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# Differentiation of integrals in higher dimensions

Edited by Charles L. Fefferman, Princeton University, Princeton, NJ, and approved February 5, 2013 (received for review November 12, 2012)

## Abstract

We prove a localization principle for directional maximal operators in *L*^{p}(ℝ^{n}), with *p* > 1. The resulting bounds, which we conjecture hold for the largest possible class of directions, imply Lebesgue-type differentiation of integrals over tubes that point in the given directions.

For a set of directions Ω in the unit sphere ^{n−1}, the directional maximal operator *M*_{Ω} is defined by

If Ω consists of a single direction and *p* > 1, the boundedness of *M*_{Ω} from *L*^{p}(ℝ^{n}) to *L*^{p}(ℝ^{n}) follows from the work of Hardy and Littlewood (see the first theorem in ref. 1). This allows one to conclude, loosely speaking, that the derivative in that direction inverts the integral. When Ω is a finite set, a fundamental problem is to find optimal bounds for the operator norm of *M*_{Ω} as a function of the cardinality of Ω and *p*. Allowing Ω to be infinite, one can also ask for geometric properties of the directions that ensure boundedness. In two dimensions, with directions in ^{1}, the questions have been answered with remarkable accuracy (see refs. 2⇓–4 for the first question; refs. 5⇓⇓–8 for the second question; or refs. 9⇓–11, which address the two questions in a unified way). However much less is known in higher dimensions (see refs. 12⇓–14 for the first question and refs. 7 and 15 for the second).

For a fixed *σ* ∈ Σ ≡ Σ(*n*) = {(*j*, *k*): 1 ≤ *j* < *k* ≤ *n*} we consider sequences {*θ*_{σ,i}}_{i∈ℤ} that satisfy 0 < *θ*_{σ,i+1} ≤ *λ*_{σ}*θ*_{σ,i} with lacunary constant 0 < *λ*_{σ} < 1, and for an orthonormal basis (*e*_{1}, …, *e*_{n}) we divide the directions into the sets

With *n* = 3, these sets consist of the directions that lie in the union of four segments that meet along the axis perpendicular to *e*_{j} and *e*_{k}. As |*i*| gets larger, the segments become thinner and accumulate at the hyperplanes perpendicular to *e*_{j} and *e*_{k}. The partition is completed by including the directions contained in these hyperplanes . Writing * = ∪ {∞}, we prove the following localization principle (see ref. 16 for a different type of one-dimensional localization).

### Theorem.

*Let n* ≥ 2 *and p* > 1. *Then*

*where C depends only on n, p, and the lacunary constants* λ_{σ}.

Note that the reverse inequality, with *C* = 1, holds trivially. This recalls the separation of dyadic frequency scales provided by the Littlewood–Paley–Stein theory (1). A difference is that we have many partitions instead of just one; however, we will see that this is unavoidable and the supremum over partitions must be taken over the whole of Σ. We will also see that the segments cannot accumulate away from the hyperplanes perpendicular to the orthonormal basis vectors.

As with the almost orthogonality principle of Alfonseca in two dimensions (17), we recover the best known results for the second question in higher dimensions. Nagel and coworkers (7) proved the *L*^{p} boundedness of the maximal operator associated to the directions , where 0 < *a*_{1} < … < *a*_{n} and 0 < *ϑ*_{i+1} ≤ *λϑ*_{i} with lacunary constant 0 < λ < 1. We can apply the theorem with , where *σ* = (*j*, *k*), reducing the problem to that of a single direction. Note that it makes no difference if the directions are normalized to live on the unit sphere or not. On the other hand, Carbery (15) proved that the maximal operator associated to the directions is *L*^{p} bounded with *p* > 1. Taking *θ*_{σ,i} = 2^{−i}, the resulting sets of directions Ω_{σ,i} are restricted to (*n* − 1)-dimensional hyperplanes, so that by choosing a suitable basis and applying Fubini’s theorem, we reduce to the (*n* − 1)-dimensional problem. Iterating the process we end up with isolated directions as before.

