Professeur des universités à l'université d'Orléans Membre du Laboratoire MAPMO CNRS, Uni... more Professeur des universités à l'université d'Orléans Membre du Laboratoire MAPMO CNRS, Université d'Orléans
Let Hn be the real hyperbolic space of dimension n, that is, the complete and simply connected Ri... more Let Hn be the real hyperbolic space of dimension n, that is, the complete and simply connected Riemannian manifold of constant curvature −1. Using the Poincare ́ model,we identify Hn with the unit ball of Rn and its boundary at infinity ∂Hn with the unit sphere S
In this paper, we prove that the product (in the distribution sense) of two functions, which are ... more In this paper, we prove that the product (in the distribution sense) of two functions, which are respectively in $ \BMO(\bR^n)$ and $\H^1(\bR^n)$, may be written as the sum of two continuous bilinear operators, one from $\H^1(\bR^n)\times \BMO(\bR^n) $ into $L^1(\bR^n)$, the other one from $\H^1(\bR^n)\times \BMO(\bR^n) $ into a new kind of Hardy-Orlicz space denoted by $\H^{\log}(\bR^n)$. More precisely, the space $\H^{\log}(\bR^n)$ is the set of distributions $f$ whose grand maximal function $\mathcal Mf$ satisfies $$\int_{\mathbb R^n} \frac {|\mathcal M f(x)|}{\log(e+|x|) +\log (e+ |\mathcal Mf(x)|)}dx <\infty.$$ The two bilinear operators can be defined in terms of paraproducts. As a consequence, we find an endpoint estimate involving the space $\H^{\log}(\bR^n)$ for the $\div$-$\curl$ lemma.
We extend to multilinear Hankel operators a result on the regularity of truncations of Hankel ope... more We extend to multilinear Hankel operators a result on the regularity of truncations of Hankel operators. We prove and use a continuity property on the bilinear Hilbert transforms on product of Lipschitz spaces and Hardy spaces. In this note, we want to extend to multilinear Hankel operators a result obtained by [BB] on the boundedness properties of truncation acting on bounded Hankel infinite matrices. Let us first recall this result. A matrix B = (bmn)m,n∈N is called of Hankel type if bmn = bm+n for some sequence b ∈ l(N). We can consider B as an operator acting on l(N). In this case, we write B = Hb. If we identify l (N) with the complex Hardy space H(D) of the unit disc then Hb can be realized as the integral operator acting on f ∈ H(D) by Hbf(z) = 1 2π ∫ T b(ζ)f(ζ) 1 − ζz dσ(ζ). In other words, Hbf = C(bg) where C denotes the Cauchy integral, g(ζ) = f(ζ) and b is the Symbol of the Hankel operator b(ζ) = ∑∞ k=0 bkζ . If f(z) = ∑ n∈N anz , one has Hbf(z) = ∑ m∈N( ∑ n∈N anbm+n)z . ...
Page 1. arXiv:1103.1822v1 [math.CA] 9 Mar 2011 PARAPRODUCTS AND PRODUCTS OF FUNCTIONS IN BMO(Rn) ... more Page 1. arXiv:1103.1822v1 [math.CA] 9 Mar 2011 PARAPRODUCTS AND PRODUCTS OF FUNCTIONS IN BMO(Rn) AND H1(Rn) THROUGH WAVELETS ALINE BONAMI, SANDRINE GRELLIER, AND LUONG DANG KY Abstract. ...
Page 1. arXiv:1103.1822v1 [math.CA] 9 Mar 2011 PARAPRODUCTS AND PRODUCTS OF FUNCTIONS IN BMO(Rn) ... more Page 1. arXiv:1103.1822v1 [math.CA] 9 Mar 2011 PARAPRODUCTS AND PRODUCTS OF FUNCTIONS IN BMO(Rn) AND H1(Rn) THROUGH WAVELETS ALINE BONAMI, SANDRINE GRELLIER, AND LUONG DANG KY Abstract. ...
