Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I just read the proof of Hardy-Littlewood-Sobolev inequality abaout fractional integral operator. Then I found the following identity (but don't understand it)

\begin{equation} \int_{% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{n}\backslash B\left( x,R\right) }\frac{1}{\left\vert x-y\right\vert ^{c(n-1)}}dy=\int_{S^{n-1}}\int_{R}^{\infty }\frac{1}{r^{^{c(n-1)}}}% r^{n-1}drd\sigma \end{equation} where B(x,R) denote n dimensional ball, centered on $x$ and radius $R$. I guess that identity is generalization of substitution formula. Could you help me to understand that identity?

share|improve this question
add comment

1 Answer 1

up vote 2 down vote accepted

The left integral is over all of space except the ball. $y$ is a point outside the ball and ranges over the volume of integration. $|x-y|$ is the distance between the points. The right integral is changing to spherical coordinates. $r$ is the distance between $x$ and $y$, so $r=|x-y|$. The inner integral is over the radius, and ranging from $R$ to $\infty$ avoids the ball nicely. $d\sigma$ represents all the angular variables ($n-$1 of them). The $r^{n-1}$ factor represents the radial variation of the surface area of an $n-1$ sphere. Presumably the next thing to happen is to do the $d\sigma$ integral getting the surface of the $n-1$ sphere ($4\pi$, for example, if $n=3$).

share|improve this answer
    
Singular variable in 2 dimension means the angle? –  beginner Jul 24 '12 at 3:24
    
@beginner: I'm not sure what singular variable you mean, but in 2 dimensions the area element is $r\;dr\; d\theta$. The $r$ is $r^{2-1}$, the $dr$ you see, and $d\theta$ is the $d\sigma$. In 3 dimensions it is $r^2\; dr\; \sin \theta\; d\theta\; d\phi$ with $\sin \theta\; d\theta\; d\phi$ being $d\sigma$ –  Ross Millikan Jul 24 '12 at 3:29
    
Do u know any book/references that can explain this substitution in detail and rigorous way? –  beginner Jul 24 '12 at 3:32
    
@beginner: Sorry, no. Maybe somebody else does. –  Ross Millikan Jul 24 '12 at 3:35
    
@beginner: In my example above, whether the factor is $\sin \theta$ or $\cos \theta$ depends whether you measure up from the $xy$ plane (sin) or down from $z$ (cos). The idea is the same, but I think down from $z$ is more standard. –  Ross Millikan Jul 24 '12 at 13:00
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.