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In this paper we investigate the dispersive properties of the solutions of the two dimensional water-waves system with surface tension. First we prove Strichartz type estimates with loss of derivatives at the same low level of regularity we were able to construct the solutions in [3]. On the other hand, for smoother initial data, we prove that the solutions enjoy the optimal Strichartz estimates (i.e, without loss of regularity compared to the system linearized at ()).
We study general continuity properties for an increasing family of Banach spaces of classes for pseudo-differential symbols, where was introduced by J.
Sjöstrand in 1993. We prove that the operators in are Schatten-von
Neumann operators of order on . We prove also that and , provided . If instead , then . By
modifying the definition of the -spaces, one also obtains symbol classes related
to the spaces.
The symbol calculus on the upper half plane is studied from the viewpoint of the Kirillov theory of orbits. The main result is the -estimates for Fuchs type pseudodifferential operators.
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