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42
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 59 (14 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
TrujilloGonzález, R.: New maximal functions and multiple weights for the multilinear CalderónZygmund theory Adv
 Math
"... Abstract. A multi(sub)linear maximal operator that acts on the product of m Lebesgue spaces and is smaller that the mfold product of the HardyLittlewood maximal function is studied. The operator is used to obtain a precise control on multilinear singular integral operators of CalderónZygmund ty ..."
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Cited by 48 (4 self)
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Abstract. A multi(sub)linear maximal operator that acts on the product of m Lebesgue spaces and is smaller that the mfold product of the HardyLittlewood maximal function is studied. The operator is used to obtain a precise control on multilinear singular integral operators of CalderónZygmund type and to build a theory of weights adapted to the multilinear setting. A natural variant of the operator which is useful to control certain commutators of multilinear CalderónZygmund operators with BMO functions is then considered. The optimal range of strong type estimates, a sharp endpoint estimate, and weighted norm inequalities involving both the classical Muckenhoupt weights and the new multilinear ones are also obtained for the commutators. 1.
On the asymptotic behavior of large radial data for a focusing nonlinear Schrödinger equation
 preprint. TERENCE TAO Department of Mathematics, UCLA, Los Angeles CA 900951555 Email address: tao@math.ucla.edu
"... Abstract. We study the asymptotic behavior of large data radial solutions to the focusing Schrödinger equation iut+∆u = −u  2 u in R 3, assuming globally bounded H 1 (R 3) norm (i.e. no blowup in the energy space). We show that as t → ±∞, these solutions split into the sum of three terms: a radiat ..."
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Cited by 35 (14 self)
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Abstract. We study the asymptotic behavior of large data radial solutions to the focusing Schrödinger equation iut+∆u = −u  2 u in R 3, assuming globally bounded H 1 (R 3) norm (i.e. no blowup in the energy space). We show that as t → ±∞, these solutions split into the sum of three terms: a radiation term that evolves according to the linear Schrödinger equation, a smooth function localized near the origin, and an error that goes to zero in the ˙ H 1 (R 3) norm. Furthermore, the smooth function near the origin is either zero (in which case one has scattering to a free solution), or has mass and energy bounded strictly away from zero, and obeys an asymptotic Pohozaev identity. These results are consistent with the conjecture of soliton resolution. 1.
The planar Cantor sets of zero analytic capacity and the local T (b)Theorem
 J. Amer. Math. Soc
, 2003
"... Abstract. In this paper we obtain rather precise estimates for the analytic capacity of a big class of planar Cantors sets. In fact, we show that analytic capacity and positive analytic capacity are comparable for these sets. The main tool for the proof is an appropriate version of the T (b)Theorem ..."
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Cited by 27 (14 self)
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Abstract. In this paper we obtain rather precise estimates for the analytic capacity of a big class of planar Cantors sets. In fact, we show that analytic capacity and positive analytic capacity are comparable for these sets. The main tool for the proof is an appropriate version of the T (b)Theorem. 1.
Analytic capacity, CalderónZygmund operators, and rectifiability
, 1999
"... For K ⊂ C compact, we say that K has vanishing analytic capacity (or γ(K) = 0) when all bounded analytic functions on C\K are constant. We would like to characterize γ(K) = 0 geometrically. Easily, γ(K)> 0 when K has Hausdorff dimension larger than 1, and γ(K) = 0 when dim(K) < 1. Thus onl ..."
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Cited by 26 (0 self)
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For K ⊂ C compact, we say that K has vanishing analytic capacity (or γ(K) = 0) when all bounded analytic functions on C\K are constant. We would like to characterize γ(K) = 0 geometrically. Easily, γ(K)> 0 when K has Hausdorff dimension larger than 1, and γ(K) = 0 when dim(K) < 1. Thus only the case when dim(K) = 1 is interesting. So far there is no characterization of γ(K) = 0 in general, but the special case when the Hausdorff measure H1(K) is finite was recently settled. In this case, γ(K) = 0 if and only if K is unrectifiable (or Besicovitchirregular), i.e., if H1(K ∩ Γ) = 0 for all C1curves Γ, as was conjectured by Vitushkin. In the present text, we try to explain the structure of the proof of this result, and present the necessary techniques. These include the introduction to Menger curvature in this context (by M. Melnikov and coauthors), and the important use of geometric measure theory (results on quantitative rectifiability), but we insist most on the role of CalderónZygmund operators and T (b)Theorems. 1.