In higher dimensions, it is not sufficient to constrain the angles between an infinite number of directions if they are to give rise to a bounded maximal operator. However, the theorem suggests a definition of lacunarity that gives rise to bounded maximal operators in general. An orthonormal basis of span(Ω) = ℝ^{d} and lacunary sequences {*θ*_{σ,i}}_{i∈ℤ} define partitions for each *σ* ∈ Σ(*d*). We call such a choice of partitions a dissection. Now if Ω consists of a single direction we call it lacunary of order 0. Recursively, we say that Ω is lacunary of order *L* if there is a dissection for which the sets Ω_{σ,i} are lacunary of order ≤ *L* − 1 for all *i* ∈ * and *σ* ∈ Σ(*d*), with uniformly bounded lacunary constants. According to this definition, the Nagel–Stein–Wainger directions are lacunary of order 1 and the Carbery directions are lacunary of order *n* − 1. By repeatedly applying the theorem as before, *M*_{Ω} is *L*^{p}(ℝ^{n}) bounded, with *p* > 1, whenever Ω is lacunary of finite order. This extends the two-dimensional result due to Sjögren–Sjölin (8) (the union of *K* sets of directions of lacunary order *L* with respect to their definition is lacunary of order 2*KL* + 1 with respect to ours).

Bateman (18) proved that, with 1 < *p* < ∞, these are the only sets that give rise to bounded maximal operators in two dimensions. We conjecture that this is also true in higher dimensions. In support of this, the orthogonal projections onto two-dimensional subspaces of the directions which give rise to bounded maximal operators must be lacunary of finite order, and one can characterize the sets of directions that give rise to bounded maximal operators using a class which is related to our finite-order lacunary class. The proofs of these results will not appear here.

After a suitably fine finite splitting of the directions, the operator *M*_{Ω} can be composed with one-dimensional Hardy–Littlewood maximal operators to dominate a constant multiple of the maximal operator ℳ_{Ω} defined by

which in turn pointwise dominates *M*_{Ω}. Here, _{Ω} denotes the family of tubes that point in a direction of Ω. Standard density arguments then yield Lebesgue-type differentiation of integrals.

### Corollary.

*If* Ω *is lacunary of finite order, then*

*for all* with *p* > 1.

Finally we note that by fixing the eccentricity (length/width) of the tubes _{Ω} and considering the associated maximal operator, directions Ω, which give rise to unbounded *M*_{Ω} and ℳ_{Ω} can be considered. Indeed, Córdoba (19) proved that such a maximal operator is bounded, with a logarithmic dependency on the eccentricity, if the directions are restricted to a curve that intersects the hyperplanes of ℝ^{n} no more than a uniformly bounded number of times.

The proof of the theorem, in the following section, is based on the Fourier methods pioneered by Nagel, Stein, and Wainger (7). The key new ingredient is a somewhat nonlinear partition of a hyperplane, covering with tensor products of two-dimensional cones, in which parts of the partition are added and subtracted many times. In the final section, we justify a number of remarks from above by constructing sets of apparently well-behaved directions for which the associated maximal operators are unbounded.

## Proof of the Theorem

By a finite splitting we can suppose that the directions Ω are contained in the first open octant of the unit sphere . We consider intersections of the segments to obtain cells of directions for each . This yields a finer partition than those of the introduction:

Note that many of the cells are empty, however we will see that this overdetermination is somehow unavoidable. Let *K*_{σ,i} denote the convolution operator associated to a Fourier multiplier *ψ*_{σ,i}, smooth on ℝ^{n}\{0}, equal to 1 on

and supported in a similar cone with *n* replaced by *n* + 1.

