Proposition: Mode kernel and regular diagonal

For $\eta >0$, the Euclidean covariance is

On a sufficiently small developing neighbourhood of every regular point, in a local parallel frame,

where

is jointly real-analytic, including the diagonal. It is a bisolution and is Hermitian under exchange of endpoints. This holds for every $0<\alpha <1$, including image-boundary parameter values.

Proof. Multiply (eq:heat) by $(4\pi s{)}^{-1/2}{e}^{-\Delta {\tau }^2/(4s)}$ and integrate in $s$. The large-$s$ integral converges on compact regular sets by (eq:heatbound); the small-$s$ integral defines a distribution. Substitute $\zeta =r{r}^{\prime }/(2s)$ and use

Absolute convergence for $\eta >0$ gives (eq:GE) Watson. This also derives the mode kernel used in I-B-VP-v2 :euclidean CheungIBVP.

The heat-integrated kernel is a fundamental distribution for $-{\Delta }_E$ in each endpoint. In a parallel Cartesian chart, the direct term in (eq:H) has the same delta source. Their difference therefore satisfies both homogeneous equations across the diagonal. It is annihilated by the elliptic operator

on the product chart. Elliptic and analytic elliptic regularity make the difference jointly real-analytic Hormander,MorreyNirenberg. Hermiticity follows from the self-adjoint heat kernel.

This argument applies to the total kernel, not separately to moving image-sector indicators.