Current perspectives on the Halo Conjecture

1 Introduction

It is a pleasure to contribute to this special issue celebrating the work of Robert Fefferman. The columnist David Brooks once remarked that, in choosing a career, we should be cognisant that our careers will change us. My research journey began as Fefferman’s student at the University of Chicago, and I believe I have benefited immeasurably by having a research career thus far largely spent working on problems motivated by Fefferman’s initial prodding and direction.

Robert Fefferman is best known for his work in multiparameter harmonic analysis. My favorite of his papers is one he wrote in the 1970’s as a graduate student, “A Geometric Proof of the Strong Maximal Theorem,” coauthored with Antonio Córdoba and published in the Annals of Mathematics. This was a breakthrough paper in the theory of differentiation of integrals that provided strategies involving covering lemmas for obtaining weak type bounds of geometric maximal operators.

Some mathematical genealogy and history are appropriate here. Robert Fefferman’s Ph. D. advisor was Elias M. Stein, whose advisor in turn was Antoni Zygmund. The original proof of the strong maximal theorem was due to Jessen, Marcinkiewicz, and Zygmund himself in the 1930’s. Zygmund also undoubtedly had a strong influence on Miguel de Guzmán, who obtained his Ph. D. at Chicago under the direction of Calderón in the late 1960’s. de Guzmán was a leading light in the theory of differentiation of integrals and the author of an indispensable monograph on the subject, Differentiation of Integrals in R n . Interestingly enough, this monograph appeared the same year, 1975, as Córdoba and Fefferman’s Annals paper, and it includes their geometric proof of the strong maximal theorem in an appendix.

Any student of Robert Fefferman is undoubtedly going to become very familiar with de Guzmán’s monograph. Now, in it de Guzmán devoted a complete chapter to the so-called Halo Problem in the theory of differentiation of integrals. The associated Halo Conjecture remains unsolved, although it has motivated several strands of research leading to rather delightful and unexpected results. In this largely expository paper, I will discuss the Halo Conjecture and describe some of these results in detail, while also providing fascinating open problems motivated by this research.

I wish to thank Professors Giorgi Oniani and Alexander Stokolos for their helpful remarks and suggestions regarding this paper.

2 Weak type estimates of geometric maximal operators

We recall the definitions of two fundamental geometric maximal operators.

The Hardy-Littlewood maximal operator is of weak type (1,1). In particular, we have the following.

This weak type estimate is readily seen to follow from the following covering lemma.

The strong maximal operator M _S is defined as follows.

That the strong maximal operator M _S is bounded on L p ( R ) for 1 < p ≤ ∞ was shown by Jessen, Marcinkiewicz, and Zygmund.

Although in [21] Jessen, Marcinkiewicz, and Zygmund also showed that the integrals of functions f for which |f| log(2 + |f|) ⁿ⁻¹ is locally integrable are strongly differentiable, to the best of my knowledge the first explicit sharp weak type estimate on M _S appeared in a 1972 Studia Math. paper by Fava.

Several asides are called for here. Close examination of the Jessen, Marcinkiewicz, and Zygmund paper indicate a proof for a restricted weak type estimate on M _S:

although the full weak type estimate appeared only decades later. We shall see shortly that the notion of extending from a restricted weak type estimate to a general weak type estimate for homothecy invariant geometric maximal operators constitutes the essence of the Halo Conjecture.

Weak type estimates like the ones above appear in different forms. Recall that for a convex increasing function Φ : 0 , ∞ → 0 , ∞ one may define the Luxemburg norm by

If Φ(x) = x log(2 + x) ⁿ⁻¹ the functions f for which ‖ f ‖ L Φ ( R n ) < ∞ is known as the Orlicz class L ( log ⁡ L ) n − 1 ( R n ) , with ‖ f ‖ L Φ ( R n ) being denoted by ‖ f ‖ L ( log ⁡ L ) n − 1 ( R n ) . In practice, however, in dealing with Orlicz classes like L ( log ⁡ L ) n − 1 ( R n ) one wishes to only consider functions restricted to, say, the unit square Q in R n , referring then to the class ‖ f ‖ L ( log ⁡ L ) n − 1 ( Q ) . If f is a measurable function supported on Q, a local analogue of 2.1 would be the estimate

which can be found in Fefferman’s paper in the Beijing Lectures in Harmonic Analysis [8].

