Results 1  10
of
11
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 21 (7 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 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 8 (0 self)
 Add to MetaCart
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
Singular integrals and elliptic boundary problems on regular SemmesKenigToro domains
, 2008
"... We develop the theory of layer potentials and related singular integral operators as a tool to study a variety of elliptic boundary problems on a family of domains introduced by Semmes [101]–[102] and Kenig and Toro [64]–[66], which we call regular SemmesKenigToro domains. This extends the classic ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
We develop the theory of layer potentials and related singular integral operators as a tool to study a variety of elliptic boundary problems on a family of domains introduced by Semmes [101]–[102] and Kenig and Toro [64]–[66], which we call regular SemmesKenigToro domains. This extends the classic work of Fabes, Jodeit, and Rivière in several ways. For one, the class of domains considered contains the class of VMO1 domains, which in turn contains the class of C 1 domains. In addition we study not only the Dirichlet and Neumann boundary problems, but also a variety of others. Furthermore, we treat not only constant coefficient operators, but also operators with variable coefficients, including operators on manifolds. Contents 1.
WEIGHTED NORM INEQUALITIES FOR FRACTIONAL OPERATORS
, 2007
"... Abstract. We prove weighted norm inequalities for fractional powers of elliptic operators together with their commutators with BMO functions, encompassing what is known for the classical Riesz potentials and elliptic operators with Gaussian domination by the classical heat operator. The method relie ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Abstract. We prove weighted norm inequalities for fractional powers of elliptic operators together with their commutators with BMO functions, encompassing what is known for the classical Riesz potentials and elliptic operators with Gaussian domination by the classical heat operator. The method relies upon a goodλ method that does not use any size or smoothness estimates for the kernels. 1.
COMPARISON OF THE CLASSICAL BMO WITH THE BMO SPACES ASSOCIATED WITH OPERATORS AND APPLICATIONS
, 2006
"... Abstract. Let L be a generator of a semigroup satisfying the Gaussian upper bounds. In this paper, we study further a new BMOL space associated with L which was introduced recently by Duong and Yan. We discuss applications of the new BMOL spaces in the theory of singular integration such as BMOL est ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Abstract. Let L be a generator of a semigroup satisfying the Gaussian upper bounds. In this paper, we study further a new BMOL space associated with L which was introduced recently by Duong and Yan. We discuss applications of the new BMOL spaces in the theory of singular integration such as BMOL estimates and interpolation results for fractional powers, purely imaginary powers and spectral multipliers of self adjoint operators. We also demonstrate that the space BMOL might coincide with or might be essentially different from the classical BMO space. 1.
OLD AND NEW MORREY SPACES VIA HEAT KERNEL BOUNDS
, 2006
"... Abstract. Given p ∈ [1, ∞) and λ ∈ (0, n), we study Morrey space Lp,λ (Rn) of all locally integrable complexvalued functions f on Rn such that for every open Euclidean ball B ⊂ Rn with radius rB there are numbers C = C(f) (depending on f) and c = c(f, B) (relying upon f and B) satisfying r −λ ..."
Abstract
 Add to MetaCart
Abstract. Given p ∈ [1, ∞) and λ ∈ (0, n), we study Morrey space Lp,λ (Rn) of all locally integrable complexvalued functions f on Rn such that for every open Euclidean ball B ⊂ Rn with radius rB there are numbers C = C(f) (depending on f) and c = c(f, B) (relying upon f and B) satisfying r −λ
Riesz transform on manifolds and heat . . .
, 2004
"... 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 s ..."
Abstract
 Add to MetaCart
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
Sharp Function Inequality for Multilinear Commutator of Singular Integral Operators with NonSmooth Kernels
"... Abstract. In this paper, we establish a sharp function estimate for some multilinear commutator of the singular integral operators with nonsmooth kernels. As the application, we obtain the L p (1 < p < ∞) norm inequality for the multilinear commutator. ..."
Abstract
 Add to MetaCart
Abstract. In this paper, we establish a sharp function estimate for some multilinear commutator of the singular integral operators with nonsmooth kernels. As the application, we obtain the L p (1 < p < ∞) norm inequality for the multilinear commutator.
Multilinear Analysis on Metric Spaces
, 2012
"... The multilinear Calderón–Zygmund theory is developed in the setting of RDspaces, namely, spaces of homogeneous type equipped with measures satisfying a reverse doubling condition. The multipleweight multilinear Calderón–Zygmund theory in this context is also developed in this work. The bilinear T1 ..."
Abstract
 Add to MetaCart
The multilinear Calderón–Zygmund theory is developed in the setting of RDspaces, namely, spaces of homogeneous type equipped with measures satisfying a reverse doubling condition. The multipleweight multilinear Calderón–Zygmund theory in this context is also developed in this work. The bilinear T1theorems for Besov and Triebel–Lizorkin spaces in the full range of exponents are among the main results obtained. Multilinear vectorvalued T1 type theorems on Lebesgue spaces, Besov spaces, and Triebel–Lizorkin spaces are also proved. Applications include the boundedness of paraproducts and bilinear multiplier operators on products of Besov and Triebel–Lizorkin spaces.