IMAGINARY POWERS OF LAPLACE OPERATORS
"... Abstract. We show that if L is a secondorder uniformly elliptic operator in divergence form on R d, then C1(1 + α) d/2 ≤ ‖Liα‖L1→L1,∞ ≤ C2(1 + α) d/2. We also prove that the upper bounds remain true for any operator with the finite speed propagation property. 1. Introduction. Assume that aij ∈ ..."
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Cited by 18 (4 self)
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Abstract. We show that if L is a secondorder uniformly elliptic operator in divergence form on R d, then C1(1 + α) d/2 ≤ ‖Liα‖L1→L1,∞ ≤ C2(1 + α) d/2. We also prove that the upper bounds remain true for any operator with the finite speed propagation property. 1. Introduction. Assume that aij ∈ C ∞ (R d), aij = aji for 1 ≤ i, j ≤ d and that κI ≤ (aij) ≤ τI for some positive constants κ and τ. We define a positive selfadjoint operator L on L 2 (R d) by the formula (1) L = − ∑ ∂iaij∂j. We refer readers to [8] for the precise definition and basic properties of L. In particular, L
A PROOF OF THE LOCAL Tb THEOREM FOR STANDARD CALDERÓNZYGMUND OPERATORS
, 705
"... Abstract. We give a proof of a socalled “local Tb ” Theorem for singular integrals whose kernels satisfy the standard CalderónZygmund conditions. The present theorem, which extends an earlier result of M. Christ [Ch], was proved in [AHMTT] for “perfect dyadic” CalderónZygmund operators. The proof ..."
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Cited by 13 (4 self)
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Abstract. We give a proof of a socalled “local Tb ” Theorem for singular integrals whose kernels satisfy the standard CalderónZygmund conditions. The present theorem, which extends an earlier result of M. Christ [Ch], was proved in [AHMTT] for “perfect dyadic” CalderónZygmund operators. The proof in [AHMTT] essentially carries over to the case considered here, with some technical adjustments. 1.
Weighted inequalities and vectorvalued CalderónZygmund operators on nonhomogeneous spaces
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Polynomial upper bounds for the instability of the nonlinear Schrödinger equation below the energy norm
 Commun. Pure Appl. Anal
"... Abstract. We continue the study (initiated in [18]) of the orbital stability of the ground state cylinder for focussing nonlinear Schrödinger equations in the H s (R n) norm for 1 − ε < s < 1, for small ε. In the L 2subcritical case we obtain a polynomial bound for the time required to move ..."
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Cited by 10 (3 self)
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Abstract. We continue the study (initiated in [18]) of the orbital stability of the ground state cylinder for focussing nonlinear Schrödinger equations in the H s (R n) norm for 1 − ε < s < 1, for small ε. In the L 2subcritical case we obtain a polynomial bound for the time required to move away from the ground state cylinder. If one is only in the H 1subcritical case then we cannot show this, but for defocussing equations we obtain global wellposedness and polynomial growth of H s norms for s sufficiently close to 1. 1.
Singular integrals on Sierpinski gaskets, Publ
 Mat
"... Abstract. We construct a class of singular integral operators associated with homogeneous CalderónZygmund standard kernels on ddimensional, d < 1, Sierpinski gaskets Ed. These operators are bounded in L2 (µd) and their principal values diverge µd almost everywhere, where µd is the natural (ddim ..."
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Cited by 9 (7 self)
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Abstract. We construct a class of singular integral operators associated with homogeneous CalderónZygmund standard kernels on ddimensional, d < 1, Sierpinski gaskets Ed. These operators are bounded in L2 (µd) and their principal values diverge µd almost everywhere, where µd is the natural (ddimensional) measure on Ed. 1.