I am looking to estimate $\int_{A\setminus B} \int_B \frac{1}{|x-y|^{n+1}}dxdy$ where $B$ is a ball and $A$ is a bounded set.

I tried to use the change of variable $r = |x-y|$ then the integral becomes: $\int_0^R \frac{Cr^{n-1}}{r^{N+1}}dr$ where $Cr^{n-1}$ is the volume of the $n$ dim sphere and $R$ be the "diameter of A". I wonder if this change of variable is correct?

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  • 1
    $\begingroup$ I think it's "correct" in the sense that it gives you an upper bound for the double integral, but note that the integral in $r$ is infinite. $\endgroup$
    – Jose27
    Mar 18 at 16:48


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