Weak integral conditions for BMO

Logunov, Alexander; Slavin, Leonid; Stolyarov, Dmitriy; Vasyunin, Vasily; Zatitskiy, Pavel B.

doi:10.1090/s0002-9939-2015-12424-0

Cited by 18 publications

(21 citation statements)

References 12 publications

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“…We address the reader to the paper [8], where it was explained how do the sharp estimates of these functionals lead to various forms of the John-Nirenberg inequality and the equivalence of the L p -type norms defining the BMO. This case is the subject of study for papers [8,7,15,22,28,29,35,36].…”

Section: Particular Casesmentioning

confidence: 99%

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Theory of locally concave functions and its applications to sharp estimates of integral functionals

Stolyarov

Zatitskiy

2016

Advances in Mathematics

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Section: Particular Casesmentioning

confidence: 99%

“…The paper [33] was the first to appear. Following this route, many authors have managed to prove various inequalities, see the papers [1,4,8,7,15,26,22,24,31,28,29,32,33,34,35,36,37,38]. Though the method turned out to be very useful, there was absolutely no theory that allowed to calculate the Bellman functions "mechanically".…”

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confidence: 99%

Theory of locally concave functions and its applications to sharp estimates of integral functionals

Stolyarov

Zatitskiy

2016

Advances in Mathematics

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“…First, the lower estimates of Theorem 2.3 allow us to obtain a range of equivalent definitions of A d 2 in terms of sequences {c Φ J } for all increasing Φ such that Φ(t) → ∞ as t → ∞. We note that this limit condition, as well as the inverted lower estimate in Part (ii) have direct analogs in a recent paper [5] concerning equivalent definitions of BMO.…”

Section: Preliminariesmentioning

confidence: 85%

“…Beyond that, one must adapt the inductive argument at the center of the proofs of Theorems 2.10 and 2.11 to the situation where one must keep track of 2 n points from the domain Ω Q on every step of the induction, as opposed to just two points before. A somewhat similar challenge, but without regard for sharpness, was successfully handled in [5].…”

Section: 2mentioning

confidence: 99%

Best constants for a family of Carleson sequences

Slavin

2016

Advances in Mathematics

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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...

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“…for an appropriate function h implies f ∈ BM O were given by Strömberg [35], Lo and Ruilin [25], and Shi and Torchinsky [32]. See also the more recent work of Logunov et al in [26]. Hence, if X = BM O and Λ(f, Q) p = h((|f (x)−f Q |) for a certain such appropriate function, then one can conclude that X p = X for 0 < p < 1.…”

Section: Sufficient Conditions For Weight Invariancementioning

confidence: 99%

John–Nirenberg Inequalities and Weight Invariant BMO Spaces

Hart

Torres

2018

J Geom Anal

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This work explores new deep connections between John-Nirenberg type inequalities and Muckenhoupt weight invariance for a large class of BM O-type spaces. The results are formulated in a very general framework in which BM O spaces are constructed using a base of sets, used also to define weights with respect to a non-negative measure (not necessarily doubling), and an appropriate oscillation functional. This includes as particular cases many different function spaces on geometric settings of interest. As a consequence the weight invariance of several BM O and Triebel-Lizorkin spaces considered in the literature is proved. Most of the invariance results obtained under this unifying approach are new even in the most classical settings.

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Weak integral conditions for BMO

Cited by 18 publications

References 12 publications

Theory of locally concave functions and its applications to sharp estimates of integral functionals

Theory of locally concave functions and its applications to sharp estimates of integral functionals

Best constants for a family of Carleson sequences

John–Nirenberg Inequalities and Weight Invariant BMO Spaces

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