# A sharp estimate for Muckenhoupt class \(A_\infty \) and BMO

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## Abstract

A classical fact in the weighted theory asserts that a weight *w* belongs to the Muckenhoupt class \(A_\infty \) if and only if its logarithm \(\log w\) 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 \(J\subset \mathbb {R}\), we provide best lower and upper bounds for the BMO norm of \(\log w\) in terms of \(A_\infty \) characteristics of *w*. The proof rests on the precise evaluation of associated Bellman functions.

## Keywords

Weight BMO Best constant Bellman function## Mathematics Subject Classification

Primary 46B20 Secondary 46A35## 1 Introduction

*BMO*, the space of functions of bounded mean oscillation, if

*Q*in \(\mathbb {R}^n\) with edges parallel to the coordinate axes and

*Q*. The

*BMO*class, introduced by John and Nirenberg in [7], plays an important role in analysis and probability, since many classical operators (maximal, singular integral, etc.) map \(L^\infty \) into

*BMO*. Another remarkable result, due to Fefferman [4], asserts that

*BMO*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

*BMO*and weights satisfying the so-called \(A_\infty \) condition of Muckenhoupt [9]. In what follows, the word ‘weight’ refers to a nonnegative, locally integrable function on some base space \(\mathcal {B}\) (which is typically \(\mathbb {R}^n\) or some cube \(\mathcal {Q}\subset \mathbb {R}^n\)). Following [8], we say that a weight

*w*satisfies the condition \(A_\infty \) (or belongs to the class \(A_\infty (\mathcal {B}\))), if

*Q*contained in the base space \(\mathcal {B}\), having edges parallel to the coordinate axes. The aforementioned connection between the space

*BMO*and the class \(A_\infty \) can be (a little informally) stated as

*w*satisfies the condition \(A_\infty \) if and only if \(\log w\) belongs to the class

*BMO*. 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 \(c\ge 1\), let \(d_\pm =d_\pm (c)\) be the constants introduced in Lemma 3.1 below.

### Theorem 1.1

*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.

## 2 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*\(A_\infty \)

*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 \(1\le p<\infty \). For a given \(c\ge 1\), introduce the domain

We will require the following well-known fact (we include an easy proof for the sake of completeness).

### Lemma 2.1

For any \((x,y)\in \Omega _c\) and any interval *J*, there is a weight \(w:J\rightarrow \mathbb {R}_+\) satisfying \([w]_{A_\infty (J)}\le c\) such that \(\langle w\rangle _J=x\) and \(\langle \log w\rangle _J=y\).

### Proof

*PR*be a line segment, passing through (

*x*,

*y*), tangent to the lower boundary of \(\Omega _c\), 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

*PR*is entirely contained in \(\Omega _c\) and there is a number \(\alpha \in (0,1)\) such that \(\alpha P+(1-\alpha )R=(x,y)\). Split the interval

*J*into two subintervals \(J_\pm \) such that \(|J_-|=\alpha |J|\) and \(|J_+|=(1-\alpha )|J|\), and define \(w=P_x\chi _{J_-}+R_x\chi _{J_+}.\) Then

*I*of

*J*and note that

*PR*, and hence is contained in \(\Omega _c\). This is exactly what we need.

Let us briefly note that if (*x*, *y*) does not lie on the upper boundary of \(\Omega _c\), then the weight *w* constructed above actually satisfies the *equality* \([w]_{A_\infty (J)}=c\). Indeed, it follows from Darboux property that if *I* is chosen appropriately, then the average \((\langle w\rangle _I,\langle \log w\rangle _I)\) is the point of tangency of *PR* to the lower boundary of \(\Omega _c\). This amounts to saying that \(\langle w\rangle _I\exp (-\langle \log w\rangle _I)=c\), which gives the desired reverse estimate \([w]_{A_\infty (J)}\ge c\). \(\square \)

*Step 2. Abstract Bellman functions*Fix \(c\ge 1\), an interval \(J\subset \mathbb {R}\) and consider the functions \(\mathbb {B}^c_{\pm }:\Omega _c\rightarrow \mathbb {R}\) given by the abstract formulas

