Off-Diagonal Decay of Bergman Kernels: On a Question of Zelditch

  • Michael ChristEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 269)


We study the orthogonal projection from \(L^2({\mathbb {C}}^d, e^{-2\lambda \phi })\) to its subspace of entire holomorphic functions, as \(\lambda \rightarrow \infty \), for weights \(\phi \) that depend only on \({\text {Re}}(z)\) and are uniformly strictly plurisubharmonic. We show that the associated Bergman kernels are \(O(e^{-c\lambda })\) away from the diagonal, if and only if \(\phi \) is real analytic.


  1. 1.
    Christ, M.: On the \(\bar{\partial }\) equation in weighted \(L^2\) norms in \({\mathbf{C}}^1\). J. Geom. Anal. 1(3), 193–230 (1991)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Christ, M.: Slow off-Diagonal Decay for Szegö Kernels Associated to Smooth Hermitian Line Bundles, Harmonic Analysis at Mount Holyoke (South Hadley, MA, 2001), Contemporary Mathematics, vol. 320, pp. 77–89. American Mathematical Society, Providence (2003)Google Scholar
  3. 3.
    Christ, M.: Upper Bounds for Bergman Kernels Associated to Positive Line Bundles with Smooth Hermitian Metrics, this volume.
  4. 4.
    Hörmander, L.: An Introduction to Complex Analysis in Several Variables. North-Holland Mathematical Library, vol. 7, 3rd edn. North-Holland Publishing Co., Amsterdam (1990)Google Scholar
  5. 5.
    Zelditch, S.: Personal CommunicationGoogle Scholar
  6. 6.
    Zelditch, S.: Off-diagonal decay of toric Bergman kernels. Lett. Math. Phys. 106(12), 1849–1864 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Zworski, M.: Personal CommunicationGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

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