on boundary integral with stationary phase

Let $f$ be a real-valued function and $\Omega$ is a simple domain with smooth boundary. We define the following operator as an extension of the Neumann-Poincaré operator, the function $A(w,z)$ is strictly positive kernel function.

$$\mathcal{K}f(z)=\Re \frac{1}{2\pi i}{\int }_{\partial \Omega }A(w,z)\frac{f(z)}{w-z}dz$$

We are interested in the asymptotic behavior of $\mathcal{K}f$ when $f(z)=z^n$.

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