*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 \(A_\infty (J_1)\) and \(A_\infty (J_2)\) in one-to-one correspondence. To describe the relation between \(\mathbb {B}_\pm ^c\) and (1.3), observe that

*w*with \([w]_{A_\infty (J)}=c\), we see that \([w]_{A_\infty (I)}\le c\) for any subinterval \(I\subseteq J\) and hence

*I*was arbitrary, this gives \(\Vert \log w\Vert _{BMO}^2\le \sup _{(x,\,y)\in \Omega _c}\big (\mathbb {B}^c_+(x,y)-y^2\big )\) and hence also \( d_+^2(c)\le \sup _{(x,\,y)\in \Omega _c}\big (B^c_+(x,y)-y^2\big )\), by taking the supremum over all

*w*. To get the reverse bound, take \(\varepsilon >0\) and a point \((x_0,y_0)\in \Omega _c\) such that \(\mathbb {B}^c_+(x_0,y_0)-y_0^2\ge \sup _{(x,\,y)\in \Omega _c}\big (B^c_+(x,y)-y^2\big )-\varepsilon \). There is a weight

*w*on

*J*satisfying \([w]_{A_\infty }\le c\), \(\langle w\rangle _J=x_0\), \(\langle \log w\rangle _J=y_0\) and \(\mathbb {B}^c_+(x_0,y_0)\le \langle \log ^2w\rangle _J+\varepsilon .\) Putting all these properties together, we see that

*w*satisfies

*Step 3. Properties of Bellman functions*Suppose that (

*x*,

*y*) belongs to the upper boundary of \(\Omega _c\). Then for any interval

*J*, by Jensen’s inequality, there is only one weight \(w:J\rightarrow \mathbb {R}_+\) satisfying \(\langle w\rangle _J=x\) and \(\langle \log w\rangle _J=y\): the constant \(w\equiv x\). Consequently, directly from the definition of \(\mathbb {B}_\pm ^c\), we have

*w*was arbitrary, this yields

*x*,

*y*, \(\lambda \) by \(\lambda x\), \(y+\log \lambda \) and \(\lambda ^{-1}\), respectively, gives the reverse bound. Consequently, setting \(\lambda =x^{-1}\), we see that

*informally*how the idea works for the function \(\mathbb {B}_+^c\). Suppose that the line segment

*PR*is entirely contained in \(\Omega _c\), \(\alpha \in (0,1)\) is a fixed parameter and let \(S=\alpha P+(1-\alpha )R\). Let \(w_-\) be a weight on the interval \([0,\alpha )\), satisfying \([w]_{A_\infty ([0,\alpha ))}\le c\), \(\langle w\rangle _{[0,\alpha )}=P_x\), \(\langle \log w\rangle _{[0,\alpha )}=P_y\). Let \(w_+\) be a weight on \([\alpha ,1]\), satisfying \([w]_{A_\infty ([\alpha ,1])}\le c\), \(\langle w\rangle _{[\alpha ,1]}=R_x\), \(\langle w\rangle _{[\alpha ,1]}=R_y\). Splicing these two weights into one weight

*w*on [0, 1], we see that

*w*satisfies \([w]_{A_\infty (0,1)}\le c\) (unfortunately, this cannot be formally proved; however, let us proceed under this assumption). This would yield

Before we proceed, let us make a crucial comment. We expect the Bellman functions \(\mathbb {B}_\pm ^c\) 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 \(\Phi _\pm ^c\) which can be solved explicitly, and hence it identifies the formulas for \(\mathbb {B}_\pm ^c\).

*Step 4. On the search of*\(\mathbb {B}_\pm ^c\) We put all the above facts together. Let us

*assume*that \(\mathbb {B}_\pm ^c\) 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

*t*. Then the requirement \({\text {*}}{det}D^2\mathbb {B}_\pm ^c=0\) yields the following ODE for \(\Phi _\pm ^c\):

*candidates*for \(\Phi _\pm ^c\), which in turn yield the candidates for the Bellman functions \(\mathbb {B}_\pm ^c\). These candidates, denoted from now on by \(\Psi _\pm ^c\) and \(B_\pm ^c\), will be explicitly introduced and studied in the next section.