The key geometric fact used in the proof of the following lemma is that the hyperplane perpendicular to ω is contained in for all ω ∈ Ω_{i}. This is no longer true if, in the definition of the cones, *n* is replaced by a constant strictly less than *n* − 1. At this point we do not use that the dividing sequences are lacunary.

### Lemma 1.

*Let p* > 1. *Then*

*where C depends only on n and p*.

##### Proof:

Fix a nonnegative, even, smooth function that is positive on [−1, 1] and with sufficient decay so that, for positive functions, *M*_{Ω}*f* is pointwise equivalent to

Throughout, ^{∧} and ^{∨} denote the Fourier transform and inverse transform, respectively. One can calculate that

It will simplify things to take *m*_{o} supported in [−1, 1], which can be arranged by choosing *m*_{o} = *ϕ*_{o}**ϕ*_{o}, where *ϕ*_{o} is an even, smooth function supported in [−1/2, 1/2]. We also fix a smooth function η_{o}, supported in the ball of radius 4*n*^{2}, centered at the origin and equal to one on the concentric ball of radius 2*n*^{2}, and consider the operator

where (*S*_{r,ω} *f*)^{∧}(ξ) = η_{o}(*r*(ω_{1}ξ_{1}, …, *ω*_{n}*ξ*_{n}))*m*_{o}(*rω* · ξ)*f*^{∧}(ξ). This is pointwise dominated by a constant multiple of the strong maximal operator ℳ_{str}, which can be bounded by iterated applications of the one-dimensional Hardy–Littlewood maximal theorem. Defining *m* by *m*(ξ) = (1 − η_{o})(ξ)*m*_{o}(**1**·ξ) with **1** = (1,…,1), we are left with the maximal operator *T*_{Ω} defined by

where (*T*_{r,ω} *f*)^{∧}(ξ) = *m*(*r*(ω_{1}ξ_{1},…,*ω*_{n}*ξ*_{n}))*f*^{∧}(ξ).

It will suffice to prove the pointwise estimate

The desired *L*^{p} estimate then follows by combining with the inequalities

and, when Ω_{i} ≠ Ø,

The final inequality is a trivial consequence of the boundedness of the strong maximal operator, combined with the fact that |*f*| ≤ *M*_{Ωi}*f*.

Before proving **[1]**, we motivate why it is reasonable to hope that it should be true. As suggested earlier, the frequency support of *T*_{r,ω} *f* is contained in the union of whenever ω ∈ Ω_{i} with **i** = (*i*_{σ})_{σ∈Σ}. If this covering were in fact a partition, we would obtain

and so, recalling that , a simplified version of **[1]**, with fewer terms on the right-hand side, would follow easily. Now the conic supports do not form a partition and so to compensate we remove the pairwise intersections of the cones and then add back the intersections of each triple of cones, and so on, until we obtain a partition. The process yields the identity

whenever ω ∈ Ω_{i}. In effect, we have expanded the polynomial , and so the remainder ℛ_{i} is given by

In contrast with the operators *K*_{σ,i}, which are essentially two-dimensional, the operators ℛ_{i} are genuinely higher dimensional objects; however, once we see that the multiplier associated to *T*_{r,ω}ℛ_{i} is identically zero whenever ω ∈ Ω_{i} and *r* > 0,

we obtain Eq. **1**. Given that the cones are invariant under scaling, by taking *r* large, this final part is little more than the assertion that the hyperplane is covered by the cones.

After the scaling *ω*_{j}*ξ*_{j} → *ξ*_{j} for 1 ≤ *j* ≤ *n*, it will suffice to prove that the region defined by

and

is empty. As ω ∈ Ω_{i}, we see that the complements of the scaled cones are contained in

