RANDOM MARTINGALES AND LOCALIZATION OF MAXIMAL INEQUALITIES

Transcription

1 RANDOM MARTINGALES AND LOCALIZATION OF MAXIMAL INEQUALITIES ASSAF NAOR AND TERENCE TAO Abstract. Let X, d, µ be a metric measure space. Hardy-Littlewood maximal operator M R fx def = sup r R f dµ. µbx, r Bx,r For = R 0, consider the We show that if there is an n > such that one has the microdoubling condition µ B x, + n r µ Bx, r for all x X and r > 0, then the weak, norm of M R has the following localization property: M R LX L, X sup r>0 MR [r,nr] LX L., X An immediate consequence is that if X, d, µ is Ahlfors-David n-regular then the weak, norm of M R is n log n, generalizing a result of Stein and Strömberg [47]. We show that this bound is sharp, by constructing a metric measure space X, d, µ that is Ahlfors-David n-regular, for which the weak, norm of M 0, is n log n. The localization property of M R is proved by assigning to each f L X a distribution over random martingales for which the associated random Doob maximal inequality controls the weak, inequality for M R. Contents. Introduction.. Microdoubling and the localization theorem 4.. Weak, norm bounds 5.3. Ultrametric approximations: deterministic and random 7.4. Lower bounds Adding more hypotheses 9.6. The example of the infinite tree 0. Doob-type maximal inequalities 3. Localization of maximal inequalities 3 4. An argument of E. Lindenstrauss 9 5. The infinite tree 4 6. Sharpness A preliminary construction The doubling example The Ahlfors-David regular example 34 References 39

2 . Introduction A metric measure space X, d, µ is a separable metric space X, d, equipped with a Radon measure µ. We assume throughout the non-degeneracy property 0 < µbx, r < for all r > 0, where Bx, r def = {y X : dx, y r}. For any locally integrable f : X C, we can then define the Hardy-Littlewood maximal function Mfx def = sup f dµ, r>0 µbx, r Bx,r which is easily verified to be measurable. We shall study the weak, operator norm of M, defined as usual to be the least quantity 0 M L X L, X for which one has the distributional inequality Mf L, X M L X L, X f L X for all f L X. Here L p X p denotes the usual Lebesgue space corresponding to the measure µ, and L p, X is the weak L p norm, def f Lp, X = sup λ>0 λ µ f > λ /p. Analogously to, the strong p, p operator norm of M is defined as usual to be the least quantity 0 M LpX L px for which Mf LpX M LpX L px f LpX for all f L p X. In most cases of interest it is probably impossible to compute M L X L, X exactly; notable exceptions to this statement are ultrametric spaces, where the weak, norm of M equals we will return to the class of ultrametric spaces presently, and the real line R, equipped with the usual metric and Lebesgue measure, where it was shown by Melas [34] that the weak, norm of M equals + 6 the case of the strong p, p norm of M, p >, when X = R, remains open, but we refer to [0, 5] for some partial results. In view of these difficulties, it seems more reasonable to ask for estimates on the asymptotic behavior of the various operator norms of maximal functions. Quite remarkably, despite the wide applicability of maximal inequalities, and significant effort by many researchers, even in the simple case when X is the n-dimensional Hilbert space l n and µ is Lebesgue measure, it is unknown whether or not the weak, norm of M is bounded independently of the dimension n. A classical application of the Vitali covering theorem see for example [7, 46,, 7] shows that for any n-dimensional normed space X, the weak, and strong p, p norms of M grow at most exponentially in n. This was greatly improved by Stein and Strömberg [47] to M L X L, X = On log n for a general n-dimensional normed space, and to the slightly better bound M L l n L, l n = On for n-dimensional Hilbert space. Until recently, there was no known example of a sequence of n-dimensional normed spaces X n for which M L X n L, X n tends to with n. A recent breakthrough of Aldaz [] showed that when X n = l n, i.e., R n equipped with the l norm whose unit ball is an axis parallel cube, M L X n L, X n must tend to with n; the best known lower bound [3] on M L l n L, l n is log n o. The best known upper estimate for M L X L, X when X = l n remains the Stein-Strömberg On log n bound.

