Let be points in . If we denote by the reflection point of with respect to the unit ball, i.e.
we then have the following well-known identity
.
The proof of the above identity comes from the fact that
.
Indeed, by squaring both sides of
we arrive at
which is obviously true. Similarly, the last identity also holds. If we replace by we also have
.
Generally, if we consider the reflection point of over a ball , i.e.
we still have the fact
.
Indeed, one gets
.
Similarly,
.
Such identity is very useful. For example, in () the following holds
.
This type of formula has been considered before when here. For a general case, Lieb and Loss introduced another method in their book published by AMS in 2001. Here we introduce a completely new proof. At first, if by the potential theory, one easily gets
.
If , one needs to make use of the reflection point of and the above identity to go back to the first case. The point here is . The integral is obviously continuous as a function of . The above argument is due to professor X.X.W.