Boundary integral of a harmonic function around a pole I have a radial harmonic function $h:\mathbb R^N\backslash\{0\}\to\mathbb R$ which has a pole of order $m$ in 0, and I would like to compute
$$
\frac{1}{\sigma_N}\int_{\partial B_r(x_0)}h(x)d\sigma(x),
$$
where $\sigma$ is the surface measure and $\sigma_N$ the surface area of the ball $B_r(x_0)=\{x\in\mathbb R^N\ |\ ||x-x_0||<r\}$.
I known that the above integral equals $h(x_0)$ if $\overline{B_r(x_0)}$ does not contain 0 (since then its just the mean value property). Moreover, in dimension $N=2$, I could show using complex analysis, that the above integral is constant if $0\in B_r(x_0)$. But what about higher dimensions? I read a paper where this constant fact is used without comment in higher dimensions, so I guess its right, but I cannot see the clue behind it.
Thx in advance!
 A: I don't think it is a constant. Take for instance, the function $$h(x)=|x|^{2-n},$$ which is harmonic in $\mathbb R^n\backslash\{0\}$. If you integrate it over the surface of radius $r$ you get
$${1 \over \sigma_N} \int_{\partial B_r(0)}h(x)\, d\sigma(x)={1 \over \sigma_N} \int_{\partial B_r(0)}|x|^{2-n}\, d\sigma(x)={1 \over \sigma_N} r^{2-n}\sigma_N=r^{2-N}.$$
Moreover, based on the section Isolated singularities on chapter 3 of Axler's Harmonic function theory (link from the author's webpage), you can compute the derivative ${d \over dr}$ of your average when the ball contains the origin. First note that, by putting $x=x_0+rz,$ 
$${1 \over |\partial B_r(x_0)|} \int_{\partial B_r(x_0)}h(x)\, d\sigma(x) =
{1 \over |\partial B_1(0)|} \int_{\partial B_1(0)}h(x_0+rz)\, d\sigma(z),$$
so that the derivative can be calculated by differentiating w.r.t. $r$ under the integral sign:
$${1 \over |\partial B_1(0)|} \int_{\partial B_1(0)} \nabla h(x_0+rz)\cdot z\, d\sigma(z).$$
Going back to the original coordinates you get ${1 \over |\partial B_r(x_0)|} \int_{\partial B_r(x_0)} \nabla h(x)\cdot \hat n\, d\sigma(z).$ Now if $\rho$ is a smaller radius which still contains $0$, by the divergence theorem, 
$$0 = \int_{\rho<|x-x_0|<r}\Delta u\, dx = \int_{B_r(x_0)}\nabla h \cdot \hat n \, d\sigma-\int_{B_\rho(x_0)} \nabla h\cdot \hat n \, d\sigma,$$
so that $\int_{\partial B_r(x_0)} \nabla h(x)\cdot \hat n\, d\sigma(z)$ is constant in $r$. This means that the derivative of your average is a constant multiple of $1/|\partial B_{r}(x_0)|$, that is, $K/r^{n-1}$. 
In the book (p.52, first paragraph) this result is used together with the fact that a radial function $u(x)$ is equal to its average on the sphere centered at the origin with radius $|x|$ to conclude that a radial harmonic function on $B_1(0)\backslash\{0\}$ is of the form $c_1+c_2|x|^{2-n}$ in its interior. I hope this answers your question.
