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32
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 22 (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
Localized Hardy spaces H 1 related to admissible functions on RDspaces and applications to Schrödinger operators
"... Abstract. Let X be an RDspace, which means that X is a space of homogenous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in X. In this paper, the authors first introduce the notion of admissible functions ρ and then develop a theory of lo ..."
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Cited by 8 (6 self)
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Abstract. Let X be an RDspace, which means that X is a space of homogenous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in X. In this paper, the authors first introduce the notion of admissible functions ρ and then develop a theory of localized Hardy spaces H1 ρ(X) associated with ρ, which includes several maximal function characterizations of H1 ρ (X), the relations between H1 ρ (X) and the classical Hardy space H1 (X) via constructing a kernel function related to ρ, the atomic decomposition characterization of H1 ρ(X), and (X) via a finite atomic the boundedness of certain localized singular integrals on H1 ρ decomposition characterization of some dense subspace of H1 ρ (X). This theory has a wide range of applications. Even when this theory is applied, respectively, to the Schrödinger operator or the degenerate Schrödinger operator on Rn, or the subLaplace Schrödinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups, some new results are also obtained. The Schrödinger operators considered here are associated with nonnegative potentials satisfying the reverse Hölder inequality. 1
The mixed problem in L p for some twodimensional Lipschitz domains
 Math. Ann
"... We consider the mixed problem, ⎪ ⎨ ∆u = 0 ..."
LittlewoodPaley theory and function spaces with . . .
, 1998
"... Characterizations via convolutions with smooth compactly supported kernels and other distinguished properties of the weighted BesovLipschitz and TriebelLizorkin spaces on IR n with weights that are locally in Ap but may grow or decrease exponentially at infinity are investigated. Squarefunct ..."
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Cited by 6 (1 self)
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Characterizations via convolutions with smooth compactly supported kernels and other distinguished properties of the weighted BesovLipschitz and TriebelLizorkin spaces on IR n with weights that are locally in Ap but may grow or decrease exponentially at infinity are investigated. Squarefunction characterizations of the weighted L p and Hardy spaces with the above class of weights are also obtained. A certain local variant of the Calder'on reproducing formula is constructed and widely used in the proofs.
TwoWeight Norm Inequalities for Maximal Operators and Fractional Integrals on NonHomogeneous Spaces
, 2000
"... Let µ be a nonnegative Borel measure on R . Fix a real number n, 0 ! n d, and assume that is "ndimensional" in the following sense: the measure of a cube is smaller than the length of its side raised to the nth power. Calder'onZygmund operators, Hardy and BMO spaces, and some other topic ..."
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Cited by 4 (1 self)
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Let µ be a nonnegative Borel measure on R . Fix a real number n, 0 ! n d, and assume that is "ndimensional" in the following sense: the measure of a cube is smaller than the length of its side raised to the nth power. Calder'onZygmund operators, Hardy and BMO spaces, and some other topics in Harmonic Analysis have been successfully handled in this setting recently, although the measure may be nondoubling. The aim of this paper is to study twoweight norm inequalities for radial fractional maximal functions associated to such . Namely, we characterize those pairs of weights for which these maximal operators satisfy strong and weak type inequalities. Sawyer and radial Muckenhoupt type conditions are respectively the solutions for these problems. Furthermore, if we strengthen Muckenhoupt conditions by adding a "powerbump" to the righthand side weight or even by introducing certain Orlicz norm, strong type inequalities can be achieved. As a consequence, twoweight norm inequalities for fractional integrals associated to are obtained. Finally, for the particular case of the HardyLittlewood radial maximal function, we show how, in contrast with the classical situation, radial Muckenhoupt weights may fail to satisfy a reverse Hölder's inequality and also strong type inequalities do not necessarily hold for them.
FREQUENCYSCALE FRAMES AND THE SOLUTION OF THE MEXICAN HAT PROBLEM
"... Abstract. We resolve a longstanding question on ..."
REAL INTERPOLATION OF SOBOLEV SPACES ASSOCIATED TO A Weight
, 2008
"... We study the interpolation property of Sobolev spaces of order 1 denoted by W 1 p,V, arising from Schrödinger operators with positive potential. We show that for 1 ≤ p1 < p < p2 < q0 with p> s0, W 1 p,V is a real interpolation space between W 1 p1,V and W 1 p2,V on some classes of manifolds and Li ..."
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Cited by 4 (2 self)
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We study the interpolation property of Sobolev spaces of order 1 denoted by W 1 p,V, arising from Schrödinger operators with positive potential. We show that for 1 ≤ p1 < p < p2 < q0 with p> s0, W 1 p,V is a real interpolation space between W 1 p1,V and W 1 p2,V on some classes of manifolds and Lie groups. The constants s0, q0 depend on our hypotheses.
Intrinsic Characterizations of Distribution Spaces on Domains
, 1998
"... We give characterizations of Besov and TriebelLizorkin spaces B s pq and F s pq in smooth domains\Omega ae R n via convolutions with compactly supported smooth kernels satisfying some moment conditions. The results for s 2 R, 0 ! p; q 1 are stated in terms of the mixed norm of a certain maxi ..."
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Cited by 2 (1 self)
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We give characterizations of Besov and TriebelLizorkin spaces B s pq and F s pq in smooth domains\Omega ae R n via convolutions with compactly supported smooth kernels satisfying some moment conditions. The results for s 2 R, 0 ! p; q 1 are stated in terms of the mixed norm of a certain maximal function of a distribution. For s 2 R, 1 p 1, 0 ! q 1 characterizations without use of maximal functions are also obtained. 1 Introduction The two scales of Besov and TriebelLizorkin spaces B s pq (R n ) and F s pq (R n ), s 2 R; 0 ! p; q 1, are wellknown scales of spaces of tempered distributions on R n , covering classical HolderZygmund spaces, fractional Sobolev spaces, local Hardy spaces and their duals. After being introduced in the 60s70s in the pioneering papers by O. V. Besov [Bes1,2] (B s pq spaces, s ? 0, 1 p; q 1), M. H. Taibleson [Tai] (B s pq spaces, s 2 R, 1 p; q 1), P. I. Lizorkin [Liz1,2] (F s pq spaces, s ? 0, 1 ! p; q ! 1), H. Triebel [Tri1...
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 ..."
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Cited by 2 (1 self)
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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.