On the analytic continuation of the Poisson kernel - Stenzel, Matthew B. (Manuscripta mathematica, 2014)
We give a “heat equation” proof of a theorem which says that for all ε sufficiently small, the map S ϵ : f ↦ exp ( - ϵ Δ ) f extends to an isomorphism from H s ( X ) to O s + ( n - 1 ) / 4 ( ∂ M ϵ ) . This result was announced by Boutet de Monvel (C R Acad Sci Paris Sér A-B 287(13):A855–A856, 1978 ) but only recently has a proof, due to Zelditch (Spectral geometry, volume 84 of proceedings of the symposium in pure mathematics, pp 299–339. American Mathematical Society, Providence, RI, 2012 ), appeared in the literature. The main tools in our proof are the subordination formula relating the Poisson kernel to the heat kernel, and an expression for the singularity of the Poisson kernel in the complex domain in terms of the Laplace transform variable s = d 2 ( z , y ) + ϵ 2 where d 2 is the analytic continuation of the distance function squared on X , z ∈ M ϵ , and y ∈ X .