Abstract
We give a “heat equation” proof of a theorem which says that for all ε sufficiently small, the map \({S_{\epsilon} \colon f \mapsto \exp(- \epsilon \sqrt{\Delta})f}\) extends to an isomorphism from H s(X) to \({\mathcal{O}^{s + (n-1)/4}(\partial M_{\epsilon})}\). 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 \({s =d^{2}(z, y) + \epsilon^{2}}\) where d 2 is the analytic continuation of the distance function squared on \({X,\,z \in M_{\epsilon}}\), and \({y \in X}\).
Similar content being viewed by others
References
Beyer W.A., Heller L.: Analytic continuation of Laplace transforms by means of asymptotic series. J. Math. Phys. 8(5), 1004–1018 (1967)
Boutet de Monvel L.: Convergence dans le domaine complexe des séries de fonctions propres. C. R. Acad. Sci. Paris Sér. A-B 287(13), A855–A856 (1978)
Boutet de Monvel, L.: Convergence dans le domaine complexe des séries de fonctions propres. In: Journées Équations aux Dérivées Partielles (Saint-Cast, 1979), Exp. No. 3, pp. 1–2. École Polytech., Palaiseau (1979)
Boutet de Monvel, L.: Complément sur le noyau de Bergman. In: Séminaire sur les équations aux dérivées partielles, 1985–1986, Exp. No. XX, pp. 1–13. École Polytech., Palaiseau (1986)
Boutet de Monvel L., Guillemin V.: The Spectral Theory of Toeplitz Operators, volume 99 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ (1981)
Boutet de Monvel, L., Sjöstrand, J.: Sur la singularité des noyaux de Bergman et de Szegő. In: Journées Équations aux Dérivées Partielles de Rennes (1975), pp. 123–164. Astérisque, No. 34–35. Soc. Math. France, Paris (1976)
Bruhat F., Whitney H.: Quelques propriétés fondamentales des ensembles analytiques-réels. Comment. Math. Helv. 33, 132–160 (1959)
Duistermaat J.J., Guillemin V.W.: The spectrum of positive elliptic operators and periodic bicharacteristics. Invent. Math. 29(1), 39–79 (1975)
Guillemin, V.: Paley-Wiener on manifolds. In: The Legacy of Norbert Wiener: A Centennial Symposium (Cambridge, MA, 1994), volume 60 of Proceedings of the Symposium in Pure Mathematics, pp. 85–91. American Mathematical Society, Providence, RI (1997)
Guillemin V., Stenzel M.: Grauert tubes and the homogeneous Monge–Ampère equation. J. Differ. Geom. 34(2), 561–570 (1991)
Guillemin V., Stenzel M.: Grauert tubes and the homogeneous Monge–Ampère equation. II. J. Differ. Geom. 35(3), 627–641 (1992)
Hadamard J.: Lectures on Cauchy’s Problem in Linear Partial Differential Equations. Dover, New York (1953)
Handelsman R., Lew J.: Asymptotic expansion of a class of integral transforms via Mellin transforms. Arch. Ration. Mech. Anal. 35, 382–396 (1969)
Hörmander, L.: The Analysis of Linear Partial Differential Operators. I, volume 256 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (1983). Distribution theory and Fourier analysis
Hörmander L.: L 2 estimates for Fourier integral operators with complex phase. Ark. Mat. 21(2), 283–307 (1983)
Jorgenson J., Lang S.: Analytic continuation and identities involving heat, Poisson, wave and Bessel kernels. Math. Nachr. 258, 44–70 (2003)
Kashiwara, M.: Analyse micro-locale du noyau de Bergman. In Séminaire Goulaouic-Schwartz (1976/1977), Équations aux dérivées partielles et analyse fonctionnelle, Exp. No. 8, pp. 1–10. Centre Math., École Polytech., Palaiseau (1977)
Leichtnam E., Golse F., Stenzel M.: Intrinsic microlocal analysis and inversion formulae for the heat equation on compact real-analytic Riemannian manifolds. Ann. Sci. École Norm. Sup. (4) 29(6), 669–736 (1996)
Lempert L., Szőke R.: Global solutions of the homogeneous complex Monge-Ampère equation and complex structures on the tangent bundle of Riemannian manifolds. Math. Ann. 290(4), 689–712 (1991)
Melin, A., Sjöstrand, J.: Fourier integral operators with complex-valued phase functions. In: Fourier integral operators and partial differential equations (Colloq. Internat., Univ. Nice, Nice, 1974), pp. 120–223. Lecture Notes in Mathematics, Vol. 459. Springer, Berlin (1975)
Trèves, F.: Introduction to Pseudodifferential and Fourier Integral Operators. Vol. 2. Plenum Press, New York (1980). Fourier integral operators, The University Series in Mathematics
Zelditch S.: Complex zeros of real ergodic eigenfunctions. Invent. Math. 167(2), 419–443 (2007)
Zelditch, S.: Pluri-potential theory on Grauert tubes of real analytic Riemannian manifolds, I. In: Spectral geometry, volume 84 of Proceedings of the Symposium in Pure Mathematics, pp. 299–339. American Mathematical Society, Providence, RI (2012)