DYADIC PRODUCT H^{1}, BMO, AND CARLESON’S COUNTEREXAMPLE

TERENCE TAO

Abstract. Some notes on the dyadic versions of the product H^{1} and BMO spaces developed by Bob Fefferman and Alice Chang, as well as Carleson’s counterexample demonstrating the necessity of non-rectangular atoms.

1. Dyadic H^{1} and atomic decomposition.

For every dyadic interval I in R, let ψ_{I} denote the Haar wavelet adapted to I

ψ_{I} = |I|^{−1/2}(χ_{I_{l}} − χ_{I_{r}})

where I_{l} and I_{r} are the left and right halves of I.

For every dyadic rectangle R = I ×J in R^{2}, let ψ_{R} denote the product Haar wavelet

ψ_{R}(x_{1}, x_{2}) = ψ_{I}(x_{1})ψ_{J}(x_{2})

adapted to R. These wavelets form an orthonormal basis of L^{2}(R^{2}):

f = ∑_{R}⟨f, ψ_{R}⟩ψ_{R}.  (1)

For any test function f, let Sf denote the dyadic square function, in vector-valued form:

Sf = (⟨f, ψ_{R}⟩ψ_{R})

where R ranges over all dyadic rectangles. From dyadic Littlewood-Paley theory we have

∥Sf∥_{p} ∼ ∥f∥_{p}  (2)

for all 1 < p < ∞. We also have the endpoint

∥Sf∥_{L^{1}(Ω)} ≲ ∥f∥_{L log^{2} L(Ω)}

for all f on Ω, where |Ω| ∼ 1. These estimates follow by decomposing S as the composition of two one-dimensional square functions, and noting from standard dyadic Littlewood-Paley theory that these functions are bounded on L^{p}, are weaktype (1, 1), and and map L log^{k+1} L(Ω) to L log^{k} L(Ω).

1991 Mathematics Subject Classification. 42B25.

For any 0 < p < ∞, we define dyadic H^{p} to be the space of all functions such that ∥Sf∥_{p} is finite, with the obvious norm. When p > 1 this is just L^{p} thanks to (2). When p = 1 the space H^{1} can be decomposed into atoms.

Definition 1.1. A dyadic H^{1} atom is any function a supported on an open set Ω with

∥a∥_{2} ≤ |Ω|^{−1/2}  (3)

and a is orthogonal to ψ_{R} for all R ⊄ Ω.

If a is a dyadic H^{1} atom on Ω, then we have

a = ∑_{R⊂Ω}⟨a, ψ_{R}⟩ψ_{R},  (4)

so we can rewrite (3) as

∑_{R⊂Ω} |⟨a, ψ_{R}⟩|^{2} ≤ |Ω|^{−1}.  (5)

From Hölder we see that ∥a∥_{1} ≤ 1. Thus every dyadic H^{1} function is in L^{1}.

Theorem 1.2. Dyadic H^{1} is the convex hull of the dyadic H^{1} atoms.

Proof Let us first prove that every dyadic H^{1} atom a is bounded in H^{1}. From (4) we see that Sa is supported on Ω, so by Hölder we have

∥Sa∥_{1} ≤ |Ω|^{1/2}∥Sa∥_{2} = |Ω|^{1/2}∥a∥_{2} ≤ 1

as desired.

Now we show that every H^{1} function f which is a finite linear combination of ψ_{R} can be broken up into atoms. The key lemma is

Lemma 1.3. For any function f, we may decompose f into orthogonal pieces f = ∑_{k} f_{k} where the f_{k} are supported on a set Ω̃_{k} of measure

|Ω̃_{k}| ∼ |{|Sf| > 2^{k}}|,  (6)

is orthogonal to all wavelets ψ_{R} such that R ⊄ Ω_{k}, and one has

∥f_{k}∥_{p} ≲ 2^{k}|Ω̃_{k}|^{1/p}  (7)

for all 1 < p < ∞.

