The sharp bound for the Hilbert transform on weighted Lebesgue spaces in terms of the classical A p characteristic

Petermichl, Stefanie

doi:10.1353/ajm.2007.0036

Cited by 153 publications

(51 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(45)] with implications to Beltrami equations, the case of the Beurling-Ahlfors transform B ∈ L (L 2 (C)) was first settled by Petermichl and Volberg [29], and with an alternative proof by Dragičević and Volberg [7]. Petermichl also obtained the sharp bounds for the Hilbert transform H ∈ L (L 2 (R)) [27] and then for the Riesz transforms R i ∈ L (L 2 (R N )) in arbitrary dimension N ∈ Z + [28]. All these results relied on ad hoc representations based on specific symmetries of the operators in question and on Bellman function arguments tailor-made for each particular situation.…”

Section: Introductionmentioning

confidence: 99%

The sharp weighted bound for general Calderón--Zygmund operators

2012

View full text Add to dashboard Cite

For a general Calderón-Zygmund operator T on R N , it is shown thatfor all Muckenhoupt weights w ∈ A2. This optimal estimate was known as the A2 conjecture. A recent result of Pérez-Treil-Volberg reduced the problem to a testing condition on indicator functions, which is verified in this paper. The proof consists of the following elements: (i) a variant of the NazarovTreil-Volberg method of random dyadic systems with just one random system and completely without "bad" parts; (ii) a resulting representation of a general Calderón-Zygmund operator as an average of "dyadic shifts;" and (iii) improvements of the Lacey-Petermichl-Reguera estimates for these dyadic shifts, which allow summing up the series in the obtained representation.

show abstract

Section: Introductionmentioning

confidence: 99%

The sharp weighted bound for general Calderón--Zygmund operators

2012

View full text Add to dashboard Cite

show abstract

“…We leave the proof as an exercise in convex analysis for the reader. Similar calculations have been done in detail in other papers, see for example [NTV2], [HTV], [W1], [Pet1]. A similar argument will be used in Section 6.4 and in Section 7.…”

Section: Existence Of Bellman Functionmentioning

confidence: 82%

“…In this paper we are interested in studying the dependence of the L p -bounds of T t w on the corresponding characteristic of the weight w, and sometimes also on the dyadic doubling constant of w. We will concentrate on the cases p = 2, and t = 1, 1/2, −1/2. This work was inspired by a string of papers that have appeared in the wake of this milenium, where sharp linear bounds in L 2 (w) for classical operators (square function, martingale transform, Beurling transform, Hilbert transform, and Riesz transforms) on weighted Lebesgue spaces have been obtained, see [HTV], [W1], [W2], [PetW], [PetV], [D], [DV], [Pet1], [Pet2]. Very recently the same linear bound has been proved to hold also for the dyadic paraproduct, see [Be].…”

Section: Introductionmentioning

confidence: 99%

Haar multipliers meet Bellman functions

Pereyra¹

2009

Rev. Mat. Iberoam.

View full text Add to dashboard Cite

Using Bellman function techniques, we obtain the optimal dependence of the operator norms in L 2 (R) of the Haar multipliers T t w on the correspondingcharacteristic of the weight w, for t = 1, ±1/2. These results can be viewed as particular cases of estimates on homogeneous spaces L 2 (vdσ), for σ a doubling positive measure and v ∈ A d 2 (dσ), of the weighted dyadic square function S d σ . We show that the operator norms of such square functions in L 2 (vdσ) are bounded by a linear function of the A d 2 (dσ) characteristic of the weight v, where the constant depends only on the doubling constant of the measure σ. We also show an inverse estimate for S d σ . Both results are known when dσ = dx. We deduce both estimates from an estimate for the Haar multiplier (2 . The estimate for the Haar multiplier adapted to the σ measure, (T σ v ) 1/2 , is proved using Bellman functions. These estimates are sharp in the sense that the rates cannot be improved and be expected to hold for all σ, since the particular case dσ = dx, v = w, correspond to the estimates for the Haar multipliers T 1/2 w proven to be sharp.

show abstract

“…Since then, the technique has proved to be extremely efficient in various contexts; consult e.g. [15,17,18,19,20] and the references therein. The rest of this paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%