We suppose for a contradiction that the region defined by **[3]** and **[4]** is not empty. It is clear by comparing the inequalities in **[3]** that the components of a vector ξ in this region cannot all have the same sign. By symmetric invariance of the conditions, we may suppose that ξ_{1},…,*ξ*_{m−1} ≥ 0 and that *ξ*_{m},…,*ξ*_{n} < 0 for some 1 < *m* ≤ *n*. We can also suppose without loss of generality that |ξ_{1}| ≥ |*ξ*_{j}| for all *j* > 1 and |*ξ*_{m}| ≥ |*ξ*_{j}| for all *j* > *m*. Then taking *j* = 1 and *k* = *m* in **[4]** we see that |ξ_{1}| ≥ *n*|*ξ*_{m}|. On the other hand, by the first condition of **[3]**,

Combining the two estimates we obtain Because for *j* > 1, this yields |ξ_{1}|+ … +|*ξ*_{n}| ≤ *n*^{2}/*r*, which contradicts the second inequality in **[3]**. Thus, *T*_{r,ω}ℛ_{i} ≡ 0 whenever *r* > 0 and ω ∈ Ω_{i}, and so we are done.

We will also require the following square function estimates, which follow easily from the two-dimensional theory.

### Lemma 2.

*Let* 1 < *p* < ∞ *and* Γ ⊂ Σ. *Then*

*and C depends only on* |Γ|, *p, and the lacunary constants λ*_{σ}.

##### Proof:

To bound directional maximal operators for *p* ≥ 2, it suffices to prove the theorem for *p* = 2, in which case the square function estimate follows directly from Plancherel’s theorem and the finite overlapping of the supports of {*ψ*_{σ,i}}_{i∈ℤ}. This is where we use the lacunarity of the sequences {*θ*_{σ,i}}_{i∈ℤ}. When *p* ≠ 2, by a standard randomizing argument, using Khintchine’s inequality, the square function estimates follow from the uniform *L*^{p} boundedness, independent of the choice of the signs, of the Fourier multiplier operators

This in turn is a consequence of the Marcinkiewicz multiplier theorem (e.g., ref. 1, p. 109), for which it suffices to check a number of conditions involving integrals of derivatives of the multipliers. After applying the product rule, the calculation reduces to the case |Γ| = 1. Applying Fubini’s theorem so as to ignore the trivial variables, this was originally checked by Córdoba and coworker (ref. 20, section 4) in their proof of a two-dimensional angular Littlewood–Paley inequality (see also ref. 7 where their result was improved).

Armed with these lemmas, the proof is completed easily as follows:

##### Case p ≥ 2:

We consider ^{Σ} = ^{Γ} × ^{Σ\Γ}, and given **i** = (*i*_{σ})_{σ∈Σ}, we write **i** = **j** × **k** where **j** = (*i*_{σ})_{σ∈Γ} and **k** = (*i*_{σ})_{σ∈Σ\Γ}. Using the inclusion and interchanging the order of the sum and the integral,

Taking , where **j** = (*i*_{σ})_{σ∈Γ}, and applying *Lemmas 1* and *2*, we obtain the desired estimate.

##### Case 1 < p < 2:

This is based on an argument of Christ used in refs. 15, 17, which refined the argument of Nagel–Stein–Wainger (7). We suppose initially that Ω is finite, so that by the triangle inequality and the Hardy–Littlewood maximal theorem, *M*_{Ω} is bounded. Then by interpolating between

and **[5]**, we see that is bounded above by

Taking , where **j** = (*i*_{σ})_{σ∈Γ}, and applying *Lemmas 1* and *2* as before, we see that

Rearranging, we obtain the desired estimate with *C* independent of Ω, so we can drop the restriction that Ω is finite. This completes the proof.

In both refs. 7 and 15, a single conic Fourier multiplier was introduced for each direction. This multiplier had to cover (the bulk of) the hyperplane perpendicular to the direction, and so was necessarily multidimensional in nature. Restrictions on the directions then ensured finite overlapping of the supports of the multipliers, yielding a bound via orthogonality as above. Instead, we introduced a number of essentially two-dimensional multipliers that added flexibility at the same time as simplifying matters. It is necessary to add and subtract a number of products of these multipliers to obtain a partition, however this came at essentially no cost. This simplifies matters because the orthogonality in two dimensions, summing over one index at a time, is trivial to check. On the other hand, our multipliers are naturally associated to partitions of the directions allowing us to introduce a multiplier for each segment instead of one for each direction.

