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The IVP for the Benjamin-Ono equation in weighted Sobolev spaces. (English) Zbl 1205.35249
Summary: We study the initial value problem associated to the Benjamin-Ono equation. The aim is to establish persistence properties of the solution flow in the weighted Sobolev spaces \(Z_{s,r} = H^s(\mathbb R) \cap L^2 (|x|^{2r}dx)\), \(s \in \mathbb R\), \(s \geqslant 1\) and \(s \geqslant r\). We also prove some unique continuation properties of the solution flow in these spaces. In particular, these continuation principles demonstrate that our persistence properties are sharp.
For Part II see [(2012; Zbl 1235.35243)].

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
35B60 Continuation and prolongation of solutions to PDEs
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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