## 3 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 \(\mathbb {B}_\pm ^c\). Throughout, \(c\ge 1\) is a fixed parameter.

### 3.1 Special functions and their properties

Introduce the auxiliary function \(G:(-1,\infty )\rightarrow \mathbb {R}\) by \(G(x)=x-\log \big (c(1+x)\big )\). One easily computes that \(G'(x)=x/(1+x)\), so *G* is strictly decreasing on \((-1,0)\) and strictly increasing on \((0,\infty )\). Furthermore, one checks immediately that \(\lim _{x\downarrow -1}G(x)=\lim _{x\uparrow \infty }G(x)=\infty \) and \(G(0)=-\log c\le 0\), so in particular we have the following fact, which we formulate as a separate statement.

### Lemma 3.1

*G*maps each of the intervals \([d_-,0]\), \([0,d_+]\) monotonically onto \([-\log c,0]\).

*G*(considered as a function from \([-\log c,0]\) to \([d_-,0]\) in the case of \(\Psi _-^c\), and from \([-\log c,0]\) to \([0,d_+]\) in the case of \(\Psi _+^c\)). In other words, the functions \(\Psi _\pm ^c\) satisfy

### Lemma 3.2

The Hessian matrix of \(B_+^c\) is nonpositive-definite. The Hessian matrix of \(B_-^c\) is nonnegative-definite.

### Proof

*x*as above, depending on whether \(\Psi _+^c\) or \(\Psi _-^c\) is studied). \(\square \)

We will also exploit the following size estimate for \(B_\pm ^c\).

### Lemma 3.3

We have \(y^2\le B_\pm ^c(x,y)\le d_\pm ^2+y^2\) for all \((x,y)\in \Omega _c\).

### Proof

### 3.2 Proof of the inequalities \(\mathbb {B}_+^c\le B_+^c\), \(\mathbb {B}_-^c\ge B_-^c\) 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 \(4_\infty \) in [17]).

### Lemma 3.4

For any \(\varepsilon >c\) and an arbitrary weight on *J* with \([w]_{A_\infty (J)}\le c\) there exists a splitting \(J = J^-\cup J^+\), \(|J^\pm |=\alpha _\pm |J|\), such that the entire interval with the endpoints \(p^\pm =(\langle w\rangle _{J^{\pm }},\langle \log w\rangle _{J^{\pm }})\) is in \(\Omega _\varepsilon \). Moreover, the splitting parameters \(\alpha _\pm \) can be chosen bounded away from 0 and 1 uniformly with respect to *w* and, therefore, with respect to *J* as well.

*Proof of* \(\mathbb {B}_+^c\le B_+^c\). Fix an \(A_\infty \) weight *w* on *J*, satisfying \([w]_{A_\infty (J)}\le c\), \(\langle w\rangle _J=x\) and \(\langle \log w\rangle _J=y\). Furthermore, pick an arbitrary \(\varepsilon >c\).

*Step 1.*Consider the following family \(\{\mathcal {J}^{n}\}_{n\ge 0}\) of partitions of

*J*, generated by the inductive use of Lemma 3.4. We start with \(\mathcal {J}^0=\{J\}\); then, given \(\mathcal {J}^n=\{J^{n,1},J^{n,2},\ldots ,J^{n,2^n}\}\), we split each \(J^{n,k}\) according to Lemma 3.4, applied to the function

*w*and the parameter \(\varepsilon \). Finally, put

*J*by

*t*; if there are two such intervals, we pick the one which has

*t*as its right endpoint. Since \([w]_{A_\infty (J)}\le c\), we have \((f_n(t),g_n(t))\in \Omega _\varepsilon \) for each

*n*and almost all \(t\in J\).

