Best constants for a family of Carleson sequences

Abstract

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...

Bellman function for extremal problems in $\mathrm{BMO}$ II: evolution

“…For the corresponding "tail"-estimate on the Muckenhoupt classes see [23]. One can also apply very similar technique to estimate the norm of the maximal operator, see [15,28], and the Carleson embedding operator, see [40,26]. We refer the reader to the lecture notes [42,43] for a more detailed scenery of the subject.…”

A multiparameter integral inequality for the dyadic maximal operator and applications

“…Also in [33], using the Monge-Ampère equation approach, a more general Bellman function than the one related to the dyadic Carleson Imbedding Theorem has been precisely evaluated thus generalizing the corresponding result in [10]. For more recent developments we refer to [1,6,7,23,24,28,29,37]. Additional results can be found in [34,35] while for the study of general theory of maximal operators one can consult [4,5] and [30].…”