Let (M,g) be a noncompact Riemannian manifold whose isometry group acts transitively on M, i.e. a (not necessarily normal) homogeneous space. Let eλ(x,y) be the integral kernel of

f↦∫λ0dEν(f) where dEν is the spectral measure of the (non-negative) Laplacian associated to (M,g). Is there a relatively simple proof that∫M|∇xeλ(x,y)|2gdy=∫Meλ(x,y)⋅Δxeλ(x,y)dy ?

Note that the differentiation is in x and the integration is in y.

In particular, it’s surely false for most non-homogeneous manifolds.I ask for `relatively simple’ because I know of a proof using the expression of the Laplacian as the generator of the heat semi-group. That proof seems to be overkill.

**Answer**

**Attribution***Source : Link , Question Author : Chris Judge , Answer Author : Community*