 
Summary: THE IVP FOR THE BENJAMINONO EQUATION IN
WEIGHTED SOBOLEV SPACES
GERM´AN FONSECA AND GUSTAVO PONCE
Abstract. 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 Zs,r = Hs(R) L2(x2rdx), s R, s 1
and s 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.
1. Introduction
This work is concerned with the initial value problem (IVP) for the Benjamin
Ono (BO) equation
(1.1)
tu + H2
xu + uxu = 0, t, x R,
u(x, 0) = u0(x),
where H denotes the Hilbert transform
(1.2)
Hf(x) =
1
