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105
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 61 (15 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
The boundedness of classical operators on variable Lp spaces
"... Abstract. We show that many classical operators in harmonic analysis —such as maximal operators, singular integrals, commutators and fractional integrals— are bounded on the variable Lebesgue space Lp(·) whenever the HardyLittlewood maximal operator is bounded on Lp(·). Further, we show that such o ..."
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Cited by 41 (2 self)
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Abstract. We show that many classical operators in harmonic analysis —such as maximal operators, singular integrals, commutators and fractional integrals— are bounded on the variable Lebesgue space Lp(·) whenever the HardyLittlewood maximal operator is bounded on Lp(·). Further, we show that such operators satisfy vectorvalued inequalities. We do so by applying the theory of weighted norm inequalities and extrapolation. As applications we prove the CalderónZygmund inequality for solutions of4u = f in variable Lebesgue spaces, and prove the Calderón extension theorem for variable Sobolev spaces.
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 33 (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.
Entropy and approximation numbers of embeddings of function spaces with Muckenhoupt weights
 I. Rev. Mat. Complut
"... We study compact embeddings for weighted spaces of Besov and TriebelLizorkin type where the weight belongs to some Muckenhoupt Ap class. For weights of purely polynomial growth, both near some singular point and at infinity, we obtain sharp asymptotic estimates for the entropy numbers and approxima ..."
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Cited by 21 (4 self)
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We study compact embeddings for weighted spaces of Besov and TriebelLizorkin type where the weight belongs to some Muckenhoupt Ap class. For weights of purely polynomial growth, both near some singular point and at infinity, we obtain sharp asymptotic estimates for the entropy numbers and approximation numbers of this embedding. The main tool is a discretization in terms of wavelet bases. Key words: wavelet bases, Muckenhoupt weighted function spaces, compact embeddings, entropy numbers, approximation numbers.
New Hardy spaces of MusielakOrlicz type and boundedness of sublinear operators
 Integral Equations Operator Theory
"... Abstract. We introduce a new class of Hardy spaces Hϕ(·,·)(Rn), called Hardy spaces of MusielakOrlicz type, which generalize the HardyOrlicz spaces of Janson and the weighted Hardy spaces of GarćıaCuerva, Strömberg, and Torchinsky. Here, ϕ: Rn × [0,∞) → [0,∞) is a function such that ϕ(x, ·) is ..."
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Cited by 20 (7 self)
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Abstract. We introduce a new class of Hardy spaces Hϕ(·,·)(Rn), called Hardy spaces of MusielakOrlicz type, which generalize the HardyOrlicz spaces of Janson and the weighted Hardy spaces of GarćıaCuerva, Strömberg, and Torchinsky. Here, ϕ: Rn × [0,∞) → [0,∞) is a function such that ϕ(x, ·) is an Orlicz function and ϕ(·, t) is a MuckenhouptA ∞ weight. A function f belongs to Hϕ(·,·)(Rn) if and only if its maximal function f ∗ is so that x 7 → ϕ(x, f∗(x)) is integrable. Such a space arises naturally for instance in the description of the product of functions in H1(Rn) and BMO(Rn) respectively (see [6]). We characterize these spaces via the grand maximal function and establish their atomic decomposition. We characterize also their dual spaces. The class of pointwise multipliers for BMO(Rn) characterized by Nakai and Yabuta can be seen as the dual of L1(Rn)+H log(Rn) where H log(Rn) is the Hardy space of MusielakOrlicz type related to the MusielakOrlicz function θ(x, t) = t log(e+ x) + log(e+ t) Furthermore, under additional assumption on ϕ(·, ·) we prove that if T is a sublinear operator and maps all atoms into uniformly bounded elements of a quasiBanach space B, then T uniquely extends to a bounded sublinear operator from Hϕ(·,·)(Rn) to B. These results are new even for the classical HardyOrlicz spaces on Rn. 1.
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 s ..."
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Cited by 18 (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.
Localized Hardy spaces H1 related to admissible functions on RDspaces and applications to Schrödinger operators
 Trans. Amer. Math. Soc
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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 13 (3 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.