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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
Functional calculus of Dirac operators and complex perturbations of Neumann and Dirichlet problems
 Journal of Functional Analysis
"... Abstract. We prove that the Neumann, Dirichlet and regularity problems for divergence form elliptic equations in the half space are well posed in L2 for small complex L ∞ perturbations of a coefficient matrix which is either real symmetric, of block form or constant. All matrices are assumed to be i ..."
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Cited by 7 (6 self)
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Abstract. We prove that the Neumann, Dirichlet and regularity problems for divergence form elliptic equations in the half space are well posed in L2 for small complex L ∞ perturbations of a coefficient matrix which is either real symmetric, of block form or constant. All matrices are assumed to be independent of the transversal coordinate. We solve the Neumann, Dirichlet and regularity problems through a new boundary operator method which makes use of operators in the functional calculus of an underlaying first order Dirac type operator. We establish quadratic estimates for this Dirac operator, which implies that the associated Hardy projection operators are bounded and depend continuously on the coefficient matrix. We also prove that certain transmission problems for kforms are
HARDY SPACES ASSOCIATED TO NONNEGATIVE SELFADJOINT OPERATORS SATISFYING DAVIESGAFFNEY ESTIMATES
"... Abstract. Let X be a metric space with doubling measure, and L be a nonnegative, selfadjoint operator satisfying DaviesGaffney bounds on L 2 (X). In this article we develop a theory of Hardy and BMO spaces associated to L, including an atomic (or molecular) decomposition, square function characte ..."
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Cited by 7 (0 self)
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Abstract. Let X be a metric space with doubling measure, and L be a nonnegative, selfadjoint operator satisfying DaviesGaffney bounds on L 2 (X). In this article we develop a theory of Hardy and BMO spaces associated to L, including an atomic (or molecular) decomposition, square function characterization, duality of Hardy and BMO spaces. Further specializing to the case that L is a Schrödinger operator on R n with a nonnegative, locally integrable potential, we establish addition characterizations of such Hardy space in terms of maximal functions. Finally, (X) for p> 1, which may or may not coincide with the space L p (X), and show that they interpolate with H 1 L(X) spaces by the complex method. we define Hardy spaces H p L The authors gratefully acknowledge support from NSF as follows: S. Hofmann (DMS
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)
New OrliczHardy spaces associated with Divergence Form Elliptic Operators
"... Abstract. Let L be the divergence form elliptic operator with complex bounded measurable coefficients, ω the positive concave function on (0, ∞) of strictly critical lower type pω ∈ (0, 1] and ρ(t) = t −1 /ω −1 (t −1) for t ∈ (0, ∞). In this paper, the authors study the OrliczHardy space Hω,L(R n) ..."
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Cited by 4 (3 self)
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Abstract. Let L be the divergence form elliptic operator with complex bounded measurable coefficients, ω the positive concave function on (0, ∞) of strictly critical lower type pω ∈ (0, 1] and ρ(t) = t −1 /ω −1 (t −1) for t ∈ (0, ∞). In this paper, the authors study the OrliczHardy space Hω,L(R n) and its dual space BMOρ,L ∗(Rn), where L ∗ denotes the adjoint operator of L in L 2 (R n). Several characterizations of Hω,L(R n), including the molecular characterization, the Lusinarea function characterization and the maximal function characterization, are established. The ρCarleson measure characterization and the JohnNirenberg inequality for the space BMOρ,L(R n) are also given. As applications, the authors show that the Riesz transform ∇L −1/2 and the LittlewoodPaley gfunction gL map Hω,L(R n) continuously into L(ω). DeL (Rn) into the clas, 1] and the corresponding fractional integral L (Rn) continuously into H q L (Rn), where 0 < p < q ≤ 1 and n/p − n/q = 2γ. All these results are new even when ω(t) = tp for all t ∈ (0, ∞) and p ∈ (0, 1). note Hω,L(Rn) by H p L (Rn) when p ∈ (0, 1] and ω(t) = tp for all t ∈ (0, ∞). The authors further show that the Riesz transform ∇L−1/2 maps H p sical Hardy space Hp (Rn) for p ∈ ( n n+1 L−γ for certain γ> 0 maps H p 1
Lp selfimprovement of generalized Poincaré inequalities in spaces of homogeneous type
 J. Funct. Anal
"... Abstract. In this paper we study selfimproving properties in the scale of Lebesgue spaces of generalized Poincaré inequalities in spaces of homogeneous type. In contrast with the classical situation, the oscillations involve approximation of the identities or semigroups whose kernels decay fast eno ..."