3 As partial evidence that when X is the n-dimensional Euclidean space l n, the weak, norm M L X L, X might be bounded, we can take Stein s theorem [45] see also the appendix of [47] which asserts that in the Euclidean case, for p > we have M LpX LpX Cp, where Cp < depends on p but not on n. For general n- dimensional normed spaces, Stein and Strömberg [47] obtained the bound M LpX LpX cpn, while Bourgain [8, 9] and Carbery [3] proved that for any n-dimensional normed space, M LpX LpX Cp < provided p > 3. It is unknown whether or not there is some < p < 3 for which there exist n-dimensional normed spaces X n such that M LpXn L px n is unbounded. This is unknown even for the case of cube averages X n = l n. It was shown by Bourgain [0] that M LpX LpX Cp, q for all p > when X = l n q and q is an even integer, and this was extended by Müller to X = l n q for all q <. A dimension independent bound on M L l n L, l n would mean that the classical Euclidean Hardy-Littlewood maximal inequality is in essence an infinite dimensional phenomenon. This statement is not quite true, since there is no Lebesgue measure on infinite dimensional Hilbert space, but nevertheless, even Stein s dimension independent bound on M Lpl n Lpln, p >, has interesting infinite dimensional consequences see for examples Tišer s work [53] on differentiation of integrals with respect to certain Gaussian measures on Hilbert space provided that the integrand is in L p for some p >. Moreover, improved bounds on M L X L, X are clearly of interest since they would yield improved quantitative estimates in the many known applications of the Hardy-Littlewood maximal inequality. As an example, such bounds are relevant for quantitative variants of Rademacher s differentiation theorem for Lipschitz functions, which are used in results on the bi-lipschitz distortion of discrete nets see [, 5]. Bounds on M L X L, X and M LpX LpX have been also intensively investigated for metric measure spaces other than finite dimensional normed spaces. Strong p, p bounds for free groups with counting measure have been established by Nevo and Stein in [40]. In Section 5 we prove the corresponding weak, inequality, which is nevertheless not sufficient for the purpose of ergodic theoretical applications as in [40]; see Conjecture below for more information. In the case of the Heisenberg group H n+, equipped with either the Carnot-Carathéodory metric or the Koranyi norm and the underlying measure being the Haar measure, dimension independent strong p, p bounds have been obtained by Zienkiewicz [56], and a weak, bound of On was obtained by Li [30]. It is unclear if these bounds generalize to other nilpotent Lie groups though perhaps similar methods could apply to certain two step nilpotent Lie groups, by replacing the use of [4] in [56] with the results of [38, 3]. The main result of the present paper implies a general bound for the weak, norm of the Hardy-Littlewood maximal function on Ahlfors-David n-regular spaces; a class of metric measure spaces that contains the examples described above as special cases except for the case of the free group, which is dealt with separately in Section 5. Specifically, assume that After presenting our work we learned from Michael Cowling that the weak, inequality for the free group can be also deduced from the work of Rochberg and Taibleson [4]. Our combinatorial proof in Section 5 is different from the proof in [4], though it is similar to the proof in an unpublished manuscript of Cowling, Meda and Setti, which adapts arguments of Strömberg [48] in the case of the hyperbolic space. We thank Michael Cowling and Lewis Bowen for showing us the Cowling-Meda-Setti manuscript. 3

4 the metric measure space X, d, µ satisfies the growth bounds x X r > 0, r n µ Bx, r Cr n, 3 where n, and C is independent of x, r. Under this assumption, we show that M L X L, X = On log n, 4 where the implied constant depends only on C. At the same time, we construct for all n an Abelian group G n, equipped with a translation invariant metric d n and a translation invariant measure µ n, that satisfies 3 with C = 8, yet We can also ensure that for all p > we have M L G n L, G n n log n. 5 M LpG n L pg n p. 6 Here, and in what follows, we use X Y, Y X to denote the estimate X CY for some absolute constant C; if we need C to depend on parameters, we indicate this by subscripts, thus X p Y means that X C p Y for some C p depending only on p. We shall also use the notation X Y for X Y Y X. Note that the bound 4 contains the Stein-Strömberg result for n-dimensional normed spaces. It also applies to, say, any translation invariant length metric on nilpotent Lie groups 3. However, it falls shy by a logarithmic factor of the two On results quoted above: for the Euclidean space l n, and the Heisenberg group H n+. Our lower bound 5 suggests that in order to improve upon the On log n bound of Stein and Strömberg, one must genuinely use the underlying geometry of the normed vector space and not just the metric properties, or the L p theory. For instance, to obtain the bound of On in the case of the Euclidean metric in [47], it was necessary to exploit the relationship between averaging on balls and the heat semigroup, in order that the Hopf-Dunford-Schwartz maximal inequality can be used. A similar strategy was used for the Heisenberg group in [30]. This type of relationship does not appear to be available for general norms on R n. The results presented above are simple corollaries of a general localization phenomenon for maximal inequalities, which we shall now describe. In fact, for the bound 4 to hold true, we need to assume a condition which is less restrictive than the Ahlfors-David regularity condition 3; in particular it need not hold for all radii r, and it thus also applies to discrete groups of polynomial growth, equipped with the word metric and the counting measure. All of these issues are explained in the following subsection... Microdoubling and the localization theorem. Let X, d, µ be a metric measure space. For R 0, we consider the maximal operator corresponding to radii in R, which is defined by M R fx def = sup r R Thus, using our previous notation, M = M 0,. f dµ. 7 µbx, r Bx,r One can modify the argument to make C arbitrarily close to, but we will not do so here as it requires more artificial constructions. 3 It seems likely however that the original Stein-Strömberg argument can be extended to this setting. 4

