A sharp estimate for Muckenhoupt class A∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_\infty $$\end{document} and BMO

A classical fact in the weighted theory asserts that a weight w belongs to the Muckenhoupt class A∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_\infty $$\end{document} if and only if its logarithm logw\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\log w$$\end{document} is a function of bounded mean oscillation. We prove a sharp quantitative version of this fact in dimension one: for a weight w defined on some interval J⊂R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J\subset \mathbb {R}$$\end{document}, we provide best lower and upper bounds for the BMO norm of logw\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\log w$$\end{document} in terms of A∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_\infty $$\end{document} characteristics of w. The proof rests on the precise evaluation of associated Bellman functions.


Introduction
A real-valued locally integrable function ϕ defined on R n is said to be in B M O, the space of functions of bounded mean oscillation, if sup Q |ϕ − ϕ Q | Q < ∞. (1.1) Here the supremum is taken over all cubes Q in R n with edges parallel to the coordinate axes and denotes the average of ϕ over Q. The B M O class, introduced by John and Nirenberg in [7], plays an important role in analysis and probability, since many classical operators B  [4], asserts that B M O is a dual to the Hardy space H 1 . It is well-known that the functions of bounded mean oscillation have very strong integrability properties (see e.g. [7]). In particular, the p-oscillation In what follows, we will work with || · || B M O 2 and denote it simply by || · || B M O . One of the reasons we choose this particular norm is that we have the identity which makes the norm very convenient to handle. We will also restrict ourselves to the localized setting. That is, if Q ⊂ R n is a fixed cube with edges parallel to the coordinate axes, we define the class B M O(Q) as the collection of all integrable ϕ : Q → R for which the expression (1.2) is finite, this time the supremum being taken over all cubes Q ⊆ Q with edges parallel to the coordinate axes. There is a well-known connection between B M O and weights satisfying the socalled A ∞ condition of Muckenhoupt [9]. In what follows, the word 'weight' refers to a nonnegative, locally integrable function on some base space B (which is typically R n or some cube Q ⊂ R n ). Following [8], we say that a weight w satisfies the condition A ∞ (or belongs to the class A ∞ (B)), if where, as above, the supremum is taken over all cubes Q contained in the base space B, having edges parallel to the coordinate axes. The aforementioned connection between the space B M O and the class A ∞ can be (a little informally) stated as More precisely, a weight w satisfies the condition A ∞ if and only if log w belongs to the class B M O. The principal goal of this paper is to provide a sharp quantitative version of this result in the case n = 1. In this particular setting, the cubes become intervals; to stress that we work in the one-dimensional case, we will use the letters I , J instead of Q.
Here is our main result. For a given c ≥ 1, let d ± = d ± (c) be the constants introduced in Lemma 3.1 below. Theorem 1.1 Let J ⊂ R be a given interval. If w is a weight on J , then we have Actually, we will prove more. Our approach will rest on the so-called Bellman function method, a powerful technique which is now used widely in various contexts of analysis and probability theory. Roughly speaking, this technique enables to extract the optimal constants in a given estimate from the existence of a certain special function enjoying an appropriate size condition and concavity. The method originates from the theory of optimal control (the theory of dynamic programming): see [1]. The connection of this approach to the estimates for martingale transforms was observed in the eighties by Burkholder [3] and then it was exploited in the study of other semimartingale inequalities (consult [12] for an overview of the results in this direction). A decisive step towards the application of the Bellman function method to general problems of harmonic analysis was made by Nazarov et al. [11] (see also [10]). Since then, the technique has been successfully applied in numerous settings: see e.g. [2,5,6,[13][14][15][16][17] and consult the references therein.
We will identify the explicit formula for the Bellman functions associated with the double inequality (1.3). As usual in this type of problems, the argumentation splits naturally into two parts. The first part, which contains an informal reasoning leading to the discovery (or the guess) of the Bellman functions, is presented in the next section. The formal verification that these guessed objects are indeed the desired Bellman functions, is the contents of Sect. 3.

