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We consider the following Hamiltonian equation on the L Hardy space on the circle, i∂tu = Π(|u|u) , where Π is the Szegö projector. This equation can be seen as a toy model for totally non dispersive evolution equations. We display a Lax pair structure for this equation. We prove that it admits an infinite sequence of conservation laws in involution, and… (More)

We derive an explicit formula for the general solution of the cubic Szegő equation and of the evolution equation of the corresponding hierarchy. As an application, we prove that all the solutions corresponding to finite rank Hankel operators are quasiperiodic.

We consider the following degenerate half wave equation on the one dimensional torus i∂tu− |D|u = |u|u, u(0, ·) = u0. We show that, on a large time interval, the solution may be approximated by the solution of a completely integrable system– the cubic Szegö equation. As a consequence, we prove an instability result for large H norms of solutions of this… (More)

We extend to multilinear Hankel operators the fact that truncation of bounded Hankel operators is bounded. We prove and use a continuity property of a kind of bilinear Hilbert transforms on product of Lipschitz spaces and Hardy spaces. 1. Statement of the results In this note, we prove that truncations of bounded multilinear Hankel operators are bounded.… (More)

In this paper, we prove Fefferman-Stein like characterizations of Hardy-Sobolev spaces of complex tangential derivatives of holomorphic functions in domains of finite type in Cn. We also study the relationship between these complex tangential Hardy-Sobolev spaces and the usual ones. We also obtain partial results on domains not necessarily of finite type.

In this thesis we consider several questions on harmonic and analytic functions spaces and some of their operators. These questions deal with Carleson-type measures in the unit ball, bi-parameter paraproducts and multipliers problem on the bitorus, boundedness of the Bergman projection and analytic Besov spaces in tube domains over symmetric cones. In part… (More)

Abstract. In this paper, we pursue the study of harmonic functions on the real hyperbolic ball started in [12]. Our focus here is on the theory of Hardy, Hardy-Sobolev and Lipschitz spaces of these functions. We prove here that these spaces admit FeffermanStein like characterizations in terms of maximal and square functionals. We further prove that the… (More)

We extend to multilinear Hankel operators the fact that some truncations of bounded Hankel operators are bounded. We prove and use a continuity property of bilinear Hilbert transforms on products of Lipschitz spaces and Hardy spaces. 1. Statement of the results We prove that some truncations of bounded multilinear Hankel operators are bounded. This extends… (More)

Given two arbitrary sequences (λj)j≥1 and (μj)j≥1 of real numbers satisfying |λ1| > |μ1| > |λ2| > |μ2| > · · · > |λj | > |μj | → 0 , we prove that there exists a unique sequence c = (cn)n∈Z+ , real valued, such that the Hankel operators Γc and Γc̃ of symbols c = (cn)n≥0 and c̃ = (cn+1)n≥0 respectively, are selfadjoint compact operators on l (Z+) and have… (More)

We consider the following Hamiltonian equation on the L Hardy space on the circle, i∂tu = Π(|u|u) , where Π is the Szegö projector. This equation can be seen as a toy model for totally non dispersive evolution equations. We display a Lax pair structure for this equation. We prove that it admits an infinite sequence of conservation laws in involution, and… (More)