Proof Let Ω_{k} denote the set

Ω_{k} = {|Sf| > 2^{k}}.  (8)

For every rectangle R, let k(R) denote the largest integer k such that |R ∩ Ω_{k}| ≥ \frac{1}{2}|R|, or k(R) = −∞ otherwise. Let R_{k} denote the set of all rectangles R such that k(R) = k. Then it is clear that we have the orthogonal decomposition

f = ∑_{k∈Z} f_{k}

where

f_{k} = ∑_{R∈R_{k}}⟨f, ψ_{R}⟩ψ_{R}

since f is orthogonal to ψ_{R} for k(R) = −∞.

Fix k. Let M_{S} denote the dyadic strong maximal function

M_{S}f(x) = sup_{R∋x} \frac{1}{|R|} ∫_{R} |f|

and let Ω̃_{k} denote the set

Ω̃_{k} = \{M_{S}χ_{Ω_{k}} ≥ \frac{1}{2}\}.

By construction we see that f_{k} is the desired support and moment properties. Since M_{S} is bounded on, say, L^{2}, we see from the Chebyshev inequality that (6) holds.

We now show (7); we remark that the following argument can be simplified substantially in the p = 2 case. By (2) we have

∥f_{k}∥_{p} ∼ ∥( ∑_{R∈R_{k}} |⟨f, ψ_{R}⟩|^{2}|R|^{−1}χ_{R})^{1/2}∥_{p}.

Since R ∈ R_{k}, we see that

χ_{R} ≲ M_{S}χ_{R∩(Ω̃_{k}\Ω_{k+1})}.

By the Fefferman-Stein vector-valued maximal inequality (applied once in either cardinal direction) we thus have

∥f_{k}∥_{p} ≲ ∥( ∑_{R∈R_{k}} |⟨f, ψ_{R}⟩|^{2}|R|^{−1}χ_{R∩(Ω̃\Ω_{k+1})})^{1/2}∥_{p}.

This is bounded by

∥Sf∥_{L^{p}(Ω̃_{k}\Ω_{k+1})} ≲ 2^{k}|Ω̃_{k}| ≲ 2^{k}|Ω_{k}|

as desired.

Using this lemma with p = 2, we see that each f_{k} is equal to O(2^{k}|{|Sf| > 2^{k}}|) times an H^{1} atom. The claim follows since Sf is in L^{1}.

By modifying the above proof, we can change the estimate (3) in the definition of atom to ∥a∥_{p} ≲ |Ω|^{−1/p′} for any 1 < p < ∞, or even to

∫_{Ω} |a(x)| log^{2}(2 + |Ω||a(x)|) ≤ 1.

Note that for p > 2 one needs the full power of (7).

2. Calderón-Zygmund decomposition

Now we give a Calderón-Zygmund decomposition of L^{p} functions into an L^{2} function and dyadic H^{1} function; this allows one to interpolate between L^{2} and dyadic H^{1}.

Proposition 2.1. Let 1 < p < 2, f be a function in L^{p}, and let α > 0. Then one can decompose f = g + b where

∥g∥_{2} ≲ α^{1−p/2}∥f∥_{p}^{p/2}  (9)

and

∥b∥_{H^{1}} ≲ α^{1−p}∥f∥_{p}^{p}.  (10)

Proof By scaling and linearity we may make α = 1 and ∥f∥_{p} = 1. Decompose f = ∑_{k} f_{k} by Lemma 1.3, and define

g = ∑_{k≤0} f_{k}, b = ∑_{k>0} f_{k}.

To prove (9), we use the orthogonality of the f_{k} and (7) to see that

∥g∥_{2}^{2} ≲ ∑_{k≤0} 2^{2k}|{|Sf| > 2^{k}}|

≲ ∑_{k} 2^{pk}|{|Sf| > 2^{k}}|

∼ ∥Sf∥_{p}^{p}

∼ ∥f∥_{p}^{p} = 1.

Similarly, to prove (10) we have

∥b∥_{H^{1}} ≲ ∑_{k>0} 2^{k}|{|Sf| > 2^{k}}|

≲ ∑_{k} 2^{pk}|{|Sf| > 2^{k}}|

∼ ∥Sf∥_{p}^{p}

∼ ∥f∥_{p}^{p} = 1.