Giorgi Oniani, Alexander Stokolos, and myself recently wondered to what extend estimates of the form 2.1, 2.2 are equivalent to one another. Given a collection B of open sets in R n of positive finite measure, define the maximal operator M B by

In [11], we showed that, provided the collection B is homothecy invariant and Φ satisfies the growth condition Φ(2x) ≤ CΦ(x), the estimates

and

are equivalent to one another. The proof involves Borel-Cantelli lemma type arguments and is strongly influenced by Stein’s paper [32] on limits of sequences of operators.

Theorem 3 was initially proven essentially by iteration of one-dimensional techniques. A. Córdoba and R. Fefferman showed that it could be proven using higher dimensional covering techniques by showing the following.

This theorem can also be generalized to Orlicz spaces, providing a covering condition that implies that a maximal operator is of weak type (Φ, Φ). To what extent boundedness of a maximal operator M B when acting on a general Orlicz space implies covering properties of the collection B is still being explored. My favorite problem along these lines is the following:

If M B is of weak type (1,1), must there exist 0 < c < C < ∞ so that, given a finite collection { R j } ⊂ B , there exists a subcollection { R ̃ j } ⊂ { R j } such that ∪ R ̃ j ≥ c ∪ R j and ∑ χ R ̃ j L ∞ ( R n ) ≤ C ?

For an example of the usefulness of this theorem, we can consider the lacunary maximal operator M _lac defined as follows.

Using Theorem 5, one can show that M _lac is bounded on L p ( R 2 ) for 2 < p ≤ ∞. (This was shown by Strömberg in [34] and Córdoba and R. Fefferman in [6]). We still do not know how to use only covering techniques to show that M _lac is bounded on L p ( R 2 ) for all 1 < p ≤ ∞. That being said, using Fourier analytic methods, Nagel, Stein, and Wainger proved in [24] that M _lac is bounded on L p ( R 2 ) for 1 < p ≤ ∞.

We now consider general directional maximal operators. Let Ω ⊂ [0, 1]. Let B Ω be the collection of all rectangles in the plane one of whose sides has slope in Ω. The associated directional maximal operator M _Ω is defined by

Sjögren and Sjölin proved in [30] that if Ω a lacunary set of finite order,^[1] then the directional maximal operator M _Ω is also bounded on L p ( R 2 ) for 1 < p ≤ ∞. As far as directional maximal operators acting on functions on R 2 are concerned, Bateman showed this is an optimal result:^[2]

Extending Bateman’s Theorem to encompass maximal operators acting on measurable functions on R n remains an area of active research, with conditions on a directional maximal operator sufficient to make it bounded on L p ( R n ) for 1 < p ≤ ∞ being provided by Parcet and Rogers [27] and conditions necessary for it to be bounded on L p ( R n ) for 1 < p ≤ ∞ being provided by Kroc and Pramanik [22]. A condition on a set of directions necessary and sufficient for the associated directional maximal operator to be bounded on L p ( R n ) has of yet not been found.

3 The Halo Conjecture

The Hardy-Littlewood and strong maximal operators satisfy the following restricted weak type and weak type estimates:

Pattern recognition suggests the following.

We recall here that a collection of sets B is said to be homothecy invariant provided it is invariant under translation and dilation.

We now consider the notion of a density basis. A collection of sets B covering R n is said to be a density basis provided, for every measurable set E ⊂ R n , for almost every x ∈ R n we have

holds for every sequence of sets {R _j } in B containing x with diameters tending to 0, and moreover that such a sequence exists. This is essentially the weakest viable condition we can place on a collection of sets in the context of differentiation of integrals.

It is helpful here to define Tauberian constants. Given a collection B of open sets of finite positive measure that covers R n , the Tauberian constant C B ( α ) is defined by

Busemann and Feller proved the following fundamental theorem involving homothecy invariant density bases.

