Poisson kernels on the half-plane are bell-shaped
Mateusz Kwaśnicki
TL;DR
The article proves that Poisson kernels for a broad class of second-order elliptic operators on the half-plane, with coefficients depending only on the vertical variable, are bell-shaped. It achieves this by factorising the Fourier transform of the Poisson kernel into a Pólya frequency function and an AM-CM component, leveraging the Eckhardt–Kostenko spectral ODE and Rogers function theory within the bell-shaped framework developed in prior work. Consequently, the Poisson kernel is (weakly) bell-shaped, yielding unimodality and a precise inflection-point structure, with probabilistic corollaries describing the boundary-hitting distribution of the associated diffusion. The results forge a link between harmonic extension methods, spectral theory, and the theory of bell-shaped functions, and include explicit homogeneous examples to illustrate the theory's reach and potential applications to boundary-hitting phenomena.
Abstract
Consider a second-order elliptic operator $L$ in the half-plane $\mathbb R \times (0, \infty)$ with coefficients depending only on the second coordinate. The Poisson kernel for $L$ is used in the representation of positive $L$-harmonic functions, that is, solutions of $L u = 0$. In probabilistic terms, the Poisson kernel is the density function of the distribution of the diffusion in $\mathbb R \times (0, \infty)$ with generator $L$ at the hitting time of the boundary. We prove that the Poisson kernel for $L$ is bell-shaped: its $n$th derivative changes sign $n$ times. In particular, it is unimodal and it has two inflection points (it is concave, then convex, then concave again).