BELLMAN FUNCTIONS AND THE JOHN-NIRENBERG INEQUALITY

TERENCE TAO

Abstract. We use a Bellman function approach to prove the John-Nirenberg inequality for dyadic BMO. We also prove a slightly sharper localized version of this inequality which will be important in applications.

1. Introduction

For any dyadic interval I, define the local BMO norm ∥f∥_{BMO(I)} by

∥f∥_{BMO(I)} = sup_{J⊂I} \frac{1}{J} ∫_{J} |f − f_{J}|^{2} dx

where the supremum is over all dyadic sub-intervals J of I.

The John-Nirenberg inequality then states that if f is supported on I and has mean zero, then

|{x ∈ I : |f(x)| ≥ λ}| ≤ C|I|e^{−ελ/∥f∥_{BMO(I)}}  (1)

for all λ ≥ 0, where C, ε > 0 are absolute constants.

Let η(x) denote a function which satisfies the bounds

η(x) ∼ 1, η′(x) ∼ 1, η′′(x) ∼ −1  (2)

on the interval [0, 1]. For instance, one could take

η(x) := 10 − (x − 2)^{2}.

In order to prove (1), it suffices to take real-valued f and estimate the set {f > λ}. It then suffices by a limiting argument to prove the following.

Proposition 1.1. Let N ≥ 0 be an integer, and let f be a real-valued BMO function on a dyadic interval I which has mean zero and which is constant on dyadic intervals of length 2^{−N}|I|. Then we have

|\{x ∈ I : f(x) ≥ λ\}| ≤ C_{0}|I|η(\frac{∥f∥_{2}^{2}}{|I|∥f∥_{BMO(I)}^{2}} )e^{−ελ/∥f∥_{BMO(I)}}  (3)

for all λ > 0, where C_{0} ≥ 1, 0 < ε < 1 are absolute constants independent of N.

The idea of the proof is to induct on N. We break f up into lower-resolution pieces, apply the induction hypothesis on each piece, and then add up. It is here that the concavity of η plays a crucial role. The purpose of the ε is to ensure that the concavity of η “beats” the convexity of e^{−ελ} in this regard. Proof We induct on N. The claim is trivial for N = 0 if C_{0} is chosen sufficiently large. Now suppose that N > 0, and the claim has been proven for N − 1.

Let f be a function obeying all the hypotheses of the proposition. We may rescale so that I = [0, 1] and ∥f∥_{BMO(I)} = 1. We may assume that λ > 1, since the claim is trivial otherwise (if C_{0} is sufficiently large).

We can then split

f = aψ + f_{l} + f_{r}  (4)

where −1 ≤ a ≤ 1, ψ is the Haar wavelet ψ := χ_{[0,1/2)} − χ_{[1/2,1)}, and f_{l}, f_{r} are functions supported on [0, 1/2], [1/2, 1] respectively with mean zero. It is clear that

|{x ∈ [0, 1] : f(x) ≥ λ}| ≤ |{x ∈ [0, 1/2] : f_{l}(x) ≥ λ+a}|+|{x ∈ [0, 1/2] : f_{r}(x) ≥ λ−a}|

and that

∥f_{l}∥_{BMO([0,1/2])}, ∥f_{r}∥_{BMO([1/2,1])} ≤ ∥f∥_{BMO([0,1])} = 1.

By the induction hypothesis (and the monotonicity of η) we thus have

|\{x ∈ [0, 1] : f(x) ≥ λ\}| ≤ \frac{C_{0}}{2} (η(2∥f_{l}∥_{2}^{2})e^{−ε(λ−a)} + η(2∥f_{l}∥_{2}^{2})e^{−ε(λ+a)}).

It thus suffices to show that

\frac{η(2∥f_{l}∥_{2}^{2})e^{εa} + η(2∥f_{r}∥_{2}^{2})e^{−εa}}{2} ≤ η(∥f∥_{2}^{2}).  (5)

Note that it is already apparent that the concavity of η will come in handy. We re-arrange the left-hand side of (5) as

\frac{η(2∥f_{l}∥_{2}^{2}) + η(2∥f_{r}∥_{2}^{2})}{2} cosh(εa) + \frac{η(2∥f_{l}∥_{2}^{2}) − η(2∥f_{r}∥_{2}^{2})}{2} sinh(εa).