Corollary 2.2. Let T be a linear operator which maps L^{2} to L^{2} and H^{1} to L^{1}. Then it maps L^{p} to L^{p} for all 1 < p < 2.

Proof It suffices to show that T is weak-type (p, p) for all such p, i.e. that

|{|T f| ≳ α}| ≲ α^{−p}∥f∥_{p}^{p}.

By splitting f = g + b as before it suffices to show that

|{|T g| ≳ α}| ≲ α^{−p}∥f∥_{p}^{p}.

and

|{|T b| ≳ α}| ≲ α^{−p}∥f∥_{p}^{p}.

The former estimate follows from Chebyshev, the boundedness of T on L^{2}, and (9). The latter follows from Chebyshev, the fact that T maps H^{1} to L^{1}, and (10).

3. Dyadic H^{p} for p < 1

We now perform the same atomic decomposition as before, but for p < 1.

Definition 3.1. A dyadic H^{p} atom is any function a supported on an open set Ω with

∥a∥_{2} ≤ |Ω|^{\frac{1}{2} − \frac{1}{p}}

and a is orthogonal to ψ_{R} for all R ⊄ Ω.

By repeating the proof of the first part of Theorem 1.2 we see that every dyadic H^{p} atom is in H^{p}. More generally, we have

∥ ∑_{Ω} λ_{Ω}a_{Ω}∥_{H^{p}} ≲ (∑_{Ω} λ_{Ω}^{p})^{1/p}

for any non-negative co-efficients λ_{Ω}; this follows from the previous observation and the elementary inequality

∥ ∑_{Ω} f_{Ω}∥_{p} ≤ (∑_{Ω} ∥f_{Ω}∥_{p}^{p})^{1/p}.

The converse to the above observation is also true, and is an easy modification of the proof of Theorem 1.2.

Theorem 3.2. Let f ∈ H^{p} be the finite linear combination of ψ_{R}. Then one can decompose f = ∑_{Ω} λ_{Ω}a_{Ω} where

(∑_{Ω} λ_{Ω}^{p})^{1/p} ≲ ∥f∥_{H^{p}}.

4. BMO and duality

Define dyadic BMO to be the dual of dyadic H^{1}, so that f is in dyadic BMO if and only if |⟨f, a⟩| ≲ 1 for all dyadic H^{1} atoms a. In other words, we have

|⟨f, ∑_{R⊂Ω}⟨a, ψ_{R}⟩ψ_{R}⟩| ≲ 1

for all open sets Ω and all a obeying (3)

∥ ∑_{R⊂Ω}⟨f, ψ_{R}⟩ψ_{R}∥_{2} ≲ |Ω|^{1/2}.  (11)

By orthogonality we may rewrite this as

∑_{R⊂Ω} |⟨f, ψ_{R}⟩|^{2} ≲ |Ω|.  (12)

If we replace (3) by L^{p} or L log^{2} L variants as before, we obtain

Proposition 4.1. (John-Nirenberg inequality) If f is in the unit ball of dyadic BMO, then for all 1 < p < ∞ we have

∥ ∑_{R⊂Ω}⟨f, ψ_{R}⟩ψ_{R}∥_{p} ≲ |Ω|^{1/p}

and

∫_{Ω} exp(| ∑_{R⊂Ω}⟨f, ψ_{R}⟩ψ_{R}|^{2}) ≲ |Ω|.

This is easiest proven by first scaling to the case |Ω| ∼ 1.

5. Subtracting off the mean

We can write the characterization (11) of BMO as

∥f − ∑_{R⊄Ω}⟨f, ψ_{R}⟩ψ_{R}∥_{L^{2}(Ω)} ≲ |Ω|^{1/2}

for all Ω.

If f is in BMO, the coefficients ⟨f, ψ_{R}⟩ obey the relationship

∑_{R:|R∩Ω|≥2^{−k}|R|} |⟨f, ψ_{R}⟩|^{2} ≲ k2^{k}|Ω|.

