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29
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 27 (10 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
Hardy spaces of differential forms on Riemannian manifolds
, 2006
"... Let M be a complete connected Riemannian manifold. Assuming that the Riemannian measure is doubling, we define Hardy spaces H p of differential forms on M and give various characterizations of them, including an atomic decomposition. As a consequence, we derive the H pboundedness for Riesz transfo ..."
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Cited by 15 (3 self)
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Let M be a complete connected Riemannian manifold. Assuming that the Riemannian measure is doubling, we define Hardy spaces H p of differential forms on M and give various characterizations of them, including an atomic decomposition. As a consequence, we derive the H pboundedness for Riesz transforms on M, generalizing previously known results. Further applications, in particular to H ∞ functional calculus and Hodge decomposition, are given.
Bounds of Riesz transforms on L p spaces for second order elliptic operators
 Ann. Inst. Fourier
"... Abstract. Let L = −div(A(x)∇) be a second order elliptic operator with real, symmetric, bounded measurable coefficients on R n or on a bounded Lipschitz domain subject to Dirichlet boundary condition. For any fixed p> 2, a necessary and sufficient condition is obtained for the boundedness of the ..."
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Cited by 11 (1 self)
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Abstract. Let L = −div(A(x)∇) be a second order elliptic operator with real, symmetric, bounded measurable coefficients on R n or on a bounded Lipschitz domain subject to Dirichlet boundary condition. For any fixed p> 2, a necessary and sufficient condition is obtained for the boundedness of the Riesz transform ∇(L) −1/2 on the L p space. As an application, for 1 < p < 3 + ε, we establish the L p boundedness of Riesz transforms on Lipschitz domains for operators with V MO coefficients. The range of p is sharp. The closely related boundedness of ∇(L) −1/2 on weighted L 2 spaces is also studied. 1.
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 10 (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 6 (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)
MAXIMAL PARABOLIC REGULARITY FOR DIVERGENCE OPERATORS INCLUDING MIXED BOUNDARY CONDITIONS
, 903
"... Abstract. We show that elliptic second order operators A of divergence type fulfill maximal parabolic regularity on distribution spaces, even if the underlying domain is highly nonsmooth, the coefficients of A are discontinuous and A is complemented with mixed boundary conditions. Applications to q ..."
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Cited by 5 (1 self)
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Abstract. We show that elliptic second order operators A of divergence type fulfill maximal parabolic regularity on distribution spaces, even if the underlying domain is highly nonsmooth, the coefficients of A are discontinuous and A is complemented with mixed boundary conditions. Applications to quasilinear parabolic equations with nonsmooth data are presented. 1.
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 5 (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
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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Cited by 5 (3 self)
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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.
The L p boundary value problems on Lipschitz domains
 Adv. Math
"... Abstract. Let Ω be a bounded Lipschitz domain in Rn. We develop a new approach to the invertibility on Lp (∂Ω) of the layer potentials associated with elliptic equations and systems in Ω. As a consequence, for n ≥ 4 and 2(n−1) − ε < p < 2 where ε> 0 depends on Ω, n+1 we obtain the solvabili ..."
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Cited by 4 (0 self)
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Abstract. Let Ω be a bounded Lipschitz domain in Rn. We develop a new approach to the invertibility on Lp (∂Ω) of the layer potentials associated with elliptic equations and systems in Ω. As a consequence, for n ≥ 4 and 2(n−1) − ε < p < 2 where ε> 0 depends on Ω, n+1 we obtain the solvability of the Lp Neumann type boundary value problems for second order elliptic systems. The analogous results for the biharmonic equation are also established.
A RELATIONSHIP BETWEEN THE DIRICHLET AND REGULARITY PROBLEMS FOR ELLIPTIC EQUATIONS
, 2006
"... Abstract. Let L = divA ∇ be a real, symmetric second order elliptic operator with bounded measurable coefficients. Consider the elliptic equation Lu = 0 in a bounded Lipschitz domain Ω of R n. We study the relationship between the solvability of the L p Dirichlet problem (D)p with boundary data in L ..."
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Cited by 4 (0 self)
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Abstract. Let L = divA ∇ be a real, symmetric second order elliptic operator with bounded measurable coefficients. Consider the elliptic equation Lu = 0 in a bounded Lipschitz domain Ω of R n. We study the relationship between the solvability of the L p Dirichlet problem (D)p with boundary data in L p (∂Ω) and that of the L q regularity problem (R)q with boundary data in W 1,q (∂Ω), where 1 < p, q < ∞. It is known that the solvability of (R)p implies that of (D) p ′. In this note we show that if (D) p ′ is solvable, then either (R)p is solvable or (R)q is not solvable for any 1 < q < ∞. 1.