I found the following interesting identity which is similar to what I have showed recent days [here]. In that entry, we showed that

where and are connected by

.

For we denote by the following

.

Then we show that

Lemma..

I found the following interesting identity which is similar to what I have showed recent days [here]. In that entry, we showed that

where and are connected by

.

For we denote by the following

.

Then we show that

Lemma..

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This short note is to prove the following

where and are connected by

.

The proof is straightforward as follows.

- Calculation of .

We see that

- Calculation of .

Similarly, we get

In this topic, we consider the Kelvin transform for Laplaction operators. Precisely, what we get is the following

.

We now consider a different situation. The detail can be found in the following paper due to X.W. Xu published in *Proc. Roy. Soc. Edinburgh Sect. A*, 2000.

Theorem. If is a sufficiently good function then satisfies the equationwhere is defined to be

.

For each point , denote and

is the inversion of with respect to the unit sphere. We have the following identities

and

.

Thus,

.

Next, think of as a system of orthogonal curvilinear coordinates for , we deduce that the metric tensor of the Euclidean space in curvilinear coordinates

.

This implies that the so-called Lame coefficients is

.

In this topic, we proved a very interesting property involving the Laplacian of the Kelvin transform of a function. Recall that, for a given function , its Kelvin transform is defined to be

.

We then have

where the inversion point of is defined to be

.

Therefore, the Kelvin transform can be defined to be

.

We also have another formula

.

The right hand side of the above identity involves , we can rewrite this one in terms of . Actually, we have the following

which gives

.

Today, we study the a more general form of the Kelvin transform. The above definition of the Kelvin transform is with respect to the origin and unit ball in . We are now interested in the case when the Kelvin transform is defined with respect to a fixed ball. The following result is adapted from a paper due to M.C. Leung published in *Math. Ann.* in 2003.

Define the reflection on the sphere with center at and radius by

.

It is direct to check that

.

We observe that

.

The Kelvin transform with center at and radius of a function is given by

.

We remark that if and are fixed, and if , then as well, where . In addition, sends a set of small diameter not too close to to a set of small diameter. We verify that double Kelvin transform is the identity map. We have