This is because the left-hand side is bounded by

∑_{R⊂{M_{S}χ_{Ω}≳2^{−k}}} |⟨f, ψ_{R}⟩|^{2}

which by (12) is bounded by

O(|{M_{S}χ_{Ω} ≳ 2^{−k}}|).

The claim then follows by observing that M_{S} is bounded by the composition of two one-dimensional Hardy-Littlewood maximal functions and thus maps L log L to L^{1} on sets of bounded measure. (The case when |Ω| ≁ 1 can then be handled by scaling).

We now consider analogues of the above characterization of dyadic BMO, when ψ_{R} is replaced by a more general function. This is the dyadic analogue of the BMO_{(b)} notion in Chang-Fefferman.

Definition 5.1. A function ψ̃_{R} is said to be a dyadic elementary particle on a rectangle R = I × J if ψ̃_{R} has mean zero on every horizontal and vertical line, and we have

|(|I|∂_{x_{1}})^{i}(|J|∂_{x_{2}})^{j}ψ̃_{R}(x)| ≲ |R|^{−1/2}

for all i, j = 0, 1 and all x in any one of the four quarters of R.

Proposition 5.2. Let f be a function such that for every open set Ω one can find co-efficients c_{R} and dyadic elementary particles ψ̃_{R} on R for each R ⊄ Ω such that

∥f − ∑_{R⊄Ω} c_{R}ψ̃_{R}∥_{L^{2}(Ω)} ≲ |Ω|^{1/2}

and

∑_{R:|R∩Ω|≥2^{−k}|R|} |c_{R}|^{2} ≲ k2^{k}|Ω|.  (13)

Then f is in dyadic BMO.

Proof It suffices to show that |⟨f, a⟩| ≲ 1 for every dyadic H^{1} atom a on an open set Ω. By scaling we may assume |Ω| ∼ 1.

Fix a, Ω, and let c_{R}, ψ̃_{R} be as above. By Cauchy-Schwarz and (3) we have

|⟨f − ∑_{R⊄Ω} c_{R}ψ̃_{R}, a⟩| ≲ 1

so it suffices to show that

|⟨ ∑_{R⊄Ω} c_{R}ψ̃_{R}, a⟩| ≲ 1.

It suffices to show that

∑_{k>0} ∑_{R∈R_{k}} |c_{R}||⟨a, ψ̃_{R}⟩| ≲ 1

where

R_{k} = {R ⊄ Ω : |R ∩ Ω| ∼ 2^{−k}|R|}.

We can split the left-hand side using (4) as

∑_{k>0} ∑_{R∈R_{k}} ∑_{R′⊂Ω} |c_{R}||⟨a, ψ_{R′}⟩||⟨ψ_{R′}, ψ̃_{R}⟩|.

Let’s now compute the inner product of ψ_{R′} and ψ̃_{R}. It turns out that this inner product is zero unless R′ is a subset of R; this is because of the moment conditions on ψ̃_{R} and the fact that ψ_{R′} is constant on each of its four quadrants. When R′ ⊂ R one can use the moment conditions on ψ_{R′} and smoothness conditions on ψ̃_{R} to obtain

|⟨ψ_{R′}, ψ̃_{R}⟩| ≲ \frac{|R′|^{3/2}}{|R|^{3/2}} .

It thus suffices to show that

∑_{k>0} ∑_{R∈R_{k}} ∑_{R′⊂Ω∩R} |c_{R}||⟨a, ψ_{R′}⟩|\frac{|R′|^{3/2}}{|R|^{3/2}} .  (14)

We now use Schur’s test. Using the above estimate and the elementary inequality ab ≲ a^{2} + b^{2}, we can estimate the left-hand side of (14) by the sum of

∑_{k>0} ∑_{R∈R_{k}} ∑_{R′⊂Ω∩R} |c_{R}|^{2} \frac{|R′|^{3/2}}{|R|^{3/2}}  (15)

and

∑_{k>0} ∑_{R∈R_{k}} ∑_{R′⊂Ω∩R} |⟨a, ψ_{R′}⟩|^{2} \frac{|R′|^{3/2}}{|R|^{3/2}}  (16)

To estimate (15) satisfactorily, it suffices by (13) to show that

∑_{R′⊂Ω∩R} \frac{|R′|^{3/2}}{|R|^{3/2}} ≲ k2^{−3k/2}

for each R ∈ R^{k}. We write the left-hand side as

|R|^{−3/2} ∫_{R∩Ω} ∑_{R′⊂Ω∩R:x∈R′} |R′|^{1/2} dx.

