Neumann-Poincare operator is in the Schatten class

Question

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.

Answer 1

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.