Let (x i ), i=1,...,n-1 be C ∞ linearly independent vector fields on an open set Φ of R n . Assume that the Lie algebra generated by these fields is of maximal rank at every point of Ω and that the volume form associated to them is of class 4 at a point x 0 of Ω. We show then that if P is the operator P=∑ i=1 n-1 x i 2 , there exists an open neighborhood V of x 0 and a function a∈C ∞ (V) such that P+a does not enjoy the uniqueness extension property.
Tópico(s): Advanced Banach Space Theory
1986 - Association of the Annals of the Fourier Institute | Annales de l’institut Fourier
Hajer Bahouri, Jean-Yves Chemin, Isabelle Gallagher,
The aim of this article is to present "refined" Hardy-type inequalities.Those inequalities are generalisations of the usual Hardy inequalities, their additional feature being that they are invariant under oscillations: when applied to highly oscillatory functions, both sides of the refined inequality are of the same order of magnitude.The proof relies on paradifferential calculus and Besov spaces.It is also adapted to the case of the Heisenberg group.
Tópico(s): Spectral Theory in Mathematical Physics
2009 - | ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE
Tópico(s): Nonlinear Partial Differential Equations
1994 - Springer Science+Business Media | Archive for Rational Mechanics and Analysis
Hajer Bahouri, Mohamed Majdoub, Nader Masmoudi,
This paper is devoted to the description of the lack of compactness of Hrad1(R2) in the Orlicz space. Our result is expressed in terms of the concentration-type examples derived by P.-L. Lions (1985) in [24]. The approach that we adopt to establish this characterization is completely different from the methods used in the study of the lack of compactness of Sobolev embedding in Lebesgue spaces and takes into account the variational aspect of Orlicz spaces. We also investigate the feature of the solutions ...
Tópico(s): Stability and Controllability of Differential Equations
2010 - Elsevier BV | Journal of Functional Analysis
Hajer Bahouri, Albert Cohen, Gabriel S. Koch,
We characterize the lack of compactness in the critical embedding of functions spaces $X\subset Y$ having similar scaling properties in the following terms : a sequence $(u_n)_{n\geq 0}$ bounded in $X$ has a subsequence that can be expressed as a finite sum of translations and dilations of functions $(\phi_l)_{l>0}$ such that the remainder converges to zero in $Y$ as the number of functions in the sum and $n$ tend to $+\infty$. Such a decomposition was established by G\'erard for the embedding of the homogeneous ...
Tópico(s): Stability and Controllability of Differential Equations
2011 - | Confluentes Mathematici
Hajer Bahouri, Jean-Yves Chemin, Raphaël Danchin,
In recent years, the Fourier analysis methods have expereinced a growing interest in the study of partial differential equations. In particular, those techniques based on the Littlewood-Paley decompos
Tópico(s): Mathematics and Applications
2011 - Springer Nature | Grundlehren der mathematischen Wissenschaften
We prove that the semilinear wave equation □u + u3 = 0 in ℝ3 is globally well-posed for data in the homogeneous Sobolev space H3/4(ℝ3). The proof is based on a nonlinear interpolation method and logarithmic Strichartz estimates.
Tópico(s): Stability and Controllability of Differential Equations
2006 - Oxford University Press | International Mathematics Research Notices
Habib Ammari, Hajer Bahouri, David Dos Santos Ferreira, Isabelle Gallagher,
We consider an inverse scattering problem and its near-field approximation at high frequencies. We first prove, for both problems, Lipschitz stability results for determining the low-frequency component of the potential. Then we show that, in the case of a radial potential supported sufficiently near the boundary, infinite resolution can be achieved from measurements of the near-field operator in the monotone case.