It is worth pausing for a moment over this theorem. Of course, if a maximal operator M B satisfies a weak type (Φ, Φ) bound, then C B ( α ) will be finite for all α > 0, with C B ( α ) ≤ C Φ ( 1 α ) . It is natural to try to reverse this procedure. Namely, if one is given a basis B and its associated C B ( α ) , one might be inclined to believe that M B will satisfy a weak type (Φ, Φ) estimate for Φ defined by Φ ( x ) = C B ( 1 x ) for x > 1 (extending the domain of Φ to all of 0 , ∞ by Φ(x) = x for 0 ≤ x ≤ 1, begging the question of whether lim α → 1 − C B ( α ) = 1 . This latter issue relates to the issue of Solyanik estimates which we do not consider here but refer the interested reader to [12], [13], [14], [31]). The Halo Conjecture asserts that this reversal indeed may be done.

de Guzmán proved the following surprising theorem involving the Halo Conjecture and density bases.

de Guzmán remarked that Theorem 8 might provide a means for disproving the Halo Conjecture. Namely, one might disprove the conjecture by constructing a homothecy invariant density basis B so that M B is unbounded on L p ( R n ) for all 1 ≤ p < ∞. At first glance this seems to be a viable thing to do. One certainly could imagine the existence of a collection B so that C B ( α ) grows very rapidly as α tends to 0 (much more rapidly than (1/α) ^p for any finite p) and with the basis B differentiating the integrals of functions locally lying in some Orlicz class strictly contained in all of the L p ( R n ) spaces for 1 ≤ p < ∞ but containing none of them. However, such a basis has never been constructed.

Alexander Stokolos and I attempted to do just that in the case of B being a homothecy invariant collection of convex sets and found that it could not be done. In particular, we proved the following.

Theorems 6 and 9 yield the following corollary.

In conjunction with the second part of Theorem 9, this corollary enables a pleasant extension of a result in the paper [5] of A. Córdoba and R. Fefferman involving L p ( R 2 ) bounds of directional maximal operators and their associated multiplier operators that we now describe.

Let θ ₁ > θ ₂ > θ ₃ > … be a decreasing sequence of angles between 0 and π/2. We define geometric maximal operator M _θ by

where the supremum is over the collection of all rectangles in the plane oriented in one of the directions θ _i . Associated to M _θ is the multiplier operator T _θ given by

where P _θ is the subset of R 2 as indicated in Figure 1.

Figure 1:

Córdoba and Fefferman proved the following:

Theorem 10, in a curious combination with Theorems 6 and 9, provides a rapid proof of the following.

I recall that, when I was a student, Fefferman told me that in formulating Theorem 10 he and Córdoba had placed the Tauberian condition on M _θ to ensure it would be a reasonable operator. Several decades later we now recognize that even previously considered “weak” conditions on a basis can have dramatic consequences in terms of the boundedness of the associated maximal operator.

To the best of my knowledge, the problem of whether the L p ( R 2 ) boundedness of T _θ for some p > 2 implies that M _θ and T _θ are bounded on L p ( R 2 ) for all 1 < p < ∞ remains unsolved.

Let us return to Theorem 9, which asserts that any homothecy invariant density basis B of convex sets is associated to a maximal operator M B that is bounded on L p ( R n ) for sufficiently large p. Note that, a priori, one cannot use Bateman’s Theorem to show that M B is bounded on L p ( R n ) for all 1 < p ≤ ∞ because, although homothecy invariant, B does not necessarily correspond to a directional basis B Ω . However, a breakthrough on this problem occurred when Gauvan recognized in [9] how to associate a homothecy invariant basis of sets in R 2 to a subset of a dyadic tree, making the problem amenable to the probabilistic techniques of Bateman and Katz [1], [2] that can show that certain types of maximal operators are unbounded on L p ( R 2 ) for 1 ≤ p < ∞. Stokolos and I were able to utilize ideas in these papers, in particular taking advantage of properties of sticky maps, to yield the following.

Theorem 12 is a two-dimensional result, in large part because certain techniques in Bateman’s paper [1] are specific to two dimensions. Currently, Stokolos, my student Blanca Radillo-Murguia, and myself are working on modifying ideas in Bateman’s paper to extend his results to the R n case for all n ≥ 3.

