Logarithmic kernel

Remarks

Example: $k(2, 0) = -\log 2 < 0$ and $k(1/2, 0) = \log 2 > 0$. For fixed $w$, the function $z \mapsto k(z, w)$ is harmonic on $\C \setminus \set{w}$. Indeed, on every disc that does not contain $w$ it is the real part of a holomorphic branch of $-\log(z - w)$. Non-example: in $\R^3$ the function $\log(1/\abs{x - y})$ is not harmonic off the pole, since for $r = \abs{x - y}$ one has $\Delta \log(1/r) = -1/r^2$; the three-dimensional analogue is the Newtonian kernel. The kernel is symmetric and lower semicontinuous, and $k(z, w) = 2\pi \, \Phi(z - w)$, where $\Phi$ is the fundamental solution of the Laplacian in the plane. So $\Delta_z k(z, w) = -2\pi \delta_w$ in the sense of distributions.