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

Moraes, Jean Carlo; Pereyra, María Christina

doi:10.5565/publmat_57213_01

Cited by 6 publications

(7 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…. Therefore we cannot compare bump conditions to the conditions in our results without the additional assumption that there is a constant q > 0 such that m J (u −1 )m J v q for all J ∈ D. If q m J (u −1 )m J v Q for all J ∈ D; the two conditions (4.6) and (4.7) become equivalent, but this assumption essentially reduces the problem to the one weight case [M,Proposition 7.4]. C v,v .…”

Section: If We Especialize To the One Weight Casementioning

confidence: 68%

On Two Weight Estimates for Dyadic Operators

Beznosova

Chung

Moraes

et al. 2017

Association for Women in Mathematics Series

View full text Add to dashboard Cite

We provide a quantitative two weight estimate for the dyadic paraproduct π b under certain conditions on a pair of weights (u, v) and b in Carlu,v, a new class of functions that we show coincides with BM O when u = v ∈ A d 2 . We obtain quantitative two weight estimates for the dyadic square function and the martingale transforms under the assumption that the maximal function is bounded from L 2 (u) into L 2 (v) and v ∈ RH d 1 . Finally we obtain a quantitative two weight estimate from L 2 (u) into L 2 (v) for the dyadic square function under the assumption that the pair (u, v) is in joint A d 2 and u −1 ∈ RH d 1 , this is sharp in the sense that when u = v the conditions reduce to u ∈ A d 2 and the estimate is the known linear mixed estimate.2010 Mathematics Subject Classification. Primary 42B20, 42B25 ; Secondary 47B38.where ∆ I v := m I + v − m I − v, and I ± are the right and left children of I.Then π b , the dyadic paraproduct associated to b, is bounded from L 2 (u) into L 2 (v). Moreover, there exists a constant C > 0 such that for all f ∈ L 2 (u)where π b f := I∈D m I f b, h I h I .

show abstract

Section: If We Especialize To the One Weight Casementioning

confidence: 68%

On Two Weight Estimates for Dyadic Operators

Beznosova

Chung

Moraes

et al. 2017

Association for Women in Mathematics Series

View full text Add to dashboard Cite

show abstract

“…This function class was introduced and studied in many papers, such as [1], [5] , [14], [15], and [16]. We now state our main results.…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 96%

On a Two Weights Estimate for the Commutator

Chung¹

2017

East Asian mathematical journal

View full text Add to dashboard Cite

Abstract. We provide quantitative two weight estimates for the commutator of the Hilbert transform under certain conditions on a pair of weights (u, v) and b in Carlu,v. In [10] and [11], Bloom's inequality is shown in a modern setting, and the boundedness of the commutators is provided by assuming both weights u, v are A 2 and b ∈ BM Oρ. In the present paper we show that the condition on b can be replaced by Carlu,v by using the joint A d 2 condition.

show abstract

“…One specific context where such need arises is when studying dyadic paraproducts in weighted settings. For example, in [1] and [9] the authors estimate the norms of paraproducts on L 2 (w), with w ∈ A d 2 . In this situation, the sequences of interest are often of the form In this notation (1.4) becomes c Φ J (w) = |J| f s J (w) R J (w).…”

Section: Preliminariesmentioning

confidence: 99%

“…The significance of such sequences in analysis stems mainly from their role in the Carleson embedding theorem and related results. A key example is the following lemma whose proof can be found in [9] (in a more general, weighted setting). We will use this lemma to derive an important corollary of our main estimates.…”

Section: Preliminariesmentioning

confidence: 99%

Best constants for a family of Carleson sequences

Slavin

2016

Advances in Mathematics

View full text Add to dashboard Cite

Abstract. We consider a general family of Carleson sequences associated with dyadic A2 weights and find sharp -or, in one case, simply best known -upper and lower bounds for their Carleson norms in terms of the A2 -characteristic of the weight. The results obtained make precise and significantly generalize earlier estimates by Wittwer, Vasyunin, Beznosova, and others. We also record several corollaries, one of which is a range of new characterizations of dyadic A2. Particular emphasis is placed on the relationship between sharp constants and optimizing sequences of weights; in most cases explicit optimizers are constructed. Our main estimates arise as consequences of the exact expressions, or explicit bounds, for the Bellman functions for the problem, and the paper contains a measure of Bellman-function innovation. PreliminariesWe will be concerned with weights on R, i.e. locally integrable functions that are positive almost everywhere. Our weights will be assumed to belong to the dyadic Muckenhoupt class A d 2 associated with a particular lattice D, i.e., the set of all dyadic intervals on the line uniquely determined by the position and size of the root interval. If an interval I is fixed, D(I) will stand for the unique dyadic lattice on I and D n (I) for the set of the dyadic subintervals of I of the n-th generation, D n (I) = {J : J ∈ D(I), |J| = 2 −n |I|}.Let w J be the average of a weight w over an interval J, The smallest such C , denoted by {c J } C , is called the Carleson norm of {c J }. The significance of such sequences in analysis stems mainly from their role in the Carleson embedding theorem and related results. A key example is the following lemma whose proof can be found in [9] (in a more general, weighted setting). We will use this lemma to derive an important corollary of our main estimates. This inequality suggests that it may be important to have good estimates of the Carleson norms of sequences one uses. One specific context where such need arises is when studying dyadic paraproducts in weighted settings. For example, in [1] and [9] the authors estimate the norms of paraproducts on L 2 (w), with w ∈ A d 2 . In this situation, the sequences of interest are often of the formwhere ∆ J (·) = · J − − · J + , and J ± are the two halves of J. In [1], Beznosova showed that the norm of {cfor a numerical constant C. In [13], Nazarov and Volberg extended this estimate, proving thatIn a recent paper [9], Moraes and Pereyra observed that one can simply combine Beznosova's result with the fact that t → t α is an increasing function for α > 0 to obtainfor an explicit r(α).The case α = 0 was considered by Wittwer in [20], where she obtained the asymptotically sharp estimatewhere {a J (w)} is the same as {cJ (w)}, except with only one of the two summands (it does not matter which one is removed). One can double Wittwer's constant to get an estimate for {c (0) J (w)}, but as we will see below, it is no longer sharp, though the logarithm remains. The same sequence {a J (w)}, but for w ∈ A ∞ , was studied by Va...

show abstract

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

Cited by 6 publications

References 34 publications

On Two Weight Estimates for Dyadic Operators

On Two Weight Estimates for Dyadic Operators

On a Two Weights Estimate for the Commutator

Best constants for a family of Carleson sequences

Contact Info

Product

Resources

About