We prove the reducibility of a class of quasi-periodically time dependent linear operators, which are derived from linearizing the dispersive third order Benjamin–Ono (BO) equation on the circle at a small amplitude quasi-periodic function, with a diophantine frequency vector ωO0Rν. It is shown that there exists a set OO0 of asymptotically full Lebesgue measure such that for any ωO, the operators can be reduced to the ones with constant coefficients by some linear transformations depending on time quasi-periodically. These transformations include a change of variable induced by a diffeomorphism of the torus, the flow of some partial differential equations and a pseudo-differential operator of order zero. We first reduce the linearized operator of order three to the one with constant coefficients plus a remainder of order zero, and then a perturbative reducibility scheme is performed. The major difficulties encountered are brought by the non-smooth character of the dispersive relation in view of the presence of the Hilbert operator H. We look for several appropriate transformations which are real, reversibility-preserving and satisfy the sharp tame bounds which are used for the reducibility. This work will be the first fundamental step in proving the existence of time quasi-periodic solutions for the dispersive third order BO equation.

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