sum\_k u\_k$ is $\sum\_k |\lambda\_k|^s\cdot |u\_k|^2$, where $\lambda\_k$ is the eigenfunction of $\Delta$ on the degree-$k$ harmonics $V\_k$. This by itself does not quite address smoothness or other pointwise-convergence issues, which need a comparison of sup-norm and $L^2$ norm on $V\_k$. It turns out that a quite general argument (for compact groups) implies that the sup of the ratio sup-norm/$L^2$-norm is essentially the square root of the dimension of the eigenspace. For spherical harmonics, this dimension is a difference of binomial coefficients ${n+d-1\choose n-1} - {n+d-3\choose n-1}$.


Using that estimate, the usual $L^2$-style argument (via Cauchy-Schwarz) for a Sobolev imbedding theorem reproves what we'd expect from the general "local" Sobolev imbedding, namely, that $H^{{n\over 2}+k+\epsilon}\subset C^k$ for every positive $\epsilon$.


