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86
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
Planar earthmover is not in l1
 In 47th Symposium on Foundations of Computer Science (FOCS
, 2006
"... We show that any L1 embedding of the transportation cost (a.k.a. Earthmover) metric on probability measures supported on the grid {0, 1,..., n} 2 ⊆ R 2 incurs distortion Ω � � log n �. We also use Fourier analytic techniques to construct a simple L1 embedding of this space which has distortion O(lo ..."
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Cited by 12 (2 self)
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We show that any L1 embedding of the transportation cost (a.k.a. Earthmover) metric on probability measures supported on the grid {0, 1,..., n} 2 ⊆ R 2 incurs distortion Ω � � log n �. We also use Fourier analytic techniques to construct a simple L1 embedding of this space which has distortion O(log n). 1
Hardy spaces and divergence operators on strongly Lipschitz domain
 of R n , J. Funct. Anal
"... Let Ω be a strongly Lipschitz domain of R n. Consider an elliptic second order divergence operator L (including a boundary condition on ∂Ω) and define a Hardy space by imposing the nontangential maximal function of the extension of a function f via the Poisson semigroup for L to be in L 1. Under su ..."
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Cited by 10 (2 self)
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Let Ω be a strongly Lipschitz domain of R n. Consider an elliptic second order divergence operator L (including a boundary condition on ∂Ω) and define a Hardy space by imposing the nontangential maximal function of the extension of a function f via the Poisson semigroup for L to be in L 1. Under suitable assumptions on L, we identify this maximal Hardy space with atomic Hardy spaces, namely with H 1 (R n) if Ω = R n, H 1 r(Ω) under the Dirichlet boundary condition, and H1 z (Ω) under the Neumann boundary condition. In particular, we obtain a new proof of the atomic decomposition for H1 z (Ω). A version for local Hardy spaces is also given. We also present an overview of the theory of Hardy spaces and BMO spaces on Lipschitz domains with proofs.
ON THE H 1 –L 1 BOUNDEDNESS OF OPERATORS
"... Abstract. We prove that if q is in (1, ∞), Y is a Banach space, and T is a linear operator defined on the space of finite linear combinations of (1,q)atoms in Rn with the property that sup{‖Ta‖Y: a is a (1,q)atom} < ∞, then T admits a (unique) continuous extension to a bounded linear operator from ..."
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Cited by 9 (0 self)
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Abstract. We prove that if q is in (1, ∞), Y is a Banach space, and T is a linear operator defined on the space of finite linear combinations of (1,q)atoms in Rn with the property that sup{‖Ta‖Y: a is a (1,q)atom} < ∞, then T admits a (unique) continuous extension to a bounded linear operator from H1 (Rn)toY. We show that the same is true if we replace (1,q)atoms by continuous (1, ∞)atoms. This is known to be false for (1, ∞)atoms. 1.
Multilinear CalderonZygmund operators on Hardy spaces
 Collect. Math
"... Abstract. It is shown that multilinear CalderónZygmund operators are bounded on products of Hardy spaces. 1. ..."
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Cited by 9 (2 self)
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Abstract. It is shown that multilinear CalderónZygmund operators are bounded on products of Hardy spaces. 1.
Harmonic measure on locally flat domains
 Duke Math. J
, 1997
"... We will review work with Tatiana Toro yielding a characterization of those domains for which the harmonic measure has a density whose logarithm has vanishing mean oscillation. ..."
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Cited by 9 (3 self)
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We will review work with Tatiana Toro yielding a characterization of those domains for which the harmonic measure has a density whose logarithm has vanishing mean oscillation.
Interpolation between H p spaces and noncommutative generalizations II
 Revista Mat. Iberoamericana
, 1993
"... Abstract We give an elementary proof that the H p spaces over the unit disc (or the upper half plane) are the interpolation spaces for the real method of interpolation between H 1 and H ∞. This was originally proved by Peter Jones. The proof uses only the boundedness of the Hilbert transform and the ..."
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Cited by 6 (3 self)
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Abstract We give an elementary proof that the H p spaces over the unit disc (or the upper half plane) are the interpolation spaces for the real method of interpolation between H 1 and H ∞. This was originally proved by Peter Jones. The proof uses only the boundedness of the Hilbert transform and the classical factorisation of a function in H p as a product of two functions in H q and H r with 1/q + 1/r = 1/p. This proof extends without any real extra difficulty to the noncommutative setting and to several Banach space valued extensions of H p spaces. In particular, this proof easily extends to the couple H p0 (ℓq0), Hp1 (ℓq1), with 1 ≤ p0, p1, q0, q1 ≤ ∞. In that situation, we prove that the real interpolation spaces and the Kfunctional are induced ( up to equivalence of norms) by the same objects for the couple Lp0 (ℓq0), Lp1 (ℓq1). In an other direction, let us denote by Cp the space of all compact operators x on Hilbert space such that tr(x  p) < ∞. Let Tp be the subspace of all upper triangular matrices relative to the canonical basis. If p = ∞, Cp is just the space of all compact operators. Our proof allows us to show for instance that the space H p (Cp) (resp. Tp) is the interpolation space of parameter (1/p, p) between H 1 (C1) (resp. T1) and H ∞ (C∞) (resp. T∞). We also prove a similar result for the complex interpolation method. Moreover, extending a recent result of KaftalLarson and Weiss, we prove that the distance to the subspace of upper triangular matrices in C1 and C ∞ can be essentially realized simultaneously by the same element.
A noncommutative version of the JohnNirenberg theorem
"... Abstract. We prove a noncommutative version of the JohnNirenberg theorem for nontracial filtrations of von Neumann algebras. As an application, we obtain an analogue of the classical large deviation inequality for elements of the associated BMO space. 1. ..."
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Cited by 6 (1 self)
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Abstract. We prove a noncommutative version of the JohnNirenberg theorem for nontracial filtrations of von Neumann algebras. As an application, we obtain an analogue of the classical large deviation inequality for elements of the associated BMO space. 1.
An elementary approach to several results on the HardyLittlewood maximal operator
 Proc. Amer. Math. Soc. 136 (2008) no 8, 28292833. BOUNDS FOR GENERAL COMMUTATORS 15
"... Abstract. We give new elementary proofs of theorems due to B. Muckenhoupt, B. Jawerth, and S. Buckley. By means of our approach we answer a question raised by J. Orobitg and C. Pérez. 1. ..."
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Cited by 5 (0 self)
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Abstract. We give new elementary proofs of theorems due to B. Muckenhoupt, B. Jawerth, and S. Buckley. By means of our approach we answer a question raised by J. Orobitg and C. Pérez. 1.