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.