5 We shall say that X, d, µ is n-microdoubling with constant K if for all x X and all r > 0 we have µ B x, + r KBx, r. 8 n The case n = in 8 is the classical K-doubling condition x X r > 0, µ B x, r KBx, r. 9 Note that 8 follows from the Ahlfors-David n-regularity condition 3, with K = ec. The microdoubling property appeared in various guises in the literature; for example, it follows from a lemma of Colding and Minicozzi [8] see also Proposition 6. in [4] that if X, d, µ is a K-doubling length space, then it is also n-microdoubling with constant O, where n = e KO. We note in passing that this exponential dependence on K is necessary, as exhibited by the interval X = [, N], with the metric inherited from R, and the measure whose density is ϕx = ; the doubling constant for this length space is of order log N, but x it can only be n-microdoubling with n a power of N. Our main result is the following localization theorem for maximal inequalities on microdoubling spaces. It deals, for any p <, with the weak p, p norm of M R, defined as the optimal number M R LpX Lp, X for which the distributional inequality holds for all f L p X and λ > 0. µ M R f > λ M R p L px L p, X λ p f p L px Theorem. Localisation. Fix n and K 5. Let X, d, µ be a metric measure space satisfying the microdoubling condition 8. Fix R 0, and p. Then we have M R LpX Lp, X K + + log log K + log n /p sup MR [r,nr] LpX L. 0 r>0 p, X Remark.. In the converse direction, one trivially has M R LpX Lp, X sup MR [r,nr] LpX L. r>0 p, X log log K +log n Note that the term in 0 is always at most log log K. Thus when K is independent of n, up to constants, in order to establish a weak p, p maximal inequality for spaces obeying 8, it suffices to do so for scales localized to an interval [r, nr]. In many cases e.g. finite-dimensional normed vector spaces we can also rescale to r =... Weak, norm bounds. To deduce some corollaries of Theorem., fix an integer m N, and note that for all f L p X and r, λ > 0 we have, µ M R [r,nr] f > λ = µ max M 0 j m R [rn j/m,rn j+/m ] f > λ m j=0 µ M R [rn j/m,rn j+/m ] f > λ m max µ M 0 j m R [rn j/m,rn j+/m ] f > λ. Thus, under the assumptions of Theorem. and specializing to p =, we have for every m N, 5

6 M R L X L, X K + m + log log K + log n sup r>0 M R [r,n /m r] L X L, X. Note that for m n log n we have n /m +, and hence for all r > 0, n M R [r,n /m r] f 8 f dµ KA µb x, r + f, n where A r is the averaging operator: A r fx def = Bx,+ nr f dµ. 3 µbx, r Bx,r Under some mild uniformity assumption on µ, the strong, norm of A r is bounded for all r > 0. For example, if µbx, r does not depend on x as is the case for invariant metrics and measures on groups, then a simple application of Fubini s theorem shows that A r L X L X. In fact, if we knew that µbx, r KµBy, r for all x X and y Bx, r which is a trivial consequence of the Ahlfors-David regularity condition 3, then we would have by the same reasoning A r L X L X K. An elegant way to combine this uniformity condition with the microdoubling condition 8, is to impose the following condition, which we call strong n-microdoubling with constant K: x X r > 0 y Bx, r, µ B y, + n r KBx, r. 4 Thus, by a combination of and, we see that if X, d, µ satisfies 4, then M L X L, X K n log n. Similarly, if R [ r, n /m r ] contains at most one point for all r > 0, then M R L X L, X K m. This happens in particular if R = Z = { k : k Z }, and m log n, proving the following corollary: Corollary.. Fix n and K 5. Let X, d, µ be a metric measure space satisfying the strong n-microdoubling condition 4. Then M L X L, X K n log n, 5 M Z L X L, X K log n. 6 The lacunary maximal function M Z was previously studied for n-dimensional normed spaces by Bourgain in [9], where he proved that its strong p, p norm is bounded by a dimension independent constant C p < recall that for the non-lacunary maximal function this is only known for p > 3. The logarithmic upper bound 6 on the weak, norm of the lacunary maximal function when X is an n-dimensional normed space was proved by Menárguez and Soria in [35]. In section 4 we present a different approach to the proof of Corollary., following an argument of E. Lindenstrauss [3]. While it gives slightly weaker results, and does not yield the localization theorem, this approach is of independent interest. Moreover, Lindenstrauss approach is based on a beautiful randomization of the Vitali covering argument, and as such complements our approach to Theorem., which is based on a random partitioning method that originated in theoretical computer science and combinatorics an overview of 6