Since the sum contains at most one rectangle of each size, and no rectangle R′ can exceed ≲ 2^{−k}|R| in area, the sum is O(k2^{−k/2}|R|^{1/2}), and the claim then follows since R ∈ R_{k}.

To estimate (16) satisfactorily, it suffices by (5) to show that

∑_{R∈R_{k}:R⊃R′} \frac{|R′|^{3/2}}{|R|^{3/2}} ≲ k2^{−3k/2}

for all R′ ⊂ Ω. However, this follows by observing that the sum contains at most one rectangle of each size, and R must have area at least ≳ 2^{k}|R′|.

6. Singular integrals

Let H_{x_{1}}, H_{x_{2}} denote the one-dimensional Hilbert transform in the x_{1}, x_{2} directions.

Proposition 6.1. Let f be a dyadic H^{1} function. Then f, H_{x_{1}}f, H_{x_{2}}f, and H_{x_{1}}H_{x_{2}}f are in L^{1}.

Proof We’ll just prove the last claim, as it is the most difficult. We only need to test this on atoms, so we need to show that

∥H_{x_{1}}H_{x_{2}}a∥_{1} ≲ 1

for all H^{1} atoms a on an open set Ω. By (4) and (5) it suffices to show that

∥ ∑_{R⊂Ω} c_{R}H_{x_{1}}H_{x_{2}}ψ_{R}∥_{1} ≲ |Ω|^{1/2}( ∑_{R⊂Ω} |c_{R}|^{2})^{1/2}  (17)

for any co-efficients c_{R}.

Fix Ω; we will always implicitly assume R ⊂ Ω in the sequel. We may normalize ∑_{R} |c_{R}|^{2} = 1 and |Ω| ∼ 1.

Decompose H_{x_{1}} = ∑_{k} H_{x_{1}}^{k} where the kernels of H_{x_{1}}^{k} kernels are odd and supported on the region |x_{1}| ∼ 2^{k}, and similarly for H_{x_{2}}. We may write the left-hand side of (17) as

∥ ∑_{s} ∑_{s′} F_{s,s′}∥_{1}  (18)

where

F_{s,s′} = ∑_{R} c_{R}H_{x_{1}}^{i_{R}+s}H_{x_{2}}^{j_{R}+s′}ψ_{R}

and R has side-lengths 2^{i_{R}} × 2^{j_{R}}.

We will need the following L^{2} estimates on F_{s,s′}.

Lemma 6.2. We have

∥F_{s,s′}∥_{2} ≲ 2^{−(s+s′)}  (19)
∥ ∑_{s} F_{s,s′}∥_{2} ≲ 2^{−s′}  (20)
∥ ∑_{s′} F_{s,s′}∥_{2} ≲ 2^{−s}  (21)
∥ ∑_{s} ∑_{s′} F_{s,s′}∥_{2} ≲ 1  (22)

for all integers s, s′.

Proof We prove (19); the other three estimates are proven similarly, noting that once one performs a summation in s then the H_{x_{1}} operator, which is bounded on L^{2} can be factored out, and similarly for the s′ summation.

We can write

ψ_{R} = |R|^{−1/2}χ_{R^{00}} ∗ (δ_{0} − δ_{2^{i_{R}−1}e_{1}}) ∗ (δ_{0} − δ_{2^{j_{R}−1}e_{2}}),

where δ_{x} is the Dirac delta mass at x, and R^{00} is the lower left quadrant of R. We can then rewrite the left-hand side of the desired inequality as

∥ ∑_{i} ∑_{j} A_{i}B_{j}f_{i,j}∥_{2}^{2}

where A_{i}f = H_{x_{1}}^{i+s}((δ_{0} − δ_{2^{i−1}e_{1}}) ∗ f)

B_{j}f = H_{x_{2}}^{j+s′}((δ_{0} − δ_{2^{j−1}e_{2}}) ∗ f)

f_{i,j} = ∑_{R:i_{R}=i,j_{R}=j} c_{R}|R|^{−1/2}χ_{R^{00}}.