... E-mail addresses: Aline.Bonami@univ-orleans.fr (A. Bonami), Sandrine.Grellier@univ-orleans. f... more ... E-mail addresses: Aline.Bonami@univ-orleans.fr (A. Bonami), Sandrine.Grellier@univ-orleans. fr (S. Grellier), bs@maths.gla.ac.uk (BF Sehba ... bounded on H 1 (B n ) and prove that there exists a constant C&gt;0 so that, for any ball B of radius on S n , 1 (B) integraldisplay B |bb B |d ...
This monograph is an expanded version of the preprint arXiv:1402.1716 or hal-00943396v1. It is de... more This monograph is an expanded version of the preprint arXiv:1402.1716 or hal-00943396v1. It is devoted to the dynamics on Sobolev spaces of the cubic Szego equation on the circle ${\mathbb S} ^1$, $$ i\partial _t u=\Pi (\vert u\vert ^2u)\ .$$ Here $\Pi $ denotes the orthogonal projector from $L^2({\mathbb S} ^1)$ onto the subspace $L^2_+({\mathbb S} ^1)$ of functions with nonnegative Fourier modes. We construct a nonlinear Fourier transformation on $H^{1/2}({\mathbb S} ^1)\cap L^2_+({\mathbb S} ^1)$ allowing to describe explicitly the solutions of this equation with data in $H^{1/2}({\mathbb S} ^1)\cap L^2_+({\mathbb S} ^1)$. This explicit description implies almost-periodicity of every solution in $H^{\frac 12}_+$. Furthermore, it allows to display the following turbulence phenomenon. For a dense $G_\delta $ subset of initial data in $C^\infty ({\mathbb S} ^1)\cap L^2_+({\mathbb S} ^1)$, the solutions tend to infinity in $H^s$ for every $s>\frac 12$ with super--polynomial growth...
ABSTRACT We prove that the uniform Carleson condition is sufficient for an analytic variety of a ... more ABSTRACT We prove that the uniform Carleson condition is sufficient for an analytic variety of a strict finite type convex domain in C n to be difined by an holomorphic function in some Hardy space Hp . This extends a result of Varopoulos for the case of strictly pseudoconvex domains.
Professeur des universités à l'université d'Orléans Membre du Laboratoire MAPMO CNRS, Uni... more Professeur des universités à l'université d'Orléans Membre du Laboratoire MAPMO CNRS, Université d'Orléans
Let Hn be the real hyperbolic space of dimension n, that is, the complete and simply connected Ri... more Let Hn be the real hyperbolic space of dimension n, that is, the complete and simply connected Riemannian manifold of constant curvature −1. Using the Poincare ́ model,we identify Hn with the unit ball of Rn and its boundary at infinity ∂Hn with the unit sphere S
In this paper, we prove that the product (in the distribution sense) of two functions, which are ... more In this paper, we prove that the product (in the distribution sense) of two functions, which are respectively in $ \BMO(\bR^n)$ and $\H^1(\bR^n)$, may be written as the sum of two continuous bilinear operators, one from $\H^1(\bR^n)\times \BMO(\bR^n) $ into $L^1(\bR^n)$, the other one from $\H^1(\bR^n)\times \BMO(\bR^n) $ into a new kind of Hardy-Orlicz space denoted by $\H^{\log}(\bR^n)$. More precisely, the space $\H^{\log}(\bR^n)$ is the set of distributions $f$ whose grand maximal function $\mathcal Mf$ satisfies $$\int_{\mathbb R^n} \frac {|\mathcal M f(x)|}{\log(e+|x|) +\log (e+ |\mathcal Mf(x)|)}dx <\infty.$$ The two bilinear operators can be defined in terms of paraproducts. As a consequence, we find an endpoint estimate involving the space $\H^{\log}(\bR^n)$ for the $\div$-$\curl$ lemma.
