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Weighted norm inequalities, offdiagonal estimates and elliptic operators, Part II: Offdiagonal estimates on spaces of homogeneous type
, 2005
"... Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the LaplaceBeltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincar ..."
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Cited by 23 (6 self)
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Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the LaplaceBeltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincaré inequalities. 1. Introduction and
Interpolation of Sobolev spaces, LittlewoodPaley inequalities and Riesz transforms on graphs
 PUBLICACIONS MATEMATIQUES
"... Abstract. Let Γ be a graph endowed with a reversible Markov kernel p, and P the associated operator, defined by Pf(x) = ∑ y p(x, y)f(y). Denote by ∇ the discrete gradient. We give necessary and/or sufficient conditions on Γ in order to compare ‖∇f‖ p and ∥ ∥ (I − P) ..."
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Cited by 5 (2 self)
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Abstract. Let Γ be a graph endowed with a reversible Markov kernel p, and P the associated operator, defined by Pf(x) = ∑ y p(x, y)f(y). Denote by ∇ the discrete gradient. We give necessary and/or sufficient conditions on Γ in order to compare ‖∇f‖ p and ∥ ∥ (I − P)
Use of abstract Hardy spaces, real interpolation and applications to bilinear operators. submitted, page available at http://fr.arxiv.org/abs/0809.4110
, 2008
"... ..."
Hardy and BMO spaces associated to divergence form elliptic operators
, 2007
"... 1 Introduction and statement of main results 2 ..."
Maximal inequalities for dual sobolev spaces W −1,p and applications to interpolation, submitted, available at http://fr.arxiv.org/abs/0812.3075
, 2008
"... We firstly describe a maximal inequality for dual Sobolev spaces W −1,p. This one corresponds to a “Sobolev version ” of usual properties of the HardyLittlewood maximal operator in Lebesgue spaces. Even in the euclidean space, this one seems to be new and we develop arguments in the general framewo ..."
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Cited by 1 (1 self)
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We firstly describe a maximal inequality for dual Sobolev spaces W −1,p. This one corresponds to a “Sobolev version ” of usual properties of the HardyLittlewood maximal operator in Lebesgue spaces. Even in the euclidean space, this one seems to be new and we develop arguments in the general framework of Riemannian manifold. Then we present an application to obtain interpolation results for Sobolev
unknown title
, 2004
"... On necessary and sufficient conditions for L pestimates of Riesz transforms associated to elliptic operators on R n and related estimates ..."
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On necessary and sufficient conditions for L pestimates of Riesz transforms associated to elliptic operators on R n and related estimates
Contents
, 901
"... The purpose of this work is to describe an abstract theory of HardySobolev spaces on doubling Riemannian manifolds via an atomic decomposition. We study the real interpolation of these spaces with Sobolev spaces and finally give applications to Riesz transforms. ..."
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The purpose of this work is to describe an abstract theory of HardySobolev spaces on doubling Riemannian manifolds via an atomic decomposition. We study the real interpolation of these spaces with Sobolev spaces and finally give applications to Riesz transforms.
(0.2) ‖∇g‖ ∞ ≤ Cα,
, 2008
"... We correct an inaccuracy in the proof of a result in [Aus1]. 2000 MSC: 42B20, 46E35 Key words: CalderónZygmund decomposition; Sobolev spaces. We recall the lemma. Lemma 0.1. Let n ≥ 1, 1 ≤ p ≤ ∞ and f ∈ D ′ (R n) be such that ‖∇f‖p < ∞. Let α> 0. Then, one can find a collection of cubes (Qi), func ..."
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We correct an inaccuracy in the proof of a result in [Aus1]. 2000 MSC: 42B20, 46E35 Key words: CalderónZygmund decomposition; Sobolev spaces. We recall the lemma. Lemma 0.1. Let n ≥ 1, 1 ≤ p ≤ ∞ and f ∈ D ′ (R n) be such that ‖∇f‖p < ∞. Let α> 0. Then, one can find a collection of cubes (Qi), functions g and bi such that (0.1) f = g + ∑ and the following properties hold:
1 Use of abstract Hardy spaces, Real interpolation and Applications to bilinear operators.
, 2008
"... This paper can be considered as the sequel of [6], where the authors have proposed an abstract construction of Hardy spaces H 1. They shew an interpolation result for these Hardy spaces with the Lebesgue spaces. Here we describe a more precise result using the real interpolation theory and we clarif ..."
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This paper can be considered as the sequel of [6], where the authors have proposed an abstract construction of Hardy spaces H 1. They shew an interpolation result for these Hardy spaces with the Lebesgue spaces. Here we describe a more precise result using the real interpolation theory and we clarify the use of Hardy spaces. Then with the help of the bilinear interpolation theory, we then give applications to study bilinear operators on Lebesgue spaces. These ideas permit us to study singular operators with singularities similar to those of bilinear CalderónZygmund operators in a far more abstract framework as in the euclidean case.
On the CalderónZygmund lemma for Sobolev functions
, 2008
"... We correct an inaccuracy in the proof of a result in [Aus1]. 2000 MSC: 42B20, 46E35 Key words: CalderónZygmund decomposition; Sobolev spaces. We recall the lemma. Lemma 0.1. Let n ≥ 1, 1 ≤ p ≤ ∞ and f ∈ D ′ (R n) be such that ‖∇f‖p < ∞. Let α> 0. Then, one can find a collection of cubes (Qi), func ..."
Abstract
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We correct an inaccuracy in the proof of a result in [Aus1]. 2000 MSC: 42B20, 46E35 Key words: CalderónZygmund decomposition; Sobolev spaces. We recall the lemma. Lemma 0.1. Let n ≥ 1, 1 ≤ p ≤ ∞ and f ∈ D ′ (R n) be such that ‖∇f‖p < ∞. Let α> 0. Then, one can find a collection of cubes (Qi), functions g and bi such that (0.1) f = g + ∑ and the following properties hold: (0.2) ‖∇g‖ ∞ ≤ Cα, i bi (0.3) bi ∈ W 1,p