The A_{i} are Fourier multipliers whose symbol a_{i}(ξ) obeys the estimates

|a_{i}(ξ)| ≲ 2^{−s} min(\frac{|ξ|}{2^{i+s}} , \frac{2^{i+s}}{|ξ|}).

By Plancherel and Cauchy-Schwarz we thus have

∥ ∑_{i} ∑_{j} A_{i}B_{j}f_{i,j}∥_{2}^{2} ≲ 2^{−2s} ∑_{i} ∥ ∑_{j} B_{j}f_{i,j}∥_{2}^{2}.

By similar reasoning involving the B_{j} we thus have

∥ ∑_{i} ∑_{j} A_{i}B_{j}f_{i,j}∥_{2}^{2} ≲ 2^{−2(s+s′)} ∑_{i} ∑_{j} ∥f_{i,j}∥_{2}^{2}.

However, the R in the definition of f_{i,j} are disjoint, so we have

∥f_{i,j}∥_{2}^{2} ≲ ∑_{R:i_{R}=i,j_{R}=j} |c_{R}|^{2}.

The claim follows.

Now let us prove (18). We first deal with the contribution when s, s′ > 0. Since F_{s,s′} is supported on the set {M_{S}χ_{Ω} ≳ 2^{−s−s′}}, which has measure O(2^{s+s′}(s+s′)), we thus have

∥ ∑_{s,s′>0} F_{s,s′}∥_{1} ≲ ∑_{s,s′>0} 2^{(s+s′)/2}(s + s′)^{1/2}∥F_{s,s′}∥_{2}.

This contribution is acceptable by (19).

Now let us look at the contribution when s ≤ 0 and s′ > 0. For each s′, the function ∑_{s≤0} F_{s,s′} is supported on the set {M_{S}χ_{Ω} ≳ 2^{−s′}}, which has measure O(2^{s′}s′). We thus have

∥ ∑_{s≤0} ∑_{s′>0} F_{s,s′}∥_{1} ≲ ∑_{s′>0} 2^{s′/2}(s′)^{1/2}∥ ∑_{s≤0} F_{s,s′}∥_{2}.

This contribution is acceptable by (20) and (19).

The case s > 0 and s′ ≤ 0 is treated similarly, so it remains to consider the case s, s′ ≤ 0. This contribution is supported on the set {M_{S}χ_{Ω} ≳ 1}, which has measure O(1). Thus

∥ ∑_{s,s′≤0} F_{s,s′}∥_{1} ≲ ∥ ∑_{s,s′≤0} F_{s,s′}∥_{2}.

This contribution is acceptable by (19)-(22).

For continuous H^{1}, the above proposition can be reversed, but I do not know of an analogue for the dyadic setting.

7. Carleson’s counterexample

Recall that a function f is in dyadic BMO if and only if (12) is true for all Ω. Based on analogy with the one-parameter theory, it is natural to ask whether one can get away with only requiring (12) when Ω is a dyadic rectangle. Unfortunately, the following argument of Carleson shows that this is impossible.

Theorem 7.1. There exists a function f which obeys (12) for all dyadic rectangles Ω, but does not obey (12) in general.

To prove this theorem we first need some notation. Define a quilt to be a finite collection R of dyadic rectangles in the unit square [0, 1]^{2} such that

∑_{R∈R} |R| = 1  (23)

and such that

∑_{R∈R:R⊂Q} |R| ≤ |Q|  (24)

for every dyadic rectangle Q ⊂ [0, 1]^{2}.

Define the area of a quilt R to be the quantity | ⋃_{R∈R} R|. The singleton set {[0, 1]^{2}} is clearly a quilt with area 1.