7 our technique is contained in Section.3. The maximal functions considered in [3] arose when taking averages over Følner sequences of an amenable group action on a measure space, and were thus not directly connected to the metric questions that are studied in the present paper. Nevertheless we consider the arguments in Section 4 to be essentially the same as those in [3]. We thank Raanan Schul for pointing out how the maximal inequality of E. Lindenstrauss implies the Hardy-Littlewood maximal inequality under strong microdoubling..3. Ultrametric approximations: deterministic and random. Doob s classical maximal inequality for martingales see Section is perhaps the simplest and most versatile maximal inequality for which the weak, norm is known exactly and is equal to. Our proof of Theorem. relates the weak, inequality for M to the maximal inequality for martingales, by allowing the martingale itself to be a random object. We show that while the weak, inequality is not itself a martingale inequality, it is possible to associate to each f L X a distribution over random martingales. These random martingales stochastically approximate Mf, in the sense that we can write down a variant of Doob s inequality for each of them, which, under the microdoubling assumption, in expectation yields theorem.. The details are presented in Section 3. An alternative interpretation of Doob s maximal inequality is that if X, d, µ is a metric measure space, and if in addition d is an ultrametric, i.e., dx, y max{dx, z, dz, y} for all x, y, z X, then M L X L, X. Indeed, restrict for simplicity to the case of a finite ultrametric, in which case we obtain an induced hierarchical family of partitions of X into balls, where each ball at a given level is the union of balls of smaller radii at the next level. This picture immediately shows that by considering the averages of f on smaller and smaller balls, in the ultrametric case we can reduce the weak, inequality for Mf to Doob s maximal inequality. Of course, not every metric is an ultrametric, or even close to an ultrametric. Nevertheless, over the previous two decades, researchers in combinatorics and computer science developed methods to associate to a general metric space X, d a distribution over random ultrametrics ρ on X, which dominate d and sufficiently approximate it in various senses depending on the application at hand. Such methods are often also called random partitioning methods, in reference to the hierarchical tree structure of ultrametrics. This approach originated in the pioneering works of Linial and Saks [3] and Alon, Karp, Peleg and West [], and has been substantially developed and refined by Bartal [4, 5]. Important contributions of Calinescu, Karloff and Rabani [] and Fakcharoenphol, Rao and Talwar [] resulted in a sharp form of Bartal s random tree method, and our work builds on these ideas. In [36, 37] such random ultrametrics were used in order to prove maximal-type inequalities of a very different nature motivated by embedding problems, as ultrametrics are isometric to subsets of Hilbert space [9]; these results also served as some inspiration for our work. One should mention here that the idea of relating metrics to ultrametric models is, of course, standard. Hierarchical partitioning schemes are ubiquitous in analysis and geometry see the discussion of Calderón-Zygmund decompositions in [45], or, say, Christ s cube construction in [6]. Proving maximal inequalities by considering certain Hierarchical partitions is extremely natural; a striking example of this type is Talagrand s majorizing measure theorem [49], which deals with sharp maximal inequalities for Gaussian processes via a construction of special ultrametrics the ultrametric approach is explicit in [49], and has an alternative later description [50] via the so called generic chaining ; see also [6]. Explicit 7

8 uses of random coverings and partitions in the context of purely analytic problems occurred in E. Lindenstrauss aforementioned randomization of the Vitali covering argument for the purpose of pointwise theorems for amenable groups [3], and in the work of Nazarov, Treil and Volberg [39] on T b theorems on non-homogeneous spaces. See also [8] for applications to extensions of Lipschitz functions..4. Lower bounds. A standard application of the Vitali covering argument see e.g. [46] or [5] yields the inequality Mf f L X, 7 L, X where Mf is the modified Hardy-Littlewood maximal operator Mfx def = sup f dµ, r>0 µbx, r, r Bx,r and Bx, r Bx, r, r Bx, r is the enlarged ball Bx, r, r def = By, r = {z X : dx, y, dy, z r for some y X}. y Bx,r In particular, if we have the doubling condition 9, then M L X L, X K. 8 The factor in 9 cannot be replaced by any smaller number while still retaining linear behavior in terms of K of the weak, operator norm; see [43]. In the absence of any further assumptions on the metric measure space, the bound 8 is close to sharp: Proposition.5 The star counterexample. Fix K. Then there exists a metric measure space obeying 9 with M L X L, X K. Proof. Without loss of generality we may take K to be an integer. Let X be the star graph formed by connecting one hub vertex v 0 to K other spoke vertices v,..., v K, with the usual graph metric thus dv 0, v i = and dv i, v j = for all distinct i, j {,..., K }. Let µ be the measure which assigns the mass K to v 0 and mass to all other vertices; one easily verifies that 9 holds. Let f L X be the function which equals on v 0 and vanishes elsewhere. Then one easily verifies that f L X = K, that µx = KK, and that Mfx K for all x X, and the claim follows. K Remark.. One can achieve a similar effect in a high-dimensional Euclidean space R n. If we let X = {0, e,..., e n } be the origin and standard basis with the usual Euclidean metric and counting measure, then 9 holds with K def = n +, while if we let f be the indicator function of 0, then Mfx for all x X, and so M L X L, X n+ = K. A more sophisticated version of this example was observed in [44]: if we take X to be the origin 0, together with a maximal.0-separated say subset of the sphere S d, then 9 holds for K = X C n for some absolute constant C >, but M L X L, X K by the same argument as before. In particular this shows that the Hardy-Littlewood weak, operator norm as well as the L p operator norm for any fixed < p < for measures in R n can grow exponentially in the dimension n. In the converse direction, a well-known 8

