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On the microlocal smoothing effect of dispersive partial differential equations. I: Second-order linear equations. (English) Zbl 0683.35010

Algebraic analysis, Pap. Dedicated to Prof. Mikio Sato on the Occas. of his Sixtieth Birthday, Vol. 2, 911-926 (1989).
[For the entire collection see Zbl 0665.00008.]
A microlocal version of the regularizing effect of some systems of second-order linear PDE is described in this paper. It is proved that certain decay of the Cauchy data in some directions implies microlocal regularity of solutions at positive time. The principal symbol is assumed to be non-characteristic or of principal type at the point. The decay of the Cauchy data is required only in some conic set. The used method is a modification of that of N. Hayashi, K. Nakamitsu and M. Tsutsumi [Math. Z. 192, 637-650 (1986; Zbl 0617.35025); and J. Funct. Anal. 71, 218-245 (1987; Zbl 0657.35033)]. The case of higher-order equations will be considered in a forthcoming paper.
Reviewer: S.Dimiev

MSC:

35B65 Smoothness and regularity of solutions to PDEs
35S10 Initial value problems for PDEs with pseudodifferential operators
35G10 Initial value problems for linear higher-order PDEs