THE LINDENSTRAUSS MAXIMAL INEQUALITY

TERENCE TAO

Abstract. We give an abstract version of a maximal inequality of Lindenstrauss, which improves upon the classical Hardy-Littlewood maximal inequality.

1. Introduction

Let (X, B_{X}, μ_{X}) and (Y, B_{Y} , μ_{Y}) be σ-finite measure spaces, let R be a totally ordered set which is at most countable, and for each r ∈ R let K_{r} : X × Y → R^{+} be a non-negative measurable kernel. We can then define the operator T_{r} on any non-negative measurable function f : X → R^{+} by

T_{r}f(y) := ∫_{X} K_{r}(x, y)f(x) dμ_{X}(x)

and the maximal function

T_{R}f(y) := sup_{r∈R} T_{r}f(y).

In this note we are interested in the problem of determining conditions on K_{n} for which we have the weak-type (1, 1) inequality

μ_{Y} (\{y ∈ Y : T_{R}f(y) > λ\}) ≤ \frac{C}{λ} ∫_{X} f(x) dμ_{X}(x)  (1)

for all non-negative measurable f : X → R^{+} and λ > 0, and some constant C > 0.

The classical Hardy-Littlewood maximal inequality corresponds to the case when X = Y is a homogeneous space, R ⊂ R^{+} is a countable set of radii, and K_{r}(x, y) := 1_{B(y,r)}(x)/μ(B(y, r)). Then, as is well known, (1) holds for some constant C which depends on the doubling constant of the homogeneous space. If one makes K_{r} a little smaller, namely K_{r}(x, y) := 1_{B(y,r)}(x)/μ(B(y, 3r)), then the standard proof of the Hardy-Littlewood inequality shows that (1) now holds with a constant of 1. (We will reprove this result shortly; in fact we can even replace B(y, 3r) with B(y, 2r) by being more careful.)

Observe in the above example that K_{r}(x, y) is supported on a “narrow” subset of X × Y, with the support getting smaller as r decreases. We shall now consider more general families of kernels with narrow support. It is convenient to introduce the following “balls” associated to any family (K_{r})_{r∈R} of kernels as follows. Given any y ∈ Y and r ∈ R, define B_{r}(y) := {x ∈ X : K_{r}(x, y) ≠ 0}, while for x ∈ X and r ∈ R define B_{r}^{∗}(x) := {y ∈ Y : K_{r}(x, y) ≠ 0}. Observe that B_{r}(y), B_{r}^{∗}(x) are measurable for almost every x, y. We also define the dilated ball

B_{≤r}^{∗}B_{r}(y) := ⋃_{x∈B_{r}(y)} ⋃_{r′∈R:r′≤r} B_{r′}^{∗}(x);

this is also measurable for almost every y.

We can then obtain the following abstract Hardy-Littlewood maximal inequality:

Theorem 1.1 (Hardy-Littlewood maximal inequality). Let (K_{r})_{r∈R} be a family of non-negative kernels which obeys the bound

K_{r}(x, y) ≤ \frac{1}{μ_{Y} (B_{≤r}^{∗}B_{r}(y))}  (2)

for almost every x, y and all r ∈ R. (We adopt the usual conventions that 1/0 = +∞ and 1/ + ∞ = 0.) Then (1) holds with constant C = 1.

Proof We shall give a slight rearrangement of the usual Vitali-type covering lemma argument, in order to motivate a variant of this theorem below.

By monotone convergence we may take R to be finite, and X, Y to have finite measure. and by dividing f by λ we may normalise λ = 1. We now induct on the cardinality |R| of R. The case |R| = 0 is vacuous, so suppose |R| > 0 and the claim has already been proven for smaller values of |R|. Let r be the largest value in R, and consider the set

E := {y ∈ Y : T_{r}f(y) > 1}  (3)

thus

{y ∈ Y : T_{R}f(y) > 1} = E ∪ {y ∈ Y \E : T_{R\{r}}f(y) > 1}.

