Let $(M,g)$ a Riemannian manifold. Further $M$ should be a harmonic space, that is $M$ is a symmetric and simply connected space of rank 1. (Example: Spheres $S^n$)
Consider the mean value operator, given by $$M_t[\phi](x)=\frac{1}{|S_t(x)|}\int_{S_t(x)}\phi do,\quad \phi\in C^\infty(M)$$ $S_t(x)$ denotes the geodesic sphere in $x\in M$ and radius $t\in\mathbb{R}$.
Now I want to calculate the expression $\Delta M_t[\phi](x)$, where $\Delta$ is the Laplacian.
My aim is to get an ODE in $M[\phi](x)$ with initial conditions $M_t[\phi](x)_{|t=0}=\phi(x)$ and $M'_t[\phi](x)_{|t=0}=0$.
But I don't know how to evaluate the above expression. In particular: Which variables are fixed? I think the Laplacian just acts on the coordinate variable x, right? Could any of you illustrate the calculation explicitly in the case of the 2-sphere $S^2$. I want to understand the calculation.
Furthermore I'm interested in knowing if there is a correspondence to differential forms. I know that one can express the Laplacian as $\Delta=\delta d+d\delta$. Is there a way to "generalize" the above problem to differential forms?
Are there any references for the above problem? One has told me that it is still an open problem to prove the fact: $M_t[\phi]$ is injective iff $M_t[\phi]=0$ for all closed geodesics of length $t$. But I didn't found papers which mentioned that problem. Many thanks!
Last but not least I want to mention that I posted the question in the Math Stack Exchange, but I didn't get an answer. According to the comment below I add the link to the question on Math Stack Exchange: https://math.stackexchange.com/questions/1490529/mean-value-operator-on-riemannian-manifold