9 application of the Besicovitch covering lemma [6, 7] shows that M L X L, X C n for some absolute constant C whenever X is a subset of R n with the Euclidean metric, and µ is an arbitrary Radon measure. In particular, as observed in [44], this shows that the constants in the Besicovitch covering lemma must grow exponentially in the dimension see also [4]..5.. Adding more hypotheses. Despite the example in Proposition.5, we know due to Corollary. that in many cases the bound 8 can be significantly improved. In particular, a more meaningful variant of Proposition.5 would be if we also impose the natural uniformity condition that µbx, r is independent of x X. As discussed in Section., this immediately implies that the averaging operators A r given in 3 are now contractions on L X. Thus in order for the weak, operator norm to be large, one needs to have contributions to the set {Mf > λ} from several scales r, rather than just a single scale as in Proposition.5. Another hypothesis that one can add, in order to make a potential counter-example more meaningful, is that the maximal operator M is already of strong-type p, p for all < p, as we know to be the case for X = l n, due to Stein s theorem [45]. Finally, we can make the task of bounding the maximal operator easier by replacing M with the lacunary maximal operator M Z. Our first main construction shows that even with all of these additional hypotheses and simplifications, we still cannot improve significantly upon 8. Theorem.3 Doubling example. Let K. Then there exists a metric measure space X, d, µ with X an Abelian group and d, µ translation-invariant, such that the doubling condition 9 holds, and M LpX L px p holds for all < p with the implied constant independent of K, but such that M Z L X L, X K We prove this theorem in Section 6.3. The basic idea is to first build a maximal operator not arising from a metric measure space which is of strong type p, p but not of weak type,, and then take an appropriate tensor product of this operator with a martingale type operator to obtain a new operator which is essentially a lacunary maximal operator associated to a metric measure space. The constant 48 in 9 can of course be improved, but we will not seek to optimize it here. As stated earlier, we also construct an example of a metric measure space that shows that Corollary. is sharp even under the stronger Ahlfors-David regularity condition 3. Theorem.4 Ahlfors-David regular example. Assume that n. Then there exists an Abelian group G, with invariant measure µ and an invariant metric d, obeying the Ahlfors- David n-regularity condition 3 with K = 8, such that and Furthermore we have for all < p. M L G L, G n log n, 0 M Z L G L, G log n. M LpG L pg p 9

10 .6. The example of the infinite tree. The above examples seem to indicate that the weak, behavior of the Hardy-Littlewood maximal function can deteriorate substantially when the doubling constant is large, even when assuming good L p bounds, as well as uniformity assumptions on the measure of balls. Nevertheless, there are some interesting examples of metric measure spaces with very poor or non-existent doubling properties, for which one still has a weak, bound. We give just one example of this phenomenon, namely the infinite regular tree. Theorem.5 Hardy-Littlewood inequality for the infinite tree. Fix an integer k, and let T be the infinite rooted k-ary tree, with the usual graph metric d and counting measure µ. Then we have M L T L, T Thus the implied constant is independent of the degree k. We prove this theorem in Section 5. We remark that the L p boundedness of this maximal function for p > was essentially established by Nevo and Stein in [40]. The argument here proceeds very differently from the usual covering type arguments, which are totally unavailable here due to the utter lack of doubling for this tree. Instead, we use a more combinatorial argument taking advantage of the expander or non-amenability properties of this tree, which roughly asserts that any given finite subset of the tree must have large boundaries at every distance scale. When k is odd, T is almost 4 identifiable with the free group on k+ generators. The above theorem then suggests that a maximal ergodic theorem in L should be available for ergodic actions of free groups on measure-preserving systems the analogous L p maximal theorems for p > being established in [40]. However, the non-amenability of the free group prevents one from applying standard arguments to transfer Theorem.5 to this setting indeed, our proof of Theorem.5 will rely heavily on this non-amenability. Thus the following conjecture remains open: Conjecture. Let F be a finitely generated free group, and let w T w be an ergodic action of F on a probability space X, B, µ. Then sup T w f n Bid, n f L X w Bid,n L, X for all f L X, where Bid, n is the collection of words in F of length less than n. We remark that by applying the pointwise convergence theorems in [40] and a standard density argument, Conjecture would imply the pointwise convergence result lim T w fx = f dµ n Bid, n w Bid,n for all f L X and almost every x X. This result is currently known for f L p X for p >, due to [40]. 4 More precisely, one needs to enlarge the tree at the root to have k + descendants instead of k. But one can easily check that this change only affects the weak, norm of the maximal function by a constant at worst. 0 X