Tópico(s): Advanced Mathematical Modeling in Engineering
2012 - Elsevier BV | Journal of Mathematical Analysis and Applications
Hajer Bahouri, Jean-Yves Chemin, Chao-Jiang Xu,
we prove in this work the trace and trace lifting theorems for the sobolev spaces associated with a system of hörmander's vectors fields of order 2. the case of non-degenerate characteristic points for left invariant vector fields on the heisenberg group is also studied.
Tópico(s): Advanced Mathematical Physics Problems
2005 - Cambridge University Press | Journal of the Institute of Mathematics of Jussieu
Hajer Bahouri, Patrick Gérard, Chao-Jiang Xu,
Tópico(s): Mathematical Analysis and Transform Methods
2000 - Springer Science+Business Media | Journal d Analyse Mathématique
Hajer Bahouri, Mohamed Majdoub, Nader Masmoudi,
This Note is devoted to the description of the lack of compactness of the Sobolev embedding of H1(R2) in the critical Orlicz space L(R2). It turns out that up to cores our result is expressed in terms of the concentration-type examples derived by J. Moser (1971) in [16] as in the radial setting investigated in Bahouri et al. (2011) [5]. However, the analysis we used in this work is strikingly different from the one conducted in the radial case which is based on an L∞ estimate far away from the origin ...
Tópico(s): Advanced Mathematical Physics Problems
2013 - Elsevier BV | Journal de Mathématiques Pures et Appliquées
Hajer Bahouri, Isabelle Gallagher,
Let X be a suitable function space and let $${\mathcal{G} \subset X}$$ be the set of divergence free vector fields generating a global, smooth solution to the incompressible, homogeneous three-dimensional Navier–Stokes equations. We prove that a sequence of divergence free vector fields converging in the sense of distributions to an element of $${\mathcal{G}}$$ belongs to $${\mathcal{G}}$$ if n is large enough, provided the convergence holds “anisotropically” in frequency space. Typically, this excludes self- ...
Tópico(s): Advanced Mathematical Physics Problems
2013 - Springer Science+Business Media | Archive for Rational Mechanics and Analysis
Hajer Bahouri, Isabelle Gallagher,
We adapt the homogeneous Littlewood–Paley decomposition on the Heisenberg group constructed by H. Bahouri, P. Gérard and C.-J Xu in [4] to the inhomogeneous case, which enables us to build paraproduct operators, similar to those defined by J.-M. Bony in [5]; although there is no simple formula for the Fourier transform of the product of two functions, some spectral localization properties of the classical case are preserved on the Heisenberg group after the product has been taken. Using the dyadic ...
Tópico(s): Advanced Mathematical Physics Problems
2001 - Royal Spanish Mathematical Society | Revista Matemática Iberoamericana
Hajer Bahouri, Clotilde Fermanian Kammerer, Isabelle Gallagher,
The present paper is dedicated to the proof of dispersive estimates on stratified Lie groups of step 2 for the linear Schrödinger equation involving a sublaplacian.It turns out that the propagator behaves like a wave operator on a space of the same dimension p as the center of the group, and like a Schrödinger operator on a space of the same dimension k as the radical of the canonical skew-symmetric form, which suggests a decay rate |t| -(k+ p-1)/2 .We identify a property of the canonical skew-symmetric ...
Tópico(s): Mathematical Analysis and Transform Methods
2016 - Mathematical Sciences Publishers | Analysis & PDE
In this paper we prove that finite energy solutions (with added regularity) to the critical wave equation q w + 1,' = 0 on R3 decay to zero in time.The proof is based on a global space-time estimate and dilation identity.0 Elsevier, Paris RI$SUM& -Dans cet article, on montre que les solutions a Cnergie finie (avec regularit ajoutee) de l'equation des ondes critique 0~ + 7b5 = 0. dans R" decroissent vers zero en temps.La demonstration est basee sur une estimee temps-espace globale et une identite de dilatation.