By deleting a set of measure zero we may assume that μ_{X}(B_{r}(y)) > 0 for all y ∈ E. Then by the kernel bounds on K_{r}, we see that for every y ∈ E we have

∫_{B_{r}(y)} f(x) dμ_{X}(x) ≥ μ_{Y} (B_{≤r}^{∗}B_{r}(y)).

Now let Σ ⊂ E be a subset of E such that the sets B_{r}(y) for y ∈ Σ are disjoint, and such that Σ is maximal with respect to set inclusion (such a set exists by Zorn’s lemma); since all the sets B_{r}(y) have positive measure we see that Σ is at most countable. Summing the previous inequality over all y ∈ Σ we conclude

∫_{F} f(x) dμ_{X}(x) ≥ μ_{Y} (E′)  (4)

where

E′ := ⋃_{y∈Σ} B_{≤r}^{∗}B_{r}(y)

and

F := ⋃_{y∈Σ} B_{r}(y).

By the maximality of Σ, we see that for all y′ ∈ E that B_{r}(y) and B_{r}(y′) intersect for some y ∈ Σ. In particular this implies that

E ⊂ E′.

Thus

μ_{Y} ({y ∈ Y : T_{R}f(y) > 1}) ≤ μ_{Y} (E′) + μ_{Y} ({y ∈ Y \E′ : T_{R\{r}}f(y) > 1}. (5)

Now from construction of E′ we observe that for y ∈ Y \E′ that

T_{R\{r}}f(y) = T_{R\{r}}(f1_{X\F})(y)

and hence by induction hypothesis

μ_{Y} ({y ∈ Y \E′ : T_{R\{r}}f(y) > 1} ≤ ∫_{X\F} f(x) dμ_{X}(x).

Combining this with (4), (5) we conclude that

μ_{Y} ({y ∈ Y : T_{R}f(y) > 1}) ≤ ∫_{X} f(x) dx

closing the induction as desired.

For instance, if X = Y is a metric measure space, R ⊂ R^{+}, and

K_{r}(x, y) := \frac{1}{μ_{X}(B̃(x, r))}1_{d(x,y)≤r}

where B(x, r) ⊂ B̃(x, r) ⊂ B(x, 2r) is the enlarged ball

B̃(x, r) := {x′ ∈ X : d(x, y), d(x′, y) ≤ r for some y ∈ X}

then the above theorem shows that the Hardy-Littlewood maximal operator

Mf(x) := sup_{r∈R} \frac{1}{μ_{X}(B̃(x, r))} ∫_{B(x,r)} |f(y)| dy

is weak-type (1, 1) with operator norm at most 1. In the case that d is an ultrametric (so that d(x, x′) ≤ max(d(x, y), d(x′, y)) for all x, x′, y) we have B̃(x, r) = B(x, r); this for instance gives the weak-type (1, 1) of the standard dyadic maximal operator on cubes, with the usual constant of 1.

In some situations this result is unsatisfactory because of failure of doubling, which may cause B̃(x, r) to be unacceptably large compared with B(x, r). However, there is a refinement of the Hardy-Littlewood maximal inequality due to Lindenstrauss which can remedy this. Roughly speaking, it allows one to replace the dilated balls B_{≤r}^{∗}B_{r}(y) by the smaller set

B_{<r}^{∗}B_{r}(y) := ⋃_{x∈B_{r}(y)} ⋃_{r′∈R:r′<r} B_{r′}^{∗}(x).

In the case where the set of radii R are lacunary, the replacement of the constraint r′ ≤ r with r′ < r can be a significant saving, and in particular one can often recover a doubling-type condition.

Theorem 1.2 (Lindenstrauss maximal inequality). Let (K_{r})_{r∈R} be a family of non-negative kernels which obeys the bounds

K_{r}(x, y) ≤ \frac{1}{μ_{Y} (B_{<r}^{∗}B_{r}(y))}  (6)

for almost every x, y and all r ∈ R, and the kernel bound

∫_{Y} K_{r}(x, y) dμ_{Y} (y) ≤ 1  (7)

for almost every y and all r ∈ R. Then (1) holds with constant C = \frac{2}{1−e^{−1}} .

