BCR algorithm and the T(b) Theorem

Auscher, Pascal; Yang, Qi

doi:10.5565/publmat_53109_08

Cited by 25 publications

(30 citation statements)

References 11 publications

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“…However, many technical assumptions appear. Furthermore, they also establish a direct proof (in the sense that it is not a reduction to the perfect dyadic case like [AY09]) of the case 1/p + 1/q ≤ 1.…”

Section: Introductionmentioning

confidence: 99%

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On general local $Tb$ theorems

Hytönen

Martikainen

2012

Trans. Amer. Math. Soc.

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Abstract. In this paper, local T b theorems are studied both in the doubling and non-doubling situation. We prove a local T b theorem for the class of upper doubling measures. With such general measures, scale invariant testing conditions are required (L ∞ or BMO). In the case of doubling measures, we also modify the general non-homogeneous method of proof to yield a new proof of the local T b theorem with L 2 type testing conditions.

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Section: Introductionmentioning

confidence: 99%

“…For doubling measures, one can also consider more general L p type testing conditions introduced by Auscher, Hofmann, Muscalu, Tao and Thiele [AHM02], and further studied by Hofmann [Hof07], Auscher and Yang [AY09] and Tan and Yan [TY09]. The most general assumption used in these papers is of the form that…”

Section: Introductionmentioning

confidence: 99%

On general local $Tb$ theorems

Hytönen

Martikainen

2012

Trans. Amer. Math. Soc.

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“…holds for every f ∈ L 2 (|ν|). Note that it must be that µ(G) > ε 0 µ(B), because otherwise (3.5) would be contradicted by the assumption (4).…”

Section: Then There Exists a Setmentioning

confidence: 99%

Non-homogeneous Square Functions on General Sets: Suppression and Big Pieces Methods

2017

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ABSTRACT. We aim to showcase the wide applicability and power of the big pieces and suppression methods in the theory of local T b theorems. The setting is new: we consider conical square functions with cones x ∈ R n \ E : |x − y| < 2 dist(x, E) , y ∈ E, defined on general closed subsets E ⊂ R n supporting a non-homogeneous measure µ. We obtain boundedness criteria in this generality in terms of weak type testing of measures on regular balls B ⊂ E, which are doubling and of small boundary. Due to the general set E we use metric space methods. Therefore, we also demonstrate the recent techniques from the metric space point of view, and show that they yield the most general known local T b theorems even with assumptions formulated using balls rather than the abstract dyadic metric cubes.

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“…The perfect case is of course very special, still the argument in [1] has been influential, although the task of lifting the proof therein to the continuous case has not proven to be easy. Auscher and Yang [4] succeeded in extending the Theorem above to the continuous case, with the duality assumption on p 1 and p 2 but the argument is an indirect reduction to the perfect case. This is less desirable, due to the interest in local Tb theorems more general settings, such as the setting of homogeneous spaces, as in Auscher and Routin [3].…”

Section: Theorem For Fixed Constants a And T Loc This Holds Suppomentioning

confidence: 99%

The perfect local $Tb$ Theorem and twisted Martingale transforms

Lacey

Vähäkangas

2014

Proc. Amer. Math. Soc.

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A local Tb Theorem provides a flexible framework for proving the boundedness of a Calderón-Zygmund operator T . One needs only boundedness of the operator T on systems of locally pseudo-accretive functions {b Q }, indexed by cubes. We give a new proof of this Theorem in the setting of perfect (dyadic) models of Calderón-Zygmund operators, imposing integrability conditions on the b Q functions that are the weakest possible. The proof is a simple direct argument, based upon an inequality for transforms of so-called twisted martingale differences, which has been noted by Auscher-Routin.

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BCR algorithm and the T(b) Theorem

Cited by 25 publications

References 11 publications

On general local $Tb$ theorems

On general local $Tb$ theorems

Non-homogeneous Square Functions on General Sets: Suppression and Big Pieces Methods

The perfect local $Tb$ Theorem and twisted Martingale transforms

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