Tópico(s): Stability and Controllability of Differential Equations
1998 - Elsevier BV | Annales de l Institut Henri Poincaré C Analyse Non Linéaire
Hajer Bahouri, Patrick Gérard,
This work is devoted to the description of bounded energy sequences of solutions to the equation (1) □ u + | u | 4 = 0 in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /], up to remainder terms small in energy norm and in every Strichartz norm. The proof relies on scattering theory for (1) and on a structure theorem for bounded energy sequences of solutions to the linear wave equation. In particular, we infer the existence of an a priori estimate of Strichartz norms of solutions to ( ...
Tópico(s): Advanced Harmonic Analysis Research
1999 - Johns Hopkins University Press | American Journal of Mathematics
Hajer Bahouri, Jean-Yves Chemin,
Dans cet article, notre but est de résoudre des équations d'ondes quasilinéaires pour des données initiales moins régulières que ce qu'imposent les méthodes d'énergie. Ceci impose de démontrer des estimées de type Strichartz pour des opérateurs d'ondes à coefficients seulement lipschitziens.
Tópico(s): Advanced Harmonic Analysis Research
1999 - Johns Hopkins University Press | American Journal of Mathematics
Hajer Bahouri, Galina Perelman,
This paper is dedicated to the study of the derivative nonlinear Schrödinger equation on the real line. The local well-posedness of this equation in the Sobolev spaces $$H^s({\mathop {{\mathbb {R}}}\nolimits })$$ is well understood since a couple of decades, while the global well-posedness is not completely settled. For the latter issue, the best known results up-to-date concern either Cauchy data in $$H^{\frac{1}{2}}({\mathop {{\mathbb {R}}}\nolimits })$$ with mass strictly less than $$4\pi $$ or general initial conditions ...
Tópico(s): Mathematical Analysis and Transform Methods
2022 - Springer Science+Business Media | Inventiones mathematicae
Hajer Bahouri, Slim Ibrahim, Galina Perelman,
In this article, we establish in the radial framework the $H^1$-scattering for the critical 2-D nonlinear Schr\"odinger equation with exponential growth. Our strategy relies on both the a priori estimate derived in \cite{CGT, PV} and the characterization of the lack of compactness of the Sobolev embedding of $H_{rad}^1(\R^2)$ into the critical Orlicz space ${\cL}(\R^2)$ settled in \cite{BMM}. The radial setting, and particularly the fact that we deal with bounded functions far away from the origin, occurs ...
Tópico(s): Nonlinear Partial Differential Equations
2014 - Khayyam Publishing | Differential and Integral Equations
Hajer Bahouri, Isabelle Gallagher,
The goal of this paper is to study the action of the heat operator on the Heisenberg group H d , and in particular to characterize Besov spaces of negative index on H d in terms of the heat kernel. That characterization can be extended to positive indexes using Bernstein inequalities. As a corollary we obtain a proof of refined Sobolev inequalities in $$\dot W^{s,p}$$ spaces.
Tópico(s): Mathematical Analysis and Transform Methods
2009 - | Progress in nonlinear differential equations and their applications
Hajer Bahouri, Clotilde Fermanian Kammerer, Isabelle Gallagher,
We construct a Littlewood–Paley decomposition associated to a Rockland operator on graded Lie groups, which allows us to deduce refined Gagliardo–Nirenberg, Sobolev and Hardy inequalities. On construit une décomposition de Littlewood–Paley associée à un opérateur de Rockland sur les groupes de Lie gradués, qui permet de déduire des inégalités précisées de type Gagliardo–Nirenberg, Sobolev et Hardy.
Tópico(s): Nonlinear Partial Differential Equations
2012 - Elsevier BV | Comptes Rendus Mathématique
Hajer Bahouri, Jean-Yves Chemin, Raphaël Danchin,
We revisit the Fourier analysis on the Heisenberg group ℍ d . Starting from the so-called Schrödinger representation and taking advantage of the projection with respect to the Hermite functions, we look at the Fourier transform of an integrable function f, as a function f ^ ℍ on the set ℍ ˜ d = def ℕ d ×ℕ d ×ℝ∖{0}. After observing that f ^ ℍ is uniformly continuous on ℍ ˜ d equipped with an appropriate distance d ^, we extend the definition of f ^ ℍ to the completion ℍ ^ d of ℍ ˜ d . This new point of view provides a simple ...
