Dimension split and the separate Poisson-disc theory

This page splits its statements by dimension, and this remark records where the split sits so that no item silently overclaims.

The $n\ge 3$ construction. The Kelvin/image construction (Dirichlet Green function of a Euclidean ball), the explicit ball kernel (Poisson kernel of a Euclidean ball) and the half-space kernel with its bounded Dirichlet problem (Poisson kernel and bounded Dirichlet problem on a half-space) are stated for $n\ge 3$. The reflection formula for the half-space kernel is the reflection of the fundamental solution, whose profile is $\mathrm{\mid }z{\mathrm{\mid }}^{2-n}$ precisely for $n\ge 3$; in the plane the corresponding profile is logarithmic and the kernel constants change. The bounded uniqueness argument requires its own planar proof. The statements above do not claim to cover the planar case.

The full planar Dirichlet theory is cited; a smooth-data lemma is used. The continuous-data disc Dirichlet theorem (The Poisson integral gives the unique continuous harmonic extension on the closed unit disc) and the full disc Poisson theory are developed on their own page. The $n=2$ branches of the interior derivative estimates and real-analyticity theorem also use Smooth sphere data have a harmonic replacement under Countable Choice, whose planar case gives the Poisson representation for smooth circle data after harmonic regularity is established. This restricted smooth-data result does not reprove the full continuous-data theorem or its boundary-convergence theorem.

Results including $n=2$. The interior derivative estimates (Interior derivative estimates for harmonic functions) and the real-analyticity theorem (Harmonic functions are real analytic) cover $n\ge 2$; their planar arguments use mean-value regularity and the smooth-data sphere lemma above. The interior $C^{2,\alpha }$ Poisson estimate and its gradient corollary (Interior estimate for the Poisson equation with Hölder data) also cover $n\ge 2$ and handle the planar case with the logarithmic Newtonian potential. These arguments use separate formulas in dimension two and dimensions at least three, without extending the image construction to the plane.

What this remark does not say. This is a statement about the scope of the items on this page, not a mathematical claim that the $n=2$ kernels fail to exist. The disc and half-plane kernels exist; they are simply developed elsewhere and referenced here.