4 Translation invariant maximal operators

My strong inclination is that the Halo Conjecture is true for homothecy invariant convex bases, but false for more general homothecy invariant bases. This motivated me to consider the problem of whether or not there exist translation invariant density bases B of sets in R n for which the associated maximal operator is not bounded on L p ( R n ) for any 1 ≤ p < ∞ (the idea being that in constructing a translation invariant basis with this property, one might get ideas on how to construct a homothecy invariant basis sharing the same property, thus disproving the Halo Conjecture via de Guzmán’s Theorem 8). I was joined in this research by Ioannis Parissis, and we found some fascinating results that I wish to describe.

A problem that emerged right off the bat was: how can one ascertain whether or not a translation invariant basis is a density basis? Note Theorem 7 does not hold for bases that are translation but not homothecy invariant.

More notation is necessary here to accurately describe translation invariant density bases. Given a collection of sets B in R n , for r > 0 we let B r denote the sets in B of diameter less than r.

Parissis and I proved the following analogue of the Busemann-Feller theorem for translation invariant bases:

A problem posed by de Guzmán, still unsolved, is whether or not, given a translation invariant density basis B , there must exist r > 0 so that C B r ( α ) < ∞ for all α > 0.

Upon having a condition on a translation invariant basis sufficient to ensure the density condition, Parissis and I returned to the agenda of constructing a translation invariant density basis M B such that M B is unbounded on L p ( R n ) for all 1 ≤ p < ∞. Now, we recognized that such a basis would necessarily lack a regularity condition with respect to balls (as is discussed in Ch. I of [33]) as otherwise the maximal operator would automatically be bounded on L p ( R n ) for 1 < p ≤ ∞. Natural directions included trying to find a translation invariant convex density basis in R n for n ≥ 2 whose corresponding maximal operator was unbounded on L p ( R n ) for all 1 < p < ∞, not an inviting prospect in light of the previously discussed results due to Bateman and with Stokolos. Alternatively we could try to construct a translation invariant nonconvex density basis lacking a regularity condition and hope the corresponding maximal operator would be unbounded on L p ( R n ) for all 1 ≤ p < ∞.

Nagel and Stein considered a related issue in the paper [23] on nontangential approach regions that extended ideas of Zygmund in [35]. In this paper they indeed constructed a translation invariant density basis lacking a regularity condition. However, the associated maximal operator happened to be not only bounded on L p ( R 2 ) for 1 < p ≤ ∞ but also of weak type (1,1)! Parissis and I tried to modify the approach regions considered by Nagel and Stein to one associated to a density basis with a corresponding maximal operator not satisfying a weak type (1,1) estimate but, using Theorem 13, found it impossible to do so. In the process, however, we found the following analogue of the Fatou Theorem that we now briefly describe.

Given f ∈ L p ( R n ) for some 1 ≤ p ≤ ∞, let u(x, y) be its Poisson extension to the upper half-space

given by

where P _y (x) is the Poisson kernel given by

In spite of these insights, however, the question of whether or not there exists a translation invariant density basis whose corresponding maximal operator is unbounded on L p ( R n ) for all 1 ≤ p < ∞ remain unsolved.

5 The Busemann-Feller condition

Our discussion of the Halo Conjecture thus far has only considered maximal operators that automatically satisfy the so-called Busemann-Feller condition. It is natural to consider analogues of the Halo-Conjecture for “centered” maximal operators not necessarily satisfying the Busemann-Feller condition, and it turns out that in this setting the Halo Conjecture does not hold.

We will need to generalize our notation and terminology to accommodate consideration of the Halo Conjecture in the context of uncentered bases. For each x ∈ R n let B ( x ) be a nonempty collection of bounded open sets in R n that contain x. We let B = { ( x , B ( x ) ) : x ∈ R n } . In [10], de Guzmán occasionally clarifies that B is not just a set of subsets of R n (as opposed to a collection of ordered pairs, with the first element of the pair being an element of R n and the second element being a subset of R n ) by writing, admittedly with an understandable abuse of notation, B = ∪ x ∈ R n B ( x ) . B is said to be a density basis if and only if, given a measurable set E in R n , for almost every x ∈ R n we have that

holds for every sequence of sets {R _j } in B ( x ) whose diameters tend to 0, and moreover that such a sequence exists. If f is a measurable function on R n , a basis is said to differentiate the integral of f whenever for a.e. x ∈ R n we have that B ( x ) contains sets of arbitrarily small diameter and that

holds for every sequence of sets {R _j } in B ( x ) whose diameters tend to 0. If L Φ ( R n ) is a class of measurable function on R n (such as an Orlicz space), we say that B differentiates L Φ ( R n ) provided Equation (5.1) holds a.e. for every f ∈ L Φ ( R n ) .