Tópico(s): Digital Filter Design and Implementation
2019 - Association of the Annals of the Fourier Institute | Annales de l’institut Fourier
Hajer Bahouri, Jean-Yves Chemin, Chao-Jiang Xu,
We prove in this work the trace and trace lifting theorem for Sobolev spaces on the Heisenberg groups for hypersurfaces with characteristics submanifolds.
Tópico(s): Geometric Analysis and Curvature Flows
2009 - Association of the Annals of the Fourier Institute | Annales de l’institut Fourier
Hajer Bahouri, Galina Perelman,
This paper is devoted to the characterization of the lack of compactness of the Sobolev embedding of $H^N(R^{2N})$ into the Orlicz space using Fourier analysis.
Tópico(s): Mathematical Analysis and Transform Methods
2014 - International Press of Boston | Mathematical Research Letters
We establish refined Sobolev inequalities between the Lorentz spaces and homogeneous Besov spaces. The sharpness of these inequalities is illustrated on several examples, in particular based on non-uniformly oscillating functions known as chirps. These results are also used to derive refined Hardy inequalities.
Tópico(s): Advanced Mathematical Physics Problems
2011 - Birkhäuser | Journal of Fourier Analysis and Applications
Hajer Bahouri, Clotilde Fermanian Kammerer, Isabelle Gallagher,
Nous définissons une classe d'opérateurs pseudo-différentiels sur le groupe de Heisenberg. Cette classe constitue une algèbre contenant les opérateurs différentiels. De plus, ces opérateurs pseudo-différentiels sont continus sur les espaces de Sobolev et l'on peut contrôler la perte de dérivée par leur ordre. Notre approche met en évidence des directions microlocales et complète, avec la théorie de Littlewood–Paley développée par Bahouri et al. une analyse microlocale du groupe de Heisenberg. Pour ...
Tópico(s): Advanced Mathematical Physics Problems
2009 - Elsevier BV | Comptes Rendus Mathématique
In this article we present the Littlewood-Paley theory and illustrate the effectiveness of this microlocal analysis tool in the study of partial differential equations, in a context which is the least technical possible. As we shall see below, the Littlewood-Paley theory provides a robust approach not only to the separate study of the various regimes of solutions to nonlinear partial differential equations, but also to the fine study of functional inequalities, and to make them accurate. 1. The ...
Tópico(s): Image and Signal Denoising Methods
2019 - | EMS Newsletter
Tópico(s): Mathematical and Theoretical Analysis
1983 - Taylor & Francis | Communications in Partial Differential Equations
Hajer Bahouri, Jean-Yves Chemin, Chao-Jiang Xu,
We prove in this work the trace and trace lifting theorem for Sobolev spaces on the Heisenberg groups for homogeneous hypersurfaces.
Tópico(s): Geometric Analysis and Curvature Flows
2006 - | Progress in nonlinear differential equations and their applications
Hajer Bahouri, Jean-Yves Chemin, Isabelle Gallagher,
Le but de cette Note est de présenter des inégalités de type Hardy « précisé ». Elles généralisent les inégalités de Hardy habituelles, et leur caractéristique est d'être invariantes par oscillations : appliqués à des fonctions très oscillantes, les deux membres de l'inégalité précisée sont du même ordre de grandeur. La démonstration repose sur le calcul paradifférentiel et les espaces de Besov. Pour citer cet article : H. Bahouri et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005). The aim of this Note is ...
Tópico(s): Differential Equations and Boundary Problems
2005 - Elsevier BV | Comptes Rendus Mathématique