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Cited by 1 (0 self)
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Abstract. In this paper we study selfimproving properties in the scale of Lebesgue spaces of generalized Poincaré inequalities in spaces of homogeneous type. In contrast with the classical situation, the oscillations involve approximation of the identities or semigroups whose kernels decay fast enough and the resulting estimates take into account their lack of localization. The techniques used do not involve any classical Poincaré or SobolevPoincaré inequalities and therefore they can be used in general settings where these estimates do not hold or are unknown. We apply our results to the case of Riemannian manifolds with doubling volume form and assuming Gaussian upper bounds for the heat kernel of the semigroup e −t ∆ with ∆ being the LaplaceBeltrami operator. We obtain generalized Poincaré inequalities with oscillations that involve the semigroup e−t ∆ and with right hand sides containing either ∇ or ∆1/2. 1.
Abstract
, 2004
"... On considère la classe des variétés riemanniennes complètes non compactes dont le noyau de la chaleur satisfait une estimation supérieure et inférieure gaussienne. On montre que la transformée de Riesz y est bornée sur L p, pour un intervalle ouvert de p audessus de 2, si et seulement si le gradien ..."
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On considère la classe des variétés riemanniennes complètes non compactes dont le noyau de la chaleur satisfait une estimation supérieure et inférieure gaussienne. On montre que la transformée de Riesz y est bornée sur L p, pour un intervalle ouvert de p audessus de 2, si et seulement si le gradient du noyau de la chaleur satisfait une certaine estimation L p pour le même intervalle d’exposants p. One considers the class of complete noncompact Riemannian manifolds whose heat kernel satisfies Gaussian estimates from above and below. One shows that the Riesz transform is L p bounded on such a manifold, for p ranging in an open interval above 2, if and only if the gradient of the heat kernel satisfies a certain L p estimate in the same
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
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
Generalized Vanishing Mean Oscillation Spaces Associated with Divergence Form Elliptic Operators
, 907
"... Abstract. Let L be a divergence form elliptic operator with complex bounded measurable coefficients, ω the positive concave function on (0, ∞) of strictly critical lower type pω ∈ (0, 1] and ρ(t) = t −1 /ω −1 (t −1) for t ∈ (0, ∞). In this paper, the authors introduce the generalized VMO spaces VMO ..."
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Abstract. Let L be a divergence form elliptic operator with complex bounded measurable coefficients, ω the positive concave function on (0, ∞) of strictly critical lower type pω ∈ (0, 1] and ρ(t) = t −1 /ω −1 (t −1) for t ∈ (0, ∞). In this paper, the authors introduce the generalized VMO spaces VMOρ,L(R n) associated with L, and characterize them via tent spaces. As applications, the authors show that (VMOρ,L(R n)) ∗ = Bω,L ∗(Rn), where L ∗ denotes the adjoint operator of L in L 2 (R n) and Bω,L ∗(Rn) the Banach completion of the OrliczHardy space Hω,L ∗(Rn). Notice that ω(t) = t p for all t ∈ (0, ∞) and p ∈ (0, 1] is a typical example of positive concave functions satisfying the assumptions. In particular, when p = 1, then ρ(t) ≡ 1 and (VMO1,L(R n)) ∗ = H 1 L ∗(Rn), where H 1 L ∗(Rn) was the Hardy space introduced by Hofmann and Mayboroda. 1