# Why is this a compact operator?

I would like to know a proof of the following result: Let $K: \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$ be an integral kernel such that there is an $\epsilon >0$ which fulfills $$\vert K(p, q)\vert \leq \min(\vert p-q \vert^{-n+\epsilon}, \vert p\vert ^{-n-\epsilon}, \vert q \vert^{-n-\epsilon})$$ Then the integral operator $\int_{\mathbb{R}^n} K(p, q)f(q) dq$ is a compact operator from $L^2(\mathbb{R}^n) \rightarrow L^2(\mathbb{R}^n)$

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Compact as an operator between which function spaces? –  Nate Eldredge Jan 22 at 1:24
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