11 Acknowledgements. We thank Raanan Schul for pointing out that the Lindenstrauss maximal inequality implies the Hardy-Littlewood maximal inequality under strong microdoubling, and Zubin Guatam for explaining the proof of the Lindenstrauss maximal inequality. A. N. was supported in part by NSF grants CCF and CCF , BSF grant , and the Packard Foundation. T. T. was supported by a grant from the MacArthur foundation, by NSF grant DMS , and by the NSF Waterman award.. Doob-type maximal inequalities Let X, d, µ be a metric measure space with µx < more generally, the arguments below extend to the σ-finite case. If F is a σ-algebra of measurable sets in X, we let L p F denote the space of L p X functions which are F -measurable. The orthogonal projection from L X to the closed subspace L F will be denoted f Ef F, and as is well known it extends to a contraction on L p X for all p. The following important inequality of Doob is classical see [9, ]. Proposition. Doob s maximal inequality. Let F 0 F F be an increasing sequence of σ-algebras. Then we have f L X = sup Ef Fk f L X, k 0 and for < p, f L p X = sup Ef Fk k 0 L, X L px p p f L px. We now establish a variant of this inequality, in which the expectations Ef F k are replaced by more general sublinear operators. Theorem. Modified Doob s inequality. Let F 0 F F be an increasing sequence of σ-algebras and fix p <. For each k N let M k be a sublinear operator 5 defined on L p X + L X such that we have the bounds and f L p X = M k f Lp, X A f LpX, 3 f L X = M k f L X B E f F k L X. 4 Suppose also that we have the localization property Then we have for all f L p X. f L p X + L X E k F k = Ek M k+ f = M k+ Ek f. 5 sup k 0 M k f L p, X A p + B p /p f LpX Remark.. Observe that the properties 4, 5 with B = are satisfied by the projection operator M k+ f def = Ef F whenever F k F F k+. Thus 4, 5 can be viewed together as a kind of assertion that M k+ lies between F k and F k+ in some sense. 5 By this we mean that M k f + g M k f + M k g and M k cf = c M k f for all functions f, g in the domain of M k and all constants c R.

12 Proof. By monotone convergence we may restrict the supremum over k 0 to a finite range, say 0 k K for some finite K N. We can then assume without loss of generality that F k is the trivial algebra {, X} for all k < 0. By homogeneity it suffices to show that f L p X = µ sup 0 k K M k f > A p + B p f p dµ. 6 Fix f L p X and note that Doob s maximal inequality implies that µ sup E f F k µ sup E f p Fk B p 0 k K B 0 k K B p Thus in order to prove 6 it will suffice to show that { } { µ sup M k f > \ sup E f } Fk A p 0 k K 0 k K B Consider the inclusion { } { sup 0 k K M k f > \ Therefore, if we introduce the sets and sup E f Fk 0 k K B K A k def = X \ Ω k def = k=0 0 j<k } X X X f p dµ. f p dµ. 7 { M k f > sup E f Fj < 0 j<k B { E f } Fj, B { E f } Fk A k. B }. 8 Then A k F k, the sets Ω k are disjoint, and using 5 we see that 8 implies the inclusion { } { sup M k f > \ sup E f } K Fk { Ak M k f > } 0 k K 0 k K B k=0 = K { M k Ak f > }. 9 On the other hand, from 4 we have Mk f Ak \Ω k L X B E f Ak \Ω k Fk L X = B E f Fk Ak \Ω k L X B B =. Hence by the sublinearity of M k we have the following inclusion up to sets of measure zero: { M k f Ak > } { M k f Ωk > k=0 }. 30