Associated to the basis B is the centered maximal operator M B given by

Also associated to B and to each 0 < α < 1 is the Tauberian constant C B ( α ) defined by^[3]

A basis B is said to be translation invariant whenever R ∈ B ( 0 ) implies R + x ∈ B ( x ) . A basis is said to be dilation invariant whenever R ∈ B ( x ) implies R ̃ ∈ B ( x ) for any R ̃ of the form R ̃ = x + c ( R − x ) , where c > 0. A basis that is both translation and dilation invariant is said to be homothecy invariant.

A Busemann-Feller basis B is one for which, if y ∈ R ∈ B ( x ) , then R ∈ B ( y ) . Frequently we refer to a Busemann-Feller basis as an “uncentered” basis, and we refer to a basis { ( x , B ( x ) ) : x ∈ R n } as a “centered” basis. This terminology is motivated by the notions of the centered and non-centered Hardy-Littlewood maximal operator, although one should recognize that for a centered basis B = { ( x , B ( x ) ) : x ∈ R n } , if R ∈ B ( x ) , the point x does not necessarily need to lie at the geometric center of R, provided such a geometric center even exists.

The Halo Conjecture asserts that a homothecy invariant Busemann-Feller density basis B differentiates the integral of any measurable function f for which ∫Φ(|f) is finite, where Φ(x) is defined on 0 , ∞ by

Note that our previous formulation of the Halo Conjecture involved not the differentiability of the class L Φ ( R n ) but the weak type (Φ, Φ) bounds of the maximal operator M B . Developing ideas in Stein’s paper [32] on limits of sequences of operators, Rubio proved in [28] that for homothecy invariant Busemann-Feller bases these two formulations are equivalent.

We have already referred to the result of de Guzmán in [10] that if the Halo Conjecture were true, then any homothecy invariant Busemann-Feller density basis of sets in R n must necessarily differentiate L p ( R n ) for sufficiently large p. We have also already mentioned that the problem of whether or not there exists translation invariant Busemann-Feller density basis whose associated maximal operator is unbounded on L p ( R n ) remains unsolved. However, Parissis and I did construct in [15] a translation invariant (non-Busemann-Feller) density basis of sets in R that does not differentiate L p ( R n ) for any finite p.

Furthermore, it turns out that there also exists a homothecy invariant density basis of sets in R that does not differentiate L p ( R ) for any finite p. Consequently, the Busemann-Feller condition in the Halo Conjecture cannot be removed. This has not been previously published so the proof will be provided here. The techniques of the associated construction differ from those in [15] because the associated basis needs to be not only translation invariant but also dilation invariant. The main result in this regard is the following.

The proof of Theorem 15 has two main parts. We first construct a homothecy invariant density basis B of sets in R so that the associated maximal operator M B is not bounded on L p ( R ) for any 1 ≤ p < ∞. Subsequently we use the result of Rubio [29] that if B is a homothecy invariant basis so that M B is not of weak type (p, p), then B necessarily does not differentiate L p ( R n ) . One should be alerted that homothecy invariance plays an important role here, as there exist translation invariant bases B that do differentiate L p ( R n ) but such that the maximal operator M B is not bounded on L p ( R n ) . Such a basis could be formed, say, by letting B ( x ) consist of sets either of the form B _r (x) or of the form B _r (x) ∪ R, where R is an arbitrary rectangular parallelepiped in R n of length greater than 1, where here B _r (x) denotes the ball in R n of radius r centered at x.

A useful consequence of the proof of Theorem 15 is the following:

Note this corollary immediately implies that we cannot remove the Busemann-Feller condition from the Halo Conjecture.

We now show there exists a homothecy invariant density basis B of sets in R so that the associated maximal operator M B is not bounded on L p ( R ) .