Associated Bellman functions
The purpose of this section is to rephrase both estimates in (1.3) in the language of the associated Bellman functions and to describe the main steps which lead to the discovery of these crucial objects. For the reader's convenience, we split the reasoning into several separate parts.
Step 1. Geometric interpretation of A ∞ weights We follow the work [17] by Vasyunin; the reader can also find in that paper the related interpretation for A p weights in the range 1 ≤ p < ∞. For a given c ≥ 1, introduce the domain Then a weight w : J → R + satisfies [w] A ∞ (J ) ≤ c if and only if for any interval I ⊆ J we have ( w I , log w I ) ∈ c . Indeed, the inequality log w I ≤ log w I + log c is guaranteed directly by the A ∞ condition, while the left estimate log w I ≤ log w I is a mere consequence of Jensen's inequality.
We will require the following well-known fact (we include an easy proof for the sake of completeness). Proof If y = log x, then the constant weight w ≡ x does the job. Suppose then, that y < log x. Let P R be a line segment, passing through (x, y), tangent to the lower boundary of c , with P, R lying on the upper boundary of this set (i.e., such that P y = log P x , R y = log R x ). (If (x, y) does not belong to the lower boundary, then there are two such line segments; take any of them). Then P R is entirely contained in c and there is a number α ∈ (0, 1) such that α P To this end, pick an arbitrary subinterval I of J and note that This means that ( w I , log w I ) lies on the line segment P R, and hence is contained in c . This is exactly what we need.
Let us briefly note that if (x, y) does not lie on the upper boundary of c , then the weight w constructed above actually satisfies the equality [w] A ∞ (J ) = c. Indeed, it follows from Darboux property that if I is chosen appropriately, then the average ( w I , log w I ) is the point of tangency of P R to the lower boundary of c . This amounts to saying that w I exp(− log w I ) = c, which gives the desired reverse Step 2. Abstract Bellman functions Fix c ≥ 1, an interval J ⊂ R and consider the functions B c ± : c → R given by the abstract formulas By the previous lemma, B c ± are welldefined objects on the whole c . It is clear that the above functions do not depend on the base interval J . Indeed, for any two intervals J 1 and J 2 , there is an affine mapping putting J 1 onto J 2 ; such a mapping preserves the averages and puts the classes A ∞ (J 1 ) and A ∞ (J 2 ) in one-to-one correspondence. To describe the relation between B c ± and (1.3), observe that where the infimum and supremum are taken over the same class as previously. In particular, this shows that d 2 Since I was arbitrary, this gives log , by taking the supremum over all w. To get the reverse bound, take ε > 0 and a point ( Putting all these properties together, we see that w satisfies This proves d 2 since ε was arbitrary. A similar reasoning establishes an analogous relation between B c − and d − (c). Thus, having found the explicit formulas for B c ± , we immediately extract the solution to our main problem. As we will see now, the identification of these formulas is possible due to certain structural properties of Bellman functions.
Step 3. Properties of Bellman functions Suppose that (x, y) belongs to the upper boundary of c . Then for any interval J , by Jensen's inequality, there is only one weight w : J → R + satisfying w J = x and log w J = y: the constant w ≡ x. Consequently, directly from the definition of B c ± , we have on the upper boundary of c . The next observation is a certain homogeneity-type condition, which allows to express B c ± in terms of some functions of one variable. Namely, fix a weight w : J → R + satisfying [w] A ∞ (J ) ≤ c, w J = x and log w J = y. Then for any λ > 0 we have [λw] A ∞ (J ) ≤ c, λw J = λx and log(λw) J = y + log λ, so Since w was arbitrary, this yields and replacing x, y, λ by λx, y + log λ and λ −1 , respectively, gives the reverse bound. Consequently, setting λ = x −1 , we see that for some function c + : [− log c, 0] → R to be found. The same reasoning shows that To find c ± , we go back to the structure of the "whole" function B c ± . In a typical situation, Bellman functions are locally convex/concave on their domains (here the phrase "local convexity/concavity" means the convexity/concavity along any line segment contained in the domain of the function). Let us describe informally how the idea works for the function B c + . Suppose that the line segment P R is entirely contained in c , α ∈ (0, 1) is a fixed parameter and let Splicing these two weights into one weight w on [0, 1], we see that Suppose that the weight w satisfies [w] A ∞ (0,1) ≤ c (unfortunately, this cannot be formally proved; however, let us proceed under this assumption). This would yield and taking the supremum over all w ± as above, we would obtain i.e., the desired local concavity of B c + . A similar heuristic argument indicates that B c − should be locally convex. Before we proceed, let us make a crucial comment. We expect the Bellman functions B c ± to yield the best constants in (1.3). In such a situation, one typically assumes that for each point (x, y) lying in the interior of the domain, the convexity/concavity assumption degenerates in some direction; that is, there is a (short) line segment passing through (x, y) along which the Bellman function is linear. As we shall see, this assumption leads to the key second-order differential equation for c ± which can be solved explicitly, and hence it identifies the formulas for B c ± .
Step 4. On the search of B c ± We put all the above facts together. Let us assume that B c ± are of class C 2 . Then the local convexity/concavity can be reformulated in terms of the corresponding Hessian matrices. Furthermore, the aforementioned degeneration condition implies that the determinant of the Hessian must vanish at each point belonging to the interior of the domain. We compute that where, for brevity, we denoted y − log x by t. Then the requirement det D 2 B c ± = 0 yields the following ODE for c ± : This equation can be easily solved, taking into account the initial condition obtained from the behavior of B c ± at the upper boundary of c . This gives us the candidates for c ± , which in turn yield the candidates for the Bellman functions B c ± . These candidates, denoted from now on by c ± and B c ± , will be explicitly introduced and studied in the next section.

Formal verification
Now we will present the formal proof of Theorem 1.1 and the rigorous identification of the explicit formulas for the Bellman functions B c ± . Throughout, c ≥ 1 is a fixed parameter.