13 Combining 9 with 30 and the assumption 3, we obtain { } µ sup 0 k K M k f > { \ K k=0 sup E f Fk 0 k K B A p This is precisely the estimate 7, as desired. } f p dµ = A p Ω k K k=0 K k=0 Ω k 3. Localization of maximal inequalities µ M k f Ωk > f p dµ A p X f p dµ. Let X, d, µ be a bounded metric measure space. Given a partition P of X and x X, we denote by Px the unique element of P containing X. We shall say that a sequence {P k } k=0 of partitions of X is a partition tree if the following conditions hold true: P 0 is the trivial partition {X}. For every x X and k {0} N we have diamp k x diamx. 3 k For every k {0} N the partition P k+ is a refinement of the partition P k, i.e., for every x X we have P k+ x P k x. For β > 0, a probability distribution Pr over partition trees {P k } k=0 is said to be β-padded if for every x X and every k N, [ Pr B x, β diamx k ] P k x. 3 Note that 3 has the following simple consequence, which we will use later: for every measurable set Ω X denote { Ω padk def β = x Ω : B x, β diamx } P k k x. 33 Thus Ω padk β [ E µ is a random subset of Ω. By Fubini s theorem we have: ] [ Ω padk β = Pr B x, β diamx ] P k k x Ω dµx 3 µω. 34 Remark 3.. In the definitions above we implicitly made the assumptions that certain events{ are measurable in the appropriate } measure spaces. Namely, for 3 we need the event B x, β diamx P k k x to be Pr-measurable for every x X and k {0} N, } and for 34 we need the event {x, {P k } k=0 : x Ω B x, β diamx P k k x to be measurable with respect to µ Pr for all k {0} N. These assumptions will be trivially satisfied in the concrete constructions below. Remark 3.. In the above definitions we made some arbitrary choices: the factor in 3 k can be taken to be some other factor r k > 0, and the lower bound on the probability in 3 can be taken to be some other probability p k. Since we will not use these additional 3

14 degrees of freedom here, we chose not to mention them for the sake of simplifying notation. But, the arguments below can be easily carried out in greater generality, which might be useful for future applications of these notions. The following lemma deals with the existence of padded random partition trees on microdoubling metric measure spaces. The argument is similar to the proof of Theorem 3.7 in [8], which is based on ideas from the theoretical computer science literature [, ]. The last part of the argument is in the spirit of the proof of the main padding inequality in [37]. Lemma 3.. Fix n and K 5. Let X, d, µ be a separable bounded metric measure space which satisfies 8. Then X admits a -padded probability distribution over 6n log K partition trees. Remark 3.3. Let X, d is a separable complete and bounded metric space which is doubling with constant λ, i.e., every ball in X can be covered by at most λ balls of half the radius. It is a classical fact, due to Vol berg and Konyagin [54] in the case of compact spaces, and Luukkainen and Saksman [33] in the case of general complete spaces see also [55] and chapter 3 in [7], that X admits a non-degenerate measure µ which is doubling with constant λ the power can be replaced here by any power bigger than. Thus the conclusion of Lemma 3. holds in this case with n = and K = λ. Proof of Lemma 3.. By rescaling the metric we may assume without loss of generality that diamx =. Since X is bounded, µx <, and we may therefore normalize µ to be a probability measure. Let x, x, x 3,... be points chosen uniformly and independently at random from X according to the measure µ, i.e., x, x,... is distributed according to the probability measure µ ℵ 0. For each k let r k be a random variable that is distributed uniformly on the interval [ k, k ]. We assume that r, r,... are independent. Let Pr denote the joint distribution of x, x,..., r, r,.... For every k N define a random variable j k : X N { } by j k x def = inf {j N { } : dx, x j r k }. Note that j k x is almost surely finite for every x X, since each x j has positive probability of falling into Bx, r k B x, k see the argument in [8] for more details. Since X is separable, it follows that the event x X k= {j kx < } has probability. From now on we will condition on this event. For every k N and l,..., l k N define P l,..., l k def = {x X : j x = l,..., j k x = l k }. Then P k def = {P l,..., l k : l,..., l k N} is a partition of X. By definition and for all k N, P l,..., l k Bx lk, r k B x lk, k, P l,..., l k, l k+ P l,..., l k. Therefore P k+ is a refinement of P k and diamp k x k for all x X. Denote β = 6n log K. 35 4

15 Since K 5, we have β <. Fix k N and x X and observe that 5 [ Pr B x, β ] [ k { P k k x = Pr y B x, β } ], j k l x = j l y l= k l= Fix l {,..., k}. Observe the following inclusion: { y B x, β }, j k l x j l y i i= j= [ Pr y B x, β ], j k l x j l y. 36 {r l β k < dx i, x r l + β k dx j, x > r l + β k }. 37 To prove 37, assume that the event on the left hand side of 37 occurs, i.e., that there is some y B x, β for which k jl x j l y. Let i N be the first index such that dx i, x r l + β. In order to prove that the event in the right hand side of 37 occurs, k it suffices to show that the event i { } j= rl β < dx k i, x r l + β dx k j, x > r l + β k occurs, which, by the minimality of i, is equivalent to showing that dx i, x > r l β. So, k assume for the sake of contradiction that dx i, x r l β. This implies in particular that k j l x = i, and moreover, since y B x, β, we have k dxi, y r l, implying that j l y i. But, d x, x jl y d y, xjl y + dx, y rl + β, and the minimality of i implies that k j l y i. Thus j l y = i = j l x, contradicting our assumption on y. Now, 37 implies that [ Pr y B x, β ], j k l x j l y e l l+ µ B x, r + β µ B x, r β l k k µ B x, r + β i dr k i= = l+ e lb µ 4 e lb B x, r β k µ B dr. 38 x, r + β k Denote ht def = log µ Bx, s. Then by Jensen s inequality we see that l l+ µ B x, r β k l µ B dr = l+ x, r β k exp l e hr β k hr+ β l l+ l l 5 k dr [ h r β h r + β ] dr. 39 k k

