In $\mathbb{R}^2$ (or more generally in $\mathbb{R}^n$), if $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ satisfies $\Delta f+ f=0$, then we can write the following regularity/derivative estimates for $f$: for any $j \in \mathbb{N}$ and any $r>0$, there are constants $C_{r,j} >0$ the following holds \begin{align*} |\nabla^j f(0)|^2 \leq C_{r,j} \int_{B(0,r)} f^2(x,y) dx dy. \end{align*} The proof of this involves considering the harmonic function $g$ defined on $\mathbb{R}^3$ by $g(x,y,z) := f(x,y) e^{z}$, getting the regularity/ derivative estimates for the harmonic function $g$ and then transfering this information to get regularity estimates for $f$. And this involves using the Mean Value Property for harmonic functions on $\mathbb{R}^3$.
Suppose now that $h:\mathbb{R}^2 \rightarrow \mathbb{R}$ satisfies $\Delta h + \lambda h=0$, then similar estimates hold on the wave length scale, more precisely the following hold with the same constants as the one above \begin{align*} |\nabla^j h(0)|^2 \leq C_{r,j} \lambda^{j+1} \int_{B(0,r/\sqrt{\lambda})} h^2(x,y) dx dy. \end{align*}
I am interested in similar regularity/derivative estimates on two dimensional Riemannian manifold $M$. My questions are as follows:
When does such an estimate hold? Is it true that whenever $M$ has constant mean curvature, one can write similar regularity estimates? My question arises from the contents of this aricle which seems to indicate that constant mean curvature is a necessary and sufficient condition for the Mean value property to hold which in turn is used in establishing the regularity/derivative estimates in $\mathbb{R}^2$.
Suppose we have regularity estimates for $\Delta$ eigenfunctions on such manifolds (eg. $\mathbb{S}^2$), then how will they read? Will it read as follows? I will formulate my question for $\mathbb{S}^2$ since the spectrum of $\Delta$ is known. The eigenvalues of $\Delta$ are $\{n(n+1): n \in \mathbb{N}\}$.
Fix $p=(0,0,1) \in \mathbb{S}^2$ to be the north pole and fix an isothermal coordinate $(x_1, x_2)$ around $p$. Then there are constants $C_{r,j}$ such that the following estimate holds $\forall n$, $\forall |\alpha| =j$ and $f$ such that $\Delta f+n(n+1)f = 0$ \begin{align*} \left|\frac{\partial^{\alpha}}{\partial x^{\alpha}}f(p)\right|^2 \leq C_{r,j} [n(n+1)]^{j+1} \int_{B(p,\frac{r}{\sqrt{n(n+1)}})} f^2 ? \end{align*}
Thanks!
