Let $\Omega$ be a bounded domain in $\mathbb{R}^d$, $d\ge 3$. We define the Neumann-Poincare operator(or double layer potential) $K: L^2(\partial\Omega)\to L^2(\partial\Omega)$ by $$(Kf)(x)=\int_{\partial \Omega}f(y)\frac{\partial}{\partial n_y}E(x,y)dS_y,$$ where $E(x,y)=||x-y||^{2-d}$ and $\frac{\partial}{\partial n_y}$ means the outer normal derivative on the boundary $\partial\Omega$.

Show that $K$ is in the Schatten class $S_p$, $p>d-1$.

Remark: (1) In the paper(http://www.math.ucsb.edu/~mputinar/poincare.pdf), on p.18 the authors only briefly mentioned the above result and a reference book(O.D. Kellogg, Foundations of Potential Theory, J. Springer, Berlin, 1929).

(2) In the paper(https://arxiv.org/pdf/1501.03627.pdf), on p.14 the authors used a kernel condition(Theorem 2.14) for the Schatten class to solve the case $d=3$. But it seems that this condition is not enough to solve higher dimensional cases.


If the boundary is Lipschitz, one can prove only that the operator is bounded, it is not necessarily even compact. If the surface is smooth the operator is in fact a pseudodifferential operator of order $-1$, and Weyl asymptotics gives the answer. The latter result holds if the boundary is just a tiny little bit better than Lipschitz, but a hard analysis and perturbation theory is neded.

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    $\begingroup$ In the paper ([1], page 15) cited above the boundary is assumed to be of class at least $C^2$, so it is presumable that the OP simply has forgotten to add this condition in his post. $\endgroup$ – Alex M. Jan 26 '18 at 18:15

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