Mathematische Annalen

, Volume 335, Issue 3, pp 645-673

First online:

A smoothing property of Schrödinger equations in the critical case

  • Michael RuzhanskyAffiliated withDepartment of Mathematics, Imperial College London
  • , Mitsuru SugimotoAffiliated withDepartment of Mathematics, Graduate School of Science, Osaka University Email author 

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This paper deals with the critical case of the global smoothing estimates for the Schrödinger equation. Although such estimates fail for critical orders of weights and smoothing, it is shown that they are still valid if one works with operators with symbols vanishing on a certain set. The geometric meaning of this set is clarified in terms of the Hamiltonian flow of the Laplacian. The corresponding critical case of the limiting absorption principle for the resolvent is also established. Obtained results are extended to dispersive equations of Schrödinger type, to hyperbolic equations and to equations of other orders.