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Dispersive estimates for principally normal pseudodifferential operators. (English) Zbl 1078.35143

Motivated by problems in unique continuation and in local solvability, the authors consider the problem of obtaining dispersive estimates for principally normal pseudodifferential operators, \(P(x,D)\), with an interest in the construction of parametrices for these operators. Assumptions are made only on the geometry of the characteristics under minimal regularity conditions for the symbols and coefficients. An obstacle in applying Carleman estimates for unique continuation problems is that the conjugated operator \(P_\phi=P(x,D+i\tau\nabla\phi)\) does not satisfy the principal normality condition. But, fortunately, the \(L^2\) estimates that follow from the pseudoconvexity condition are strong enough to allow spatial localization on a much smaller scale which is precisely the largest scale on which the principal normality holds. Another obstacle is that for principally normal operators the characteristic set is a codimension-2 manifold, thus preventing uniformly strong kernel decay estimates in all directions for the parametrix. Fortunately, a factorization of the parametrix leads to the dispersive estimates without studying the kernel decay in bad regions. Initially, assumptions and results are stated in a dyadic setting and elliptic arguments are used to reduce the problems to some canonical formulation. A set of increasingly complex problems, with real symbols and in the symplectic case, are considered.

MSC:

35S05 Pseudodifferential operators as generalizations of partial differential operators
47G30 Pseudodifferential operators
35A17 Parametrices in context of PDEs
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