## Concluding Remarks

The theorem is remarkably sharp in the sense that the supremum in σ must be taken over the whole of Σ, and the partitions must accumulate at the hyperplanes perpendicular to the basis vectors. To see this, we construct sets of relatively well-behaved directions for which the associated maximal operators are unbounded. We will see that the directions need only be badly spaced after projecting onto a two-dimensional subspace, and so, in contrast with the two-dimensional case, it is not enough to constrain the angles between the directions if they are to give rise to a bounded maximal operator in higher dimensions.

First, we enumerate and consider Ω defined by

Then the angles between the directions form a lacunary sequence converging to 0 with lacunary constant 1/2. Taking *θ*_{σ,i} = 2^{−i}, the sets Ω_{σ,i} consist of at most one direction for all *i* ∈ * and *σ* ∈ Σ\{(1, 2)}. Despite this, *M*_{Ω} is unbounded. Indeed, consider the set of rectangles ℛ in Π = span(*e*_{1}, *e*_{2}) with longest side parallel to the orthogonal projection of some ω ∈ Ω on Π. Then, as observed by Fefferman (21), the construction of Besicovitch provides finite subsets ℛ_{N} ⊂ ℛ, for all *N* ≥ 1, that satisfy

For each rectangle *R* in ℛ_{N}, we set

where ω is a direction of Ω whose orthogonal projection points in the direction of *R*, and let α ≡ α(*N*) to be 10 times the maximum β(*R*) with *R* ∈ ℛ_{N}. Taking

we then have

Using **[6]**, we see that for all *N* ≥ 1,

so that *M*_{Ω} is unbounded on *L*^{p}(ℝ^{n}) when *p* is finite.

Second, we let and be orthogonal unit vectors in span(*e*_{2}, *e*_{n}), close to *e*_{2} and *e*_{n}, with in the first quadrant determined by *e*_{2} and *e*_{n}. We construct a set of directions, accumulating rapidly at , for which the angles between the orthogonal projections onto are badly spaced. Indeed, we take Ω = {ω_{ℓ}}_{ℓ}_{≥1} so that . This does not yet completely determine ω_{ℓ}. Supposing that we have chosen ω_{ℓ}_{−1} we can choose the direction ω_{ℓ} sufficiently close to so that the angle between ω_{ℓ}_{−1} and is at least double that between ω_{ℓ} and . We can also choose the directions so that

for all (*j*, *k*) ∈ Σ(*n*)\{(*2, n*)}. Taking *θ*_{σ,i} = 2^{−i}, the sets Ω_{σ,i}, defined with respect to the orthonormal basis (*e*_{1}, …, *e*_{n}), consist of at most one direction for all *i* ∈ ℤ* and *σ* ∈ Σ(*n*)\{(2, *n*)}. On the other hand, if we define the final segments by

accumulating at then they also consist of at most one direction for all *i* ∈ ℤ. Despite this, *M*_{Ω} is unbounded as before. Indeed, consider the set of rectangles ℛ in with longest side parallel to the orthogonal projection on Π of some ω_{ℓ}. Then there are finite subsets ℛ_{N} ⊂ ℛ, for all *N* ≥ 1, that satisfy **[6]**. Considering , defined as before but with respect to the basis , we again find *M*_{Ω} unbounded on *L*^{p}(ℝ^{n}) for finite *p*.

## Acknowledgments

Authors acknowledge support by European Research Council Grants 256997 and 277778, and Spanish grants MTM2010-16518 and SEV-2011-0087.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. E-mail: keith.rogers{at}icmat.es.

Author contributions: J.P. and K.M.R. designed research, performed research, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

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