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I am trying to understand a line of the proof of part I-A of the Sobolev Embedding Theorem in Sobolev Spaces by Adams (section 4.16, page 89). Specifically, the following value integral is presented

$$\int_{C_{x,\rho}} |x - y|^{(m-n)p'} dy$$

where $C_{x,\rho}$ is a cone in $\mathbb{R}^n$. The next line of the proof states that because $(m - n)p' > -n$, the integral is finite. I don't see how this follows. In particular, if $n = 2$, couldn't $(m-n)p' = -1$ so $|x - y|$ would be infinite for $x = y$?

esquire70
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1 Answers1

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There is a hypothesis of the Sobolev Imbedding Theorem relating $m$, $n$, and $p$ that you are neglecting to take into account.

Umberto P.
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  • The relevant assumptions seem to be (1) the cone condition, (2) $m \ge 1$ is an integer, (3) and $1 \le p < \infty$, (4) $mp > n$. The cone condition has $x \in C_{x,\rho}$ so we get 0 raised to something greater than $-n$ for $y = x$ (or at least I think). This leaves that $-n$ must be non-negative which isn't possible because the $n$ is the dimension of $\Omega$. Is there an assumption I am missing? – esquire70 Oct 22 '18 at 15:16
  • $(m-n)p' > -n$ is an immediate consequence of $mp > n$. – Umberto P. Oct 22 '18 at 16:30
  • Thanks. I don't think I am stuck on that though. I am stuck on how $\int_{C_{x,\rho}}|x - y|^{-n}dy$ is guaranteed to be finite for any number greater than $-n$. In particular, if $x = 0$ and the cone is a half-disk of radius $\rho$ (letting $\kappa = \pi$ in defining the cone) then $\int_{C_{x,\rho}}|x - y|^{-n}dy$ is undefined. I must be wrong about this but there is something I am missing. – esquire70 Oct 22 '18 at 20:42
  • Your second sentence doesn't make sense. Did you mean to say $\int_{C_{x,\rho}} |x-y|^t , dy$ is guaranteed to be finite for any number $t > -n$? – Umberto P. Oct 22 '18 at 21:31
  • My apologies. Yes, that is what I meant. – esquire70 Oct 22 '18 at 21:36