We extend to multilinear Hankel operators a result on the regularity of truncations of Hankel ope... more We extend to multilinear Hankel operators a result on the regularity of truncations of Hankel operators. We prove and use a continuity property on the bilinear Hilbert transforms on product of Lipschitz spaces and Hardy spaces. In this note, we want to extend to multilinear Hankel operators a result obtained by [BB] on the boundedness properties of truncation acting on bounded Hankel infinite matrices. Let us first recall this result. A matrix B = (bmn)m,n∈N is called of Hankel type if bmn = bm+n for some sequence b ∈ l(N). We can consider B as an operator acting on l(N). In this case, we write B = Hb. If we identify l (N) with the complex Hardy space H(D) of the unit disc then Hb can be realized as the integral operator acting on f ∈ H(D) by Hbf(z) = 1 2π ∫ T b(ζ)f(ζ) 1 − ζz dσ(ζ). In other words, Hbf = C(bg) where C denotes the Cauchy integral, g(ζ) = f(ζ) and b is the Symbol of the Hankel operator b(ζ) = ∑∞ k=0 bkζ . If f(z) = ∑ n∈N anz , one has Hbf(z) = ∑ m∈N( ∑ n∈N anbm+n)z . ...
Page 1. arXiv:1103.1822v1 [math.CA] 9 Mar 2011 PARAPRODUCTS AND PRODUCTS OF FUNCTIONS IN BMO(Rn) ... more Page 1. arXiv:1103.1822v1 [math.CA] 9 Mar 2011 PARAPRODUCTS AND PRODUCTS OF FUNCTIONS IN BMO(Rn) AND H1(Rn) THROUGH WAVELETS ALINE BONAMI, SANDRINE GRELLIER, AND LUONG DANG KY Abstract. ...
Page 1. arXiv:1103.1822v1 [math.CA] 9 Mar 2011 PARAPRODUCTS AND PRODUCTS OF FUNCTIONS IN BMO(Rn) ... more Page 1. arXiv:1103.1822v1 [math.CA] 9 Mar 2011 PARAPRODUCTS AND PRODUCTS OF FUNCTIONS IN BMO(Rn) AND H1(Rn) THROUGH WAVELETS ALINE BONAMI, SANDRINE GRELLIER, AND LUONG DANG KY Abstract. ...
... E-mail addresses: Aline.Bonami@univ-orleans.fr (A. Bonami), Sandrine.Grellier@univ-orleans. f... more ... E-mail addresses: Aline.Bonami@univ-orleans.fr (A. Bonami), Sandrine.Grellier@univ-orleans. fr (S. Grellier), bs@maths.gla.ac.uk (BF Sehba ... bounded on H 1 (B n ) and prove that there exists a constant C&gt;0 so that, for any ball B of radius on S n , 1 (B) integraldisplay B |bb B |d ...
This monograph is an expanded version of the preprint arXiv:1402.1716 or hal-00943396v1. It is de... more This monograph is an expanded version of the preprint arXiv:1402.1716 or hal-00943396v1. It is devoted to the dynamics on Sobolev spaces of the cubic Szego equation on the circle ${\mathbb S} ^1$, $$ i\partial _t u=\Pi (\vert u\vert ^2u)\ .$$ Here $\Pi $ denotes the orthogonal projector from $L^2({\mathbb S} ^1)$ onto the subspace $L^2_+({\mathbb S} ^1)$ of functions with nonnegative Fourier modes. We construct a nonlinear Fourier transformation on $H^{1/2}({\mathbb S} ^1)\cap L^2_+({\mathbb S} ^1)$ allowing to describe explicitly the solutions of this equation with data in $H^{1/2}({\mathbb S} ^1)\cap L^2_+({\mathbb S} ^1)$. This explicit description implies almost-periodicity of every solution in $H^{\frac 12}_+$. Furthermore, it allows to display the following turbulence phenomenon. For a dense $G_\delta $ subset of initial data in $C^\infty ({\mathbb S} ^1)\cap L^2_+({\mathbb S} ^1)$, the solutions tend to infinity in $H^s$ for every $s>\frac 12$ with super--polynomial growth...
ABSTRACT We prove that the uniform Carleson condition is sufficient for an analytic variety of a ... more ABSTRACT We prove that the uniform Carleson condition is sufficient for an analytic variety of a strict finite type convex domain in C n to be difined by an holomorphic function in some Hardy space Hp . This extends a result of Varopoulos for the case of strictly pseudoconvex domains.
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