Estimating cosh(εa) = 1 + O(ε^{2}a^{2}) and sinh(εa) = O(εa) (recalling that ε, a = O(1)) and using (2), we can write this as

\frac{η(2∥f_{l}∥_{2}^{2}) + η(2∥f_{r}∥_{2}^{2})}{2} + O(ε^{2}a^{2}) + O(εa |∥f_{l}∥_{2}^{2} − ∥f_{r}∥_{2}^{2}|).

The three terms in (4) are clearly orthogonal, so it is not hard to see the relationship

∥f∥_{2}^{2} = a^{2} + ∥f_{l}∥_{2}^{2} + ∥f_{r}∥_{2}^{2}.

From this and (2) we see that

\frac{η(2∥f_{l}∥_{2}^{2}) + η(2∥f_{r}∥_{2}^{2})}{2} ≤ η(∥f∥_{2}^{2} − a^{2}) − c |∥f_{l}∥_{2}^{2} − ∥f_{r}∥_{2}^{2}|^{2}
≤ η(∥f∥_{2}^{2}) − ca^{2} − c |∥f_{l}∥_{2}^{2} − ∥f_{r}∥_{2}^{2}|^{2}

for some absolute constant c > 0 independent of ε. Inserting this into the previous and using the elementary estimate

a |∥f_{l}∥_{2}^{2} − ∥f_{r}∥_{2}^{2}| ≤ a^{2} + |∥f_{l}∥_{2}^{2} − ∥f_{r}∥_{2}^{2}|^{2}

we see that we can prove (5) as desired of ε is sufficiently small.

The function B(A, λ) = C_{0}η(A)e^{−ελ} is an example of a Bellman function - a function comparable to the desired bound, but carefully chosen to have a certain “concavity” property which allows the bound to be proven by induction.

In principle, one could try to find the optimum Bellman function, and thus obtain extremely sharp values of C and ε in (1). We will not attempt to do so here.

The idea of inducting on scale and exploiting concavity is a useful one for delicate endpoint estimates, especially if these estimates can be transplanted into a dyadic setting. The techniques used to perform the induction are different from the usual techniques in that one cannot afford to lose any constants in the main terms (although error terms can still be estimated fairly crudely using e.g. big-O notation as we did above). Recent examples of this philosophy are in e.g. [3], [4], [2], [1]. Non-endpoint versions of this idea also exist, in which one can now afford lose constants and logarithmic powers in the main terms, but nothing greater; see [6], [7].

One may dismiss these techniques as variants of standard theory, however they are more precise and can be used to obtain results which seem to be beyond the usual techniques such as stopping time arguments. We present an example below.

2. A somewhat localized version of John-Nirenberg

We now give a stronger, somewhat localized version of John-Nirenberg.

Theorem 2.1. Let M ≥ 0 be a fixed integer, and let N ≥ 0. Let f be a realvalued BMO function supported on a dyadic interval I with mean zero and which is constant on dyadic intervals of length 2^{−N}|I|. Then for any λ > 0, one can find a collection J = J(f, λ, I, N) of dyadic intervals J ⊂ I such that

{f > λ} ⊂ ⋃_{J∈J} J,  (6)

∥f∥_{BMO(J)} > 0 for all J ∈ J,  (7)

and

∑_{J∈J} |J|(\frac{λ}{∥f∥_{BMO(J)}} )^{M} ≤ C_{M}η(\frac{∥f∥_{2}^{2}}{|I|∥f∥_{BMO(I)}^{2}} )e^{−ελ/∥f∥_{BMO(I)}}|I|  (8)

for some absolute constants C_{M}, ε > 0.

Of course, one can easily strengthen f > λ to |f| > λ in (6) if desired.

Note that the M = 0 version this inequality easily implies the ordinary John-Nirenberg inequality. The formulations for higher M basically say that one can cover the exceptional set {f > λ} by intervals J on which the oscillation ∥f∥_{BMO(J)} is still very large on the average. This inequality will be useful in the treatment of uniform estimates for the bilinear Hilbert transform [5]. Proof We begin with the recursive construction of the collection J(f, λ, I, N). If N = 0, or more generally if ∥f∥_{BMO(I)} = 0, we set J = ∅.

Now suppose N > 0 and ∥f∥_{BMO(I)} > 0. By rescaling we may set I = [0, 1] and ∥f∥_{BMO(I)} = 1. If λ ≤ C_{1}M we set J = {I}. Here C_{1} is a large constant to be chosen later.

