Geodesic
Sharpness and the local Petrovskii condition for strictly hyperbolic operators with constant coefficients
Izvestiya. Mathematics , Tome 28 (1987) no. 2, pp. 233-273

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It is proved that for almost all hyperbolic operators with constant coefficients analytic sharpness of the fundamental solution everywhere is equivalent to the local Petrovskii condition. In the proximity of simple ($O$-modal) singularities of wave front sets the author finds all domains from one side of which there is sharpness. Bibliography: 24 titles. 
@article{IM2_1987_28_2_a1,
     author = {V. A. Vassiliev},
     title = {Sharpness and the local {Petrovskii} condition for strictly hyperbolic operators with constant coefficients},
     journal = {Izvestiya. Mathematics },
     pages = {233--273},
     publisher = {mathdoc},
     volume = {28},
     number = {2},
     year = {1987},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1987_28_2_a1/}
}
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V. A. Vassiliev. Sharpness and the local Petrovskii condition for strictly hyperbolic operators with constant coefficients. Izvestiya. Mathematics , Tome 28 (1987) no. 2, pp. 233-273. http://geodesic.mathdoc.fr/item/IM2_1987_28_2_a1/