is harmonic in , and suppose . If , then
I like this theorem, it is beautiful and concise. The proof was kind of trivial if apply Poisson Kernel bravely. Let’s make it more general.
Set , thus define , if we are given that , then we are going to prove:
consider unit vector , by Poisson Integral,
since , thus we assume , we know that is homogeneous, hence
The inner integral is .