Now suppose λ > C_{1}M. We perform the decomposition (4) as before, and set

J(f, λ, [0, 1], N) := J_{l} ∪ J_{r}

where

J_{l} := J(f_{l}, λ − a, [0, 1/2], N − 1), J_{r} := J(f_{r}, λ + a, [1/2, 1], N − 1).

This completes the recursive construction of J(f, λ, I, N) for all f, λ, I, N.

It is easy to verify (6) and (7) inductively. Now we turn to (8). The claim is easily verified if N = 0 or ∥f∥_{BMO(I)} = 0. Now suppose inductively that N > 0 and that the claim has already been proven for N − 1.

Again we rescale set I = [0, 1] and ∥f∥_{BMO(I)} = 1. If λ ≤ C_{1}M then the claim is easy to verify (if C_{M} is chosen sufficiently large depending on C_{1}, M), so we assume λ > C_{1}M.

Let a, f_{l}, f_{r}, J_{l}, J_{r} be as above. By the inductive hypothesis and some monotonicity, we have

∑_{J∈J_{l}} |J|(\frac{λ − a}{∥f∥_{BMO(J)}} )^{M} ≤ \frac{1}{2}C_{M}η(2∥f_{l}∥_{2}^{2})e^{−ε(λ−a)}

and

∑_{J∈J_{r}} |J|(\frac{λ + a}{∥f∥_{BMO(J)}} )^{M} ≤ \frac{1}{2}C_{M}η(2∥f_{l}∥_{2}^{2})e^{−ε(λ+a)}.

It thus suffices to show that

\frac{η(2∥f_{l}∥_{2}^{2})F (a) + η(2∥f_{r}∥_{2}^{2})F (−a)}{2} ≤ η(∥f∥_{2}^{2})

where

F (a) = e^{εa}(1 − a/λ)^{M}.

As before, we rewrite the left-hand side as

\frac{η(2∥f_{l}∥_{2}^{2}) + η(2∥f_{r}∥_{2}^{2})}{2} \frac{F (a) + F (−a)}{2} + \frac{η(2∥f_{l}∥_{2}^{2}) − η(2∥f_{r}∥_{2}^{2})}{2} \frac{F (a) − F (−a)}{2}.

We can repeat the previous argument provided that we can show that

|\frac{F (a) + F (−a)}{2} − 1| ≤ Cε^{2}a^{2} + CC_{1}^{−2}a^{2}

and

|\frac{F (a) − F (−a)}{2}| ≤ Cεa + CC_{1}^{−1}a,

where the C are absolute constants independent of ε, C_{1}, M. But these are simple consequences of Taylor’s theorem and the bounds |a| ≤ 1, λ > C_{1}M.

It is tempting to try to complete the localization in (8) and conjecture that this bound can be improved (for instance) to

∑_{J∈J} |J|e^{ελ/∥f∥_{BMO(J)}} ≤ C|I|.

However, this bound is false. A counter-example is the function

f(x) = ∑_{n=1}^{K} ψ(2^{n}x)/n

supported on [0, 1/2], where ψ is the Haar wavelet. If one sets λ ∼ log K, then the collection J must contain at least one interval of the form J = [0, 2^{−n}] for some 1 ≤ n ≤ K. But on this interval we have ∥f∥_{BMO(J)} ∼ 1/n, and one can easily see that the left-hand side is therefore too large, since K is unbounded.

The author thanks John Garnett and Christoph Thiele for comments.

References

[1] T. Tao, Endpoint bilinear restriction theorems for the cone, and some sharp null form estimates, submitted, Math Z.

[2] T. Tao, J. Wright, L^{p} improving estimates for averages along curves, in preparation.

[3] F. Nazarov, S. Triel, The hunt for a Bellman function: applications to estimates for singular integral operators and to other classical problems in harmonic analysis, Algebra i Analiz 8 (1996), 32–162.

[4] F. Nazarov, S. Triel, A. Volberg, The Bellman function and two-weight inequalities for Haar multipliers, to appear, J. Amer. Math. Soc.

[5] C. Muscalu, T. Tao, C. Thiele, Multilinear operators with singular multipliers II. Uniform estimates, in preparation.

[6] T. H. Wolff, A sharp bilinear cone restriction estimate, to appear, Annals of Math.

[7] T. H. Wolff, Local smoothing type estimates on L^{p} for large p, to appear, GAFA.

Department of Mathematics, UCLA, Los Angeles, CA 90024

  E-mail address: tao@math.ucla.edu