For any quilt R, the function

f = ∑_{R∈R} |R|^{1/2}ψ_{R}

clearly obeys (12) for every dyadic rectangle Ω. However, it will fail to obey (12) for Ω = ⋃_{R∈R} R as soon as we prove

Lemma 7.2. There exists quilts of arbitrarily small area.

Proof Let R be a quilt of area σ. We shall construct a new quilt R′ of area σ − \frac{σ^{2}}{4} . The claim then follows by starting with the singleton quilt and iterating this construction as often as one pleases.

Fix σ, R. Let N ≫ 1 be such that every rectangle in R has length and width at least 2^{−N}. For every integer 0 ≤ j < 2^{N}, let A_{j}^{1}, A_{j}^{2} be the affine transformations

A_{j}^{1}(x, y) = (\frac{j}{2^{N}} + \frac{x}{2^{N+1}} , y)

A_{j}^{2}(x, y) = (x, \frac{j}{2^{N}} + \frac{y}{2^{N+1}} ).

and define R′ by

R′ = ⋃_{i=1}^{2} ⋃_{j=0}^{2^{N} −1} A_{j}^{i}(R).

It is easy to verify that the collections A_{j}^{i}(R) of rectangles are disjoint, and that (23) holds for R′. Now we verify (24). Suppose Q has side-lengths 2^{−k_{1}} × 2^{−k_{2}}. If k_{1} > N then there is at most one collection A_{j}^{1}(R) which contributes to (24), and none of the A_{j}^{2}(R) contribute. The claim follows by rescaling. Similarly if k_{2} > N. Now suppose that k_{1}, k_{2} ≤ N. Then there are at most 2^{N−k_{1}} collections A_{j}^{1}(R) which contribute to (24), and each contributes at most 2^{−N−1−k_{2}} by a scaling argument. A similar statement holds for the A_{j}^{2}(R); adding up the contributions one obtains the result.

12 TERENCE TAO

It remains to compute the area of R′. Since

| ⋃_{j=0}^{2^{N} −1} ⋃_{R∈A_{j}^{1}(R)} R| = ∑_{j=0}^{2^{N} −1} | ⋃_{R∈A_{j}^{1}(R)} R| = ∑_{j=0}^{2^{N} −1} 2^{−N−1}σ = \frac{σ}{2}

and similarly for the A_{j}^{2}(R), it suffices by the inclusion-exclusion principle to show that

|( ⋃_{j_{1}=0}^{2^{N} −1} ⋃_{R∈A_{j_{1}}^{1} (R)} R) ∩ ( ⋃_{j_{2}=0}^{2^{N} −1} ⋃_{R∈A_{j_{2}}^{2} (R)} R)| = \frac{σ^{2}}{4} .

We can rewrite the left-hand side as

∑_{j_{1}=0}^{2^{N} −1} ∑_{j_{2}=0}^{2^{N} −1} |(I_{j_{1}} × I_{j_{2}} ∩ ⋃_{R∈A_{j_{1}}^{1} (R)} R) ∩ (I_{j_{1}} × I_{j_{2}} ∩ ⋃_{R∈A_{j_{2}}^{2} (R)} R)|,

where I_{j} = [j/2^{N}, (j + \frac{1}{2})/2^{N}]. The first set in parentheses is of the form E_{j_{1},j_{2}} × I_{j_{2}}, and the second set is of the form I_{j_{1}} × F_{j_{1},j_{2}}, for some sets E_{j_{1},j_{2}}, F_{j_{1},j_{2}} satisfying

∑_{j_{2}} |E_{j_{1},j_{2}}| = ∑_{j_{1}} |F_{j_{1},j_{2}}| = σ/2

Since the previous expression simplifies to

∑_{j_{1}=0}^{2^{N} −1} ∑_{j_{2}=0}^{2^{N} −1} |E_{j_{1},j_{2}}||F_{j_{1},j_{2}}|,

the claim then follows. The author thanks Jim Wright for providing a copy of Carleson’s counterexample.

Department of Mathematics, UCLA, Los Angeles CA 90095-1555

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