Special functions and their properties
Introduce the auxiliary function G : . One easily computes that G (x) = x/(1 + x), so G is strictly decreasing on (−1, 0) and strictly increasing on (0, ∞). Furthermore, one checks immediately that lim x↓−1 G(x) = lim x↑∞ G(x) = ∞ and G(0) = − log c ≤ 0, so in particular we have the following fact, which we formulate as a separate statement.
The (candidates for) Bellman functions are given explicitly by Let us verify formally the local convexity/concavity of these objects.

Lemma 3.2 The Hessian matrix of B c
Proof Actually, we have already performed all the necessary calculations in the preceding section, but we rewrite the computations for the sake of completeness. Setting t := y − log x, the Hessian matrix of B c ± is given by (with the same restrictions on the range of x as above, depending on whether c + or c − is studied).
We will also exploit the following size estimate for B c ± .

Lemma 3.3 We have y
Proof As shown in the proof of the previous lemma, the functions c + and c − are decreasing on [− log c, 0]. Consequently,

Proof of the inequalities B c + ≤ B c + , B c − ≥ B c − and (1.3)
The argument rests on the inductive use of the local convexity/concavity of the Bellman functions. We will need the following technical fact (see Lemma 4 ∞ in [17]).

Lemma 3.4 For any ε > c and an arbitrary weight on J with [w]
A ∞ (J ) ≤ c there exists a splitting J = J − ∪ J + , |J ± | = α ± |J |, such that the entire interval with the endpoints p ± = ( w J ± , log w J ± ) is in ε . Moreover, the splitting parameters α ± can be chosen bounded away from 0 and 1 uniformly with respect to w and, therefore, with respect to J as well.
x and log w J = y. Furthermore, pick an arbitrary ε > c.
Next, we define the sequences ( f n ) n≥0 , (g n ) n≥0 of functions on J by log w(s)ds.
Here J n (t) ∈ J n is an interval containing t; if there are two such intervals, we pick the one which has t as its right endpoint. Since [w] A ∞ (J ) ≤ c, we have ( f n (t), g n (t)) ∈ ε for each n and almost all t ∈ J .
Step 2. Let B ε + be the Bellman function studied in the previous section, corresponding to the parameter ε. We will prove that for any nonnegative integer n and any J n,k ∈ J n we have To this end, observe that the pair ( f n , g n ) is constant on J n,k (equal to p = ( w J n,k , log w J n,k ) there), while ( f n+1 , g n+1 ) takes two values on this interval: p ± = ( w J n,k ± , log w J n,k ± )). By Lemma 3.4, the entire interval with the endpoints p ± is contained within ε , and hence B ε + is concave along this interval. It remains to note that (3.5) follows immediately from this concavity.
Step 3. Summing (3.5) and (3.7) over k, we get Consequently, for any nonnegative integer n we have Combining this estimate with Lemma 3.3, we get y) . (3.6) However, recall that the splitting ratios α ± of Lemma 3.4 were bounded away from 0 and 1. Therefore, the diameter of J n tends to 0 as n → ∞, i.e., we have lim n→∞ sup 1≤k≤2 n |J n,k | = 0. Consequently, by Lebesgue's differentiation theorem, we have g n (t) → log w(t) for almost all t ∈ J . By Fatou's lemma, (3.6) yields and since w was arbitrary, we get B c + (x, y) ≤ B ε + (x, y). It remains to let ε → c to get the desired bound. Proof of the inequality B c − ≥ B c − . The argument is very similar to that used above, however, there are some small differences, so we have decided to write the proof separately. Let w be an A ∞ weight with [w] A ∞ (J ) ≤ c. Set x = w J , y = log w J and let ε > c. Construct the partitions {J n } n≥0 of J and the functional sequences ( f n ) n≥0 , (g n ) n≥0 using the same formulas as previously. Let B ε − be the Bellman function corresponding to the parameter ε. Then Lemma 3.2 shows that for any n and k, Now we use Lebesgue's differentiation theorem and Lebesgue's dominated convergence theorem. The function B ε − is continuous on ε , so we have for almost all t ∈ J . In addition, by Lemma 3.3, where M is the Hardy-Littlewood maximal operator. The expression on the right is integrable on J : indeed, log w is square-integrable as a function from the class B M O and M is L 2 -bounded. Therefore, letting n → ∞ in (3.8) yields, by Lebesgue's dominated convergence theorem, Since w was arbitrary, this implies B c − (x, y) ≥ B ε − (x, y). It remains to let ε → c to get the claim.
where in the last passage we have exploited the fact that c − is a decreasing function. This shows that log  The estimate (3.11) is equivalent to showing that p a,b ∈ c . By Jensen's inequality, p a,b cannot lie above the upper boundary of c , i.e., we have y a,b ≤ log x a,b . To show that y a,b ≥ log x a,b − log c, it suffices to note that the lower boundary of c is a graph of a concave function y = log x − c and This proves the first equality in (3.9); actually, it proves a little more (which will be useful later): the point p a,b lies above the line which is tangent to the lower boundary of c at the point p b . See Fig. 1. and, by (3.12), log w (0,κ) = 1 κ log 2 w (0,1) = 2d 2 κ . This, directly from the definition of B c ± , yields the estimates