Haar multipliers meet Bellman functions

Pereyra, María

doi:10.4171/rmi/584

Cited by 10 publications

(12 citation statements)

References 23 publications

(7 reference statements)

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“…The single variable version of the following proposition first appeared in [20]. In [14], one can also find a Bellman function proof of a similar result which can be extended to the doubling measure case. We are going to prove Theorem 1.4 only when p = 2 , and following the one-dimensional proof discovered by Beznosova [1].…”

Section: Embedding Theorems and Weighted Inequalities In R Nmentioning

confidence: 86%

Weighted inequalities for multivariable dyadic paraproducts

Chung¹

2011

Publicacions Matemàtiques

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Using Wilson's Haar basis in R n , which is different than the usual tensor product Haar functions, we define its associated dyadic paraproduct in R n . We can then extend "trivially" Beznosova's Bellman function proof of the linear bound in L 2 (w) with respect to [w] A2 for the 1-dimensional dyadic paraproduct.Here trivial means that each piece of the argument that had a Bellman function proof has an n-dimensional counterpart that holds with the same Bellman function. The lemma that allows for this painless extension we call the good Bellman function Lemma. Furthermore the argument allows to obtain dimensionless bounds in the anisotropic case.

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Section: Embedding Theorems and Weighted Inequalities In R Nmentioning

confidence: 86%

Weighted inequalities for multivariable dyadic paraproducts

Chung¹

2011

Publicacions Matemàtiques

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“…The Haar multipliers T w are closely related to the resolvent of the dyadic paraproduct [Pe1], and appeared in the study of Sobolev spaces on Lipschitz curves [Pe3]. It was proved in [Pe2] that the L 2 -norm for the Haar multiplier T w depends linearly on the C d 2 -characteristic of the weight w. We show the following theorem that generalizes a result of Beznosova for T t w [Be1, Chapter 5].…”

Section: Introductionmentioning

confidence: 62%

“…The result is optimal for T ±1/2 w , see [Be1], [Pe2] and [BMP]. We expect that, for both the paraproducts and t-Haar multipliers with complexity (m, n), the dependence on the complexity can be strengthened to be linear, in line with the best results for the Haar shift operators.…”

Section: Introductionmentioning

confidence: 80%

Weighted estimates for dyadic paraproducts and $t$-Haar multipliers with complexity $(m,n)$

Moraes¹,

Pereyra²

2013

Publicacions Matemàtiques

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We extend the definitions of dyadic paraproduct and t-Haar multipliers to dyadic operators that depend on the complexity (m, n), for m and n natural numbers. We use the ideas developed by Nazarov and Volberg to prove that the weighted L 2 (w)-norm of a paraproduct with complexity (m, n), associated to a function b ∈ BMO d , depends linearly on the A d 2-characteristic of the weight w, linearly on the BMO d-norm of b, and polynomially on the complexity. This argument provides a new proof of the linear bound for the dyadic paraproduct due to Beznosova. We also prove that the L 2-norm of a t-Haar multiplier for any t ∈ R and weight w is a multiple of the square root of the C d 2t-characteristic of w times the square root of the A d 2-characteristic of w 2t , and is polynomial in the complexity.

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“…See [63] for Bellman function extensions of the results for dyadic square functions to homogeneous spaces, [10] for a neat Bellman function transference lemma that allows to use Bellman functions in R to deduce results in R n with no sweat, similar considerations are used in [75]. There are now simpler Bellman function proofs that recover the estimates for the dyadic shift operators [59,75], and for the dyadic paraproduct [54].…”

Section: Chronology Of First Linear Estimates On L 2 (W)mentioning

confidence: 99%

Weighted Inequalities and Dyadic Harmonic Analysis

Pereyra

2012

Excursions in Harmonic Analysis, Volume 2

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We survey the recent solution of the so-called A 2 conjecture, all Calderón-Zygmund singular integral operators are bounded on L 2 (w) with a bound that depends linearly on the A 2 characteristic of the weight w, as well as corresponding results for commutators. We highlight the interplay of dyadic harmonic analysis in the solution of the A 2 conjecture, especially Hytönen's representation theorem for Calderón-Zygmund singular integral operators in terms of Haar shift operators. We describe Chung's dyadic proof of the corresponding quadratic bound on L 2 (w) for the commutator of the Hilbert transform with a BMO function, and we deduce sharpness of the bounds for the dyadic paraproduct on L p (w) that were obtained extrapolating Beznosova's linear bound on L 2 (w). We show that if an operator T is bounded on the weighted Lebesgue space L r (w) and its operator norm is bounded by a power α of the A r characteristic of the weight, then its commutator [T, b] with a function b in BMO will be bounded on L r (w) with an operator norm bounded by the increased power α + max{1, 1 r−1 } of the A r characteristic of the weight. The results are sharp in terms of the growth of the operator norm with respect to the A r characteristic of the weight for all 1 < r < ∞.

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Haar multipliers meet Bellman functions

Cited by 10 publications

References 23 publications

Weighted inequalities for multivariable dyadic paraproducts

Weighted inequalities for multivariable dyadic paraproducts

Weighted estimates for dyadic paraproducts and $t$-Haar multipliers with complexity $(m,n)$

Weighted Inequalities and Dyadic Harmonic Analysis

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