Proof. Let B ( 0 ) consist of all dilates about the origin of sets of the form

and let B ( x ) = { x + R : R ∈ B ( 0 ) } . Observe that M B χ [ − 1,1 ] ( x ) ≥ 2 − j − 1 for every x in ( 0 , 2 2 j ) . Accordingly, M B is not of weak type (p, p) for any 1 ≤ p < ∞.

To show that B is a density basis, it suffices to show that C B ( α ) < ∞ for every 0 < α < 1. As C B ( α ) is nonincreasing in α, it suffices to show that, given m ∈ N , we have C B ( 2 − m ) < ∞ . Accordingly, we fix m ∈ N . Let E ⊂ R be of finite measure, and suppose that M B χ E ( 0 ) > 2 − m . Then there exists a set S ∈ B ( 0 ) for which 1 | S | ∫ S χ E > 2 − m . Let k ∈ N be such that S is a dilate about the origin of the set − 1 / 2,1 / 2 ∪ − ( 2 2 k ) − 2 − k , − ( 2 2 k ) . For specificity, we let γ > 0 so that S = − γ / 2 , γ / 2 ∪ γ ( − ( 2 2 k ) − 2 − k ) , − γ ( 2 2 k ) . Now, if k ≤ m, then the average of χ _E over γ ( − ( 2 2 k ) − 2 − k ) , γ / 2 exceeds 2 − m ⋅ 1 + 2 − k 1 + 2 2 k , which in turn exceeds 2 − 2 m − m . If k > m, then the average of χ _E over (−γ/2, γ/2) exceeds 2^−m − 2^−k + 2^−m−k which in turn exceeds 2^−m−1. Either way, we have that M H L χ E ( 0 ) ≥ 2 − 2 m − m , where M _HL denotes the uncentered Hardy-Littlewood maximal operator acting on measurable functions on R . The basis B being translation invariant, we then have

As M _HL has a weak type (1,1) bound of 3, we have

as desired. □

The following result of Rubio indicates that if B is a homothecy invariant basis of sets in R n whose associated maximal operator M B is not of weak type (p, p), then B does not differentiate L p ( R n ) . We remark that in [29], Rubio has the hypothesis that B is a Busemann-Feller basis, although close inspection of the proof indicates that the Busemann-Feller condition is never used.

Proof of Theorem 15. The proof follows immediately from Propositions 1 and 2. □

Proof of Corollary 1. In Theorem VIII.4.1 of [10], de Guzmán provides a proof that if the Halo Conjecture were true, then each homothecy invariant Busemann-Feller density basis of sets in R n must differentiate L p ( R n ) for some finite p. Close inspection of the proof (and antecedent results in Chapters III and VIII of [10]) reveals that the theorem carries through for homothecy invariant density bases that do not necessarily enjoy the Busemann-Feller condition. Accordingly, we realize that, if the Halo Conjecture were reformulated with the Busemann-Feller condition removed, the validity of that conjecture would imply that any homothecy invariant density basis of sets in R would necessarily differentiate L p ( R ) for sufficiently large p. As we have shown this is not the case we recognize that the Halo Conjecture with the Busemann-Feller condition removed is false. Hence there exists a homothecy invariant basis B = ∪ x ∈ R B ( x ) that fails to differentiate a.e. the integral of a function f for which ∫ R Φ ( | f | ) is finite. □

It is worthwhile to contrast Corollary 1 with the result of Oniani in [26] that, provided B is a translation (not necessarily homothecy) invariant density basis consisting of convex sets, the Busemann-Feller condition may be removed from the Halo Conjecture.

I have an increasing appreciation of the late E. M. Stein’s attention to the underlining algebraic properties of operators in harmonic analysis. Indeed, harmonic analysis has been described as the study of translation invariant operators. But the Halo Conjecture involves geometric maximal operators associated to density bases with three conditions: not only translation invariance but also dilation invariance and the Busemann-Feller condition. We now know that the Busemann-Feller condition cannot be removed. The role of the dilation condition is very mysterious to me. Underneath the Halo Conjecture is a potentially much more difficult conjecture, namely that if B is a translation invariant Busemann-Feller density basis with associated halo function Φ(x), then B necessarily differentiates the integral of all functions f lying in the class L Φ ( R n ) . Either a proof or counterexample of this would be very interesting.