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

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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

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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

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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

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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.