Sphere inversion in Riesz potential

Question

I am reading the paper: ``ON THE DISTRIBUTION OF FIRST HITS FOR THE SYMMETRIC STABLE PROCESSES" by Blumenthal, Getoor and Ray, (Trans. Amer. Math. Soc. 99 (1961), 540-554).

On page 546, the authors talk about the idea of Riesz regarding spherical inversions. In particular they say, for the sphere $\{u:|x-u| =r\}$, inversion is the change of coordinates $$ u \mapsto v = x+r^2\frac{(u-x)}{|u-x|^2}. $$ Then the authors explain ``Riesz noted that if $f(u)$ is a potential of exponent $\alpha$ in $R^N$, then after inversion in a sphere with center $x$, $(x —v)^{\alpha-N}f(v)$ is a potential."

I dont understand this sentence. Also later they use this idea to write some identities which I do not understand at all. For example, in page 547, they write ``if $|y| >1$, inverting along the sphere $\{u: |y-u|^2 = |y|^2-1\}$ yields $$ \int _{|u| \le 1} (1-|u^2|)^{-\alpha/2} |u-y|^{\alpha-N}du = (|y|^2 -1)^{\alpha/2} \int_{|v| \le 1} (1-|v|^2)^{-\alpha/2} |v-y|^{-N}dv " $$

I would be obliged if anyone can shed some light on these mysterious identities.

Answer 1

Well, you may like to have a look at the original M. Riesz's 1938 paper: it is a fantastic read!

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In the language of these papers, a "potential" of exponent $\alpha$ is a function $f$ of the form $$ f(x) = \int_{\mathbb{R}^N} |y - x|^{\alpha - N} \mu(dy) , $$ where $\mu$ is a non-negative measure. M. Riesz observed that if $f$ is a potential, then its Kelvin transform $$ \mathcal{K} f(x) = |x|^{\alpha - N} u(x / |x|^2) $$ is again a potential. This corresponds to inversion in the unit sphere $B(0, 1)$, that is, the change of coordinates: $$ x \mapsto y = x / |x|^2 . $$

Obviously, the above observation extends (by translation and change of scale) to arbitrary spheres, and this is precisely the result that Blumenthal, Getoor and Ray refer to.

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The above observation is particularly useful when inversion in the corresponding sphere preserves the unit ball. Such a sphere have a form given in the second part of your question, and the identity you are asking about is proved by the corresponding change of variable.

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All authors mentioned above (M. Riesz, Blumenthal, Getoor and Ray) used these kind of calculations a lot, and rarely provided details. I believe, mostly because these are direct extensions of the well-studied case corresponding to $\alpha = 2$.

I do not know good references which give all details. A book by Bliedtner and Hansen, and perhaps a book by Landkoff, have them, if I remember correctly, but I find them both rather unfriendly.