*Step 2.*Let \(B_+^\varepsilon \) be the Bellman function studied in the previous section, corresponding to the parameter \(\varepsilon \). We will prove that for any nonnegative integer

*n*and any \(J^{n,k}\in \mathcal {J}^n\) we have

*Step 3.*Summing (3.5) and (3.7) over

*k*, we get

*n*we have

*w*was arbitrary, we get \(\mathbb {B}_+^c(x,y)\le B_+^\varepsilon (x,y)\). It remains to let \(\varepsilon \rightarrow c\) to get the desired bound. \(\square \)

*Proof of*(1.3),

*the right estimate.*Let

*w*be an \(A_\infty \) weight on

*J*with \([w]_{A_\infty (J)}\le c\). Then for any subinterval \(I\subseteq J\), we have \([w]_{A_\infty (I)}\le c\) and hence, by Lemma 3.3 and the inequality \(\mathbb {B}_+^c\le B_+^c\),

*I*was arbitrary, this gives the desired upper bound for \(\Vert \log w\Vert _{BMO(J)}\). \(\square \)

*Proof of the inequality*\(\mathbb {B}_-^c\ge B_-^c\). 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 \(A_\infty \) weight with \([w]_{A_\infty (J)}\le c\). Set \(x=\langle w\rangle _J\), \(y=\langle \log w\rangle _J\) and let \(\varepsilon >c\). Construct the partitions \(\{\mathcal {J}^n\}_{n\ge 0}\) of

*J*and the functional sequences \((f_n)_{n\ge 0}\), \((g_n)_{n\ge 0}\) using the same formulas as previously. Let \(B_-^\varepsilon \) be the Bellman function corresponding to the parameter \(\varepsilon \). Then Lemma 3.2 shows that for any

*n*and

*k*,

*k*,

*J*: indeed, \(\log w\) is square-integrable as a function from the class

*BMO*and \(\mathcal {M}\) is \(L^2\)-bounded. Therefore, letting \(n\rightarrow \infty \) in (3.8) yields, by Lebesgue’s dominated convergence theorem,

*w*was arbitrary, this implies \(\mathbb {B}_-^c(x,y)\ge B_-^\varepsilon (x,y)\). It remains to let \(\varepsilon \rightarrow c\) to get the claim. \(\square \)

*Proof of*(1.3),

*the left estimate.*Pick a weight

*w*on

*J*such that \(\log w\) belongs to the class

*BMO*. Then

*w*is an \(A_\infty \) weight. Set \(c=[w]_{A_\infty }\); we may assume that \(c>1\), since otherwise there is nothing to prove (indeed, if \(c=1\), then

*w*, and hence also \(\log w\), are constant). Pick \(c'\in (1,c)\) and choose \(I\subset J\) such that \(\langle w\rangle _I\exp (-\langle \log w\rangle _I)\ge c'\); such a choice is possible by the very definition of the \(A_\infty \) condition. Since \(\mathbb {B}_-^c\ge B_-^c\), we get, by the very definition of \(B_-^c\),

### 3.3 Sharpness of (1.3) and proofs of the inequalities \(\mathbb {B}_+^c\ge B_+^c\), \(\mathbb {B}_-^c\le B_-^c\)

Let us split this subsection into two parts.

*The constants*\(d_\pm (c)\)

*in*(1.3)

*cannot be improved.*Fix \(c\ge 1\) and consider the weight

*w*on (0, 1) given by \(w(s)=s^d\), where \(d\in \{d_-(c),d_+(c)\}\). We will prove that

*w*is at least

*c*and all we need is the estimate \([w]_{A_\infty (0,1)}\le c\). We will check that

*a*and

*b*were arbitrary, we obtain \(||\log w||_{BMO}\le d\); furthermore, taking \(a=0\) we see that actually equality holds here. This is precisely the desired assertion. \(\square \)

*Proof of*\(\mathbb {B}_+^c\ge B_+^c\), \(\mathbb {B}_-^c\le B_-^c\). Fix \(c\ge 1\), \(d\in \{d_-(c),d_+(c)\}\), \(\kappa \ge 1\) and consider a modification of the above weight, given by \(w(s)=s^d\chi _{(0,1]}(s)+\chi _{[1,\kappa )}(s)\). This weight satisfies \([w]_{A_\infty (0,\kappa )}=c\). Indeed, we have \([w]_{A_\infty (0,\kappa )}\ge [w]_{A_\infty (0,1)}=c\) and, as we will prove now,

## Notes

### Acknowledgements

The author would like to thank an anonymous referee for the careful reading of the paper and several helpful suggestions. The research was supported by Narodowe Centrum Nauki (Poland) Grant DEC-2014/14/E/ST1/00532.

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