Helmholtz operators and symmetric space duality
We consider the property of vanishing logarithmic term (VLT) for the fundamental solution of the shifted Laplace-d'Alembert operators □ + b (b a constant), on pseudo-Riemannian reductive symmetric spaces M. Our main result is that such an operator on the c-dual or Flensted-Jensen dual of M has the VLT property if and only if a corresponding operator on M does. For Lorentzian spaces, where the □ + b are hyperbolic, VLT is known to be equivalent to the strong Huygens principle. We use our results to construct a large supply of new (space, operator) pairs satisfying Huygens' principle.
Publication Source (Journal or Book title)
Branson, T., & Ólafsson, G. (1997). Helmholtz operators and symmetric space duality. Inventiones Mathematicae, 129 (1), 63-74. https://doi.org/10.1007/s002220050158