We remark that the hypothesis (7) is very natural, as it asserts that T_{r} is of strongtype (1, 1) with operator norm at most 1. It is not hard to see that both (7) and (6) are consequences of (2), so the Lindenstrauss maximal inequality is stronger than the Hardy-Littlewood maximal inequality except for the small loss in the constant C. One can optimise the constant \frac{2}{1−e^{−1}} a little bit more but we will not attempt to do so here.

Proof As before we can take R to be finite, X, Y to have finite measure, f to be bounded, and λ = 1. For technical reasons we shall need to assume the existence of an ε > 0 such that μ_{X}(B_{r}(y)) > ε and μ_{Y} (B_{r}^{∗}(x)) > ε for all x, y, r; one can reduce to this case by adding a small amount of measure to X and Y and increasing K_{r} slightly, and then sending ε → 0 and using monotone convergence; we omit the details.

We again induct on R, thus |R| ≥ 1 and we assume the claim has already been proven for smaller |R|. We take r to be the largest value of R, and define the set E by (3) as before. Thus

∫_{Y} K_{r}(x, y)f(x) dμ_{X}(x) ≥ 1  (8)

for all y ∈ E. In particular this forces B_{r}(y) to have positive measure.

We now deviate from the previous argument by selecting Σ randomly rather than greedily. Indeed, we apply a Poisson process with intensity p(y) for every y ∈ E, where p : E → R^{+} is a bounded strictly positive density function to be chosen later. This creates a random subset^{1} Σ ⊂ E with the property that for any non-negative measurable w : E → R^{+}, the quantity ∑_{y∈Σ} w(y) is a Poisson random variable with expectation ∫_{E} w(y)p(y) dμ_{Y} (y). In particular (setting w = 1_{E}) we see that Σ is almost surely finite.

Now we define the random sets

E′ := ⋃_{y∈Σ} B_{<r}^{∗}B_{r}(y)

and

F := ⋃_{y∈Σ} B_{r}(y).

Observe that

μ_{Y} ({T_{R}f > 1}) ≤ μ(E) + μ(E′) + μ_{Y} ({y ∉ E′ : T_{R\{r}}f(y) > 1}).

1If E contains atoms, then Σ may contain multiplicity, i.e. it is a multiset rather than a set.

One way to create Σ is to let N be a Poisson random variable with expectation P := ∫_{E} p(y)dμ_{Y} (y)

and then let Σ = {y_{1}, …, y_{N} } where y_{1}, …, y_{N} are iid elements of E chosen using the probability distribution p(y)dμ_{Y} (y)/P.

LINDENSTRAUSS MAXIMAL INEQUALITY 5

But as before we have for y ∉ E′

T_{R\{r}}f(y) = T_{R\{r}}(f1_{X\F} (y))

so by induction hypothesis

μ_{Y} ({y ∉ E′ : T_{R\{r}}f(y) > 1}) ≤ C ∫_{X\F} f(x) dx.

where C = \frac{2}{1−e^{−1}} . We take expectations and conclude

μ_{Y} ({T_{∗}f > 1}) ≤ μ(E) + Eμ(E′) + CE ∫_{X\F} f(x) dx.

To close the induction it will thus suffice to show that

μ(E) + Eμ(E′) ≤ \frac{2}{1 − e^{−1}} E ∫_{F} f(x) dx.

Now using the weight function w(x) := 1/p(x) we see that

μ(E) = E ∑_{y∈Σ} \frac{1}{p(x)}

while from definition of E′ we have

Eμ(E′) ≤ E ∑_{y∈Σ} μ_{Y} (B_{<r}^{∗}B_{r}(y)).

It is now natural to set

p(x) := \frac{1}{μ_{Y} (B_{<r}^{∗}B_{r}(y))}

(note that p is bounded by 1/ε) and so

μ(E) + Eμ(E′) ≤ 2E ∑_{y∈Σ} μ_{Y} (B_{<r}^{∗}B_{r}(y)).

Applying (8) followed by Fubini’s theorem we conclude

μ(E) + Eμ(E′) ≤ 2E ∑_{y∈Σ} μ_{Y} (B_{<r}^{∗}B_{r}(y)) ∫_{B_{r}(y)} K_{r}(x, y)f(x) dμ_{X}(x)

= 2 ∫_{X}[E ∑_{y∈Σ} μ_{Y} (B_{<r}^{∗}B_{r}(y))K_{r}(x, y)]f(x) dμ_{X}(x).

