## 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
Full Text:

### References:

 [1] ; ; The Schrödinger equation on a compact manifold: Strichartz estimates and applications. Journées ?Équations aux Dérivées Partielles? (Plestin-les-Grèves, 2001), Exp. 5. Université de Nantes, Nantes, 2001. [2] Inegalités de Carleman Lp pour des indices critiques. Doctoral dissertation, Universite de Rennes I, 2002. [3] Escauriaza, Duke Math J 104 pp 113– (2000) [4] Ginibre, J Funct Anal 133 pp 50– (1995) [5] Hörmander, Math Ann 140 pp 124– (1960) [6] The analysis of linear partial differential operators. III. Pseudodifferential operators. Grundlehren der Mathematischen Wissenschaften, 274. Springer, Berlin, 1985. [7] The analysis of linear partial differential operators. IV. Fourier integral operators. Grundlehren der Mathematischen Wissenschaften, 275. Springer, Berlin, 1985. [8] Isakov, J Differential Equations 105 pp 217– (1993) [9] Jerison, Ann of Math (2) 121 pp 463– (1985) [10] Kapitanski??, J Soviet Math 56 pp 2348– (1991) [11] Keel, Amer J Math 120 pp 955– (1998) [12] Kenig, Duke Math J 55 pp 329– (1987) [13] Koch, Comm Pure Appl Math 54 pp 339– (2001) [14] Mockenhaupt, J Amer Math Soc 6 pp 65– (1993) [15] Smith, Ann Inst Fourier (Grenoble) 48 pp 797– (1998) · Zbl 0974.35068 [16] Sogge, Ark Mat 28 pp 159– (1990) [17] Sogge, Amer J Math 112 pp 943– (1990) [18] Fourier integrals in classical analysis. Cambridge Tracts in Mathematics, 105. Cambridge University, Cambridge, 1993. · Zbl 0783.35001 [19] Staffilani, Comm Partial Differential Equations 27 pp 1337– (2002) [20] Tataru, Comm Partial Differential Equations 21 pp 841– (1996) [21] Tataru, Amer J Math 123 pp 385– (2001) [22] Tataru, Comm Partial Differential Equations 27 pp 2101– (2002) [23] Tataru, J Amer Math Soc 15 pp 419– (2002) [24] Wolff, Geom Funct Anal 2 pp 225– (1992) [25] Recent work on sharp estimates in second order elliptic unique continuation problems. Fourier analysis and partial differential equations (Miraflores de la Sierra, 1992), 99-128. Studies in Advanced Mathematics. CRC, Boca Raton, Fla., 1995.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.