Results 1  10
of
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 ..."
Abstract

Cited by 61 (15 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 59 (12 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 47 (3 self)
 Add to MetaCart
(Show Context)
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
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) ..."
Abstract

Cited by 23 (12 self)
 Add to MetaCart
(Show Context)
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
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 ..."
Abstract

Cited by 15 (5 self)
 Add to MetaCart
(Show Context)
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.
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) ..."
Abstract

Cited by 13 (5 self)
 Add to MetaCart
(Show Context)
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.
, 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 ..."
Abstract

Cited by 12 (6 self)
 Add to MetaCart
(Show Context)
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.
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 ..."
Abstract

Cited by 11 (1 self)
 Add to MetaCart
(Show Context)
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.
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 ..."
Abstract

Cited by 10 (0 self)
 Add to MetaCart
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.
CORRECTION TO “HARDY AND BMO SPACES ASSOCIATED TO DIVERGENCE FORM ELLIPTIC OPERATORS”
, 907
"... We present here a correction to an error in our paper [5]. We are grateful to Dachun Yang for bringing the error to our attention, and we thank Alan McIntosh for helpful discussions which have led to this correction of the error. In particular, our approach here follows that of Auscher, McIntosh and ..."
Abstract

Cited by 9 (1 self)
 Add to MetaCart
(Show Context)
We present here a correction to an error in our paper [5]. We are grateful to Dachun Yang for bringing the error to our attention, and we thank Alan McIntosh for helpful discussions which have led to this correction of the error. In particular, our approach here follows that of Auscher, McIntosh and Russ [1], as explained to us by McIntosh. In [5] we develop a theory of H1 (Hardy type) and BMO spaces adapted to a second order, divergence form elliptic (aka accretive) operator L inR n, with complex, L ∞ coefficents. This had been done previously in work of Duong and Yan [2, 3], assuming a pointwise Gaussian bound on the heat kernel associated to L, which need not hold for arbitrary operators of the type that we consider. The point of our work, then, was to develop a theory analogous to that of [2, 3], in the absence of the Gaussian assumption. In particular we establish the equivalence of several different H1 type spaces, based on membership in L1 of various square functions and nontangential maximal functions adapted to L (if L is the Laplacian, then this theory reduces to that of the classical Hardy and BMO spaces). Among these (and of central importance) is the “square function Hardy space ” H1 S, defined as the completion of the set h with respect to the norm where S h f (x):= { f ∈ L 2 (R n) : S h f ∈ L 1 (R n)}, ‖ f ‖ H 1 S h {(y,t):x−y<t}:=‖S h f‖1,