Meanwhile, we have

E ∫_{F} f(x) dx = ∫_{X}[E1_{F} (x)]f(x) dx.

Thus it will suffice to show that

E ∑_{y∈Σ} μ_{Y} (B_{<r}^{∗}B_{r}(y))K_{r}(x, y) ≤ \frac{1}{1 − e^{−1}} E1_{F} (x)

for almost every x ∈ X. Fix x. The left-hand side is just

∫_{E} K_{r}(x, y) dμ_{Y} (y).

Applying (6), (7), we can bound this by min(1, α), where

α := ∫_{E∩B_{r}^{∗}(x)} \frac{1}{μ_{Y} (B_{<r}^{∗}B_{r}(y))} dμ_{Y} (y).

On the other hand, observe that x ∈ F if and only if |Σ ∩ E ∩ B_{r}^{∗}(x)| ≥ 1. But |Σ ∩ B_{r}^{∗}(x)| is a Poisson random variable with expectation α. Thus

E1_{F} (x) = 1 − e^{−α}.

The claim now follows from the elementary inequality

min(1, α) ≤ \frac{1 − e^{−α}}{1 − e^{−1}}

for all α > 0 (this follows from the concavity of 1 − e^{−x} for 0 ≤ x ≤ 1, and the monotonicity of 1 − e^{−x} for x ≥ 1).

The above theorem is an abstraction of a slightly more concrete inequality of Lindenstrauss [1], which is as follows. Let G be a group with a bi-invariant Haar measure^{2} μ, and let F_{1}, …, F_{n} be a finite tempered Følner sequence, in the sense that

μ( ⋃_{r′<r} F_{r′}^{−1}F_{r}) ≤ Aμ(F_{r})

for some A ≥ 1 and all 1 ≤ r ≤ n. Then by applying the above theorem to X = Y = G, R := {1, …, n} and the kernels

K_{r}(x, y) := \frac{1}{Aμ(F_{r})}1_{y^{−1}x∈F_{r}}

we conclude that the maximal operator

Mf(y) := sup_{1≤r≤n} \frac{1}{μ(F_{r})} ∫_{F_{r}} f(gy) dμ(g)

is of weak-type (1, 1) with norm at most \frac{2}{1−e^{−1}} A.

The inequality also shows that on a metric measure space X, the maximal operator

M̃f(x) := sup_{r∈R} \frac{1}{V (x)} ∫_{B(x,r)} f(y) dμ(y)

is of weak-type (1, 1) with norm at most \frac{2}{1−e^{−1}} , where V (x) is the quantity

V (x) := max(μ({x′ ∈ X : d(x′, y) ≤ r′ for some r′ < r, y ∈ B(x, r)}), sup_{y∈B(x,r)} μ(B(y, r))).

One can gauge the strength of these inequalities by looking at the usual Euclidean Hardy-Littlewood inequality (with X = R^{d}) in the high-dimensional limit. The usual Hardy-Littlewood argument gives a weak-type (1,1) bound of 2^{d}. If one restricts the radii to powers of d, then the Lindenstrauss argument gives a weaktype (1,1) bound of O(1); one can then refine this to powers of d^{1/d log d} to get a bound of O(d log d). But all balls are comparable in measure to a ball of radius equal to a power of d^{1/d log d}. We thus see that the full Hardy-Littlewood maximal inequality is also weak-type (1, 1) with a bound of O(d log d). This is not as strong as the bound of O(d) established by Stein and Stömberg [2] but is not too far off.

2It is also possible to tackle the non-unimodular case without difficulty.

The author thanks Zubin Guatam for explaining the argument of Lindenstrauss.

References

[1] E. Lindenstrauss, Pointwise Theorems for Amenable Groups, Invent. Math. 146 (2001), no. 2, 259–295.

[2] E. M. Stein, J. Strömberg, Behaviour of maximal functions in R^{n} for large n, Ark. Mat. 21 (1983), 259–269.

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

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