16 The expression in the exponent in 39 can be estimated as follows: l l [ h r β h r + β ] l +β k l +β k dr = h s ds h s ds k k l β k l β k β k+ [ h l βe k h l + β k]. 40 By recalling the definition of h, a combination of 38, 39, 40 yields the bound, [ Pr y B x, β ] µ B x, l β k β k l+3, j k l x j l y. 4 µ B x, l + β k Note that since l k and β we know that 5 l + β k + n n+ l β k. Hence, combining the assumption 8 with 4, we see that Pr [ y B x, β k, j l x j l y Plugging 4 into 36 we see that [ ] Pr B x, 6n log K k P k x ] K n+β k l+3 n + β k l+3 log K 35 k l. 4 [ = Pr B x, β ] P k k x This is precisely the statement that the partition tree {P k } k=0 is 6n log K -padded. k k l. The connection between the existence of padded random partition trees and the Hardy- Littlewood maximal inequality is established in the proof of Theorem.. Proof of Theorem.. By a standard monotone convergence argument we may assume that R is bounded, say R [0, D] for some D >. Fix f L p X. By homogeneity it suffices to show that µ M R f > C p + log log K Q p + K p f p dµ, + log n X where C > 0 is a universal constant and Q def = sup MR [r,nr] LpX L. 43 p, X r>0 By monotone convergence we may assume that f and hence also M R f has bounded support. We would like to apply Theorem., but unfortunately there are no obvious candidates for F k with which we have either 4 or 5. Nevertheless, we shall be able to proceed by replacing M R with a slightly modified variant. Let E be the support of f and denote and E def = {x X : dx, E D}, E def = {x X : dx, E D}. 6 l=

17 Then E E E and diame 4D + diame <. Moreover the support of M R f is contained in E. It will therefore suffice to prove that M R LpE L p, E log log K + Q + K. + log n By rescaling the metric we may assume that diame =. Once this is achieved we may also assume that R 0, ], since the operator M R,, viewed as an operator on L p E, is pointwise bounded by the averaging operator on E. Using Lemma 3., let {P k } k=0 be a random partition tree on E which is β-padded, where β = 6n log K. Let m be the largest integer such that m β. Denote for k 0 and i {,, 3}, def = R [ 3k+im, 3k +im] and R i def = Rk. i R i k Thus R = R R R 3, which implies that µ M R f > = µ max {M R f, M R f, M R 3f} > Fix i {,, 3} and k N {0}, and define E i k Then the sets E i k def = Recalling 33, we denote Ẽ i k { } x E : M R i k fx > \ are disjoint and µ M R if > = µ def = E i k pad3k +i+m β = Then by 34 we know that k N {0} µ M R f > + µ M R f > + µ M R 3f >. 44 k j=0 sup M R i k f > k N {0} { x Ek i : B x, [ E µ k=0 ] Ẽi k { } x E : M R i j fx >. = µ Ek i. 45 k=0 } β P 3k +i+m 3k +i+m x. µ Ei k. 46 Plugging 46 into 45 we see that [ µ M R if > E µ Ẽi k ] [ ] = E µ sup M R i k f >, 47 k N {0} where M R i k is the sublinear operator M R i k g def = Ẽi k M R i k g. 7

18 Write r = 3k+im log log K and let v + be an integer such that +log n m/v n. By the definition of Q, for every g L p E and t > 0 we have µ MR i g > t µ M k R i k g > t = µ M R [r, m r]g > t Thus, v µ M R [r um v,nr um v ] g > t vq p g p L pe. t p u=0 g L p E = M R i k g v /p Q g LpE. 48 Lp, E For every k N {0} we let F k def = σp k be the σ-algebra generated by the partition P k. Then F 0 F F. We claim that for every k N {0}, if F F 3k+i+m then F MR i k+ g = M R i k+ F g. 49 By the definition of M R i k, in order to prove 49 we have to show that for almost every x E we have F x Ẽi k+ x M R i k+ gx = Ẽi k+ x M R i k+ F gx. 50 It is non-trivial to check 50 only when x Ẽi k+, in which case we are guaranteed that B x, β 3k+i+m P 3k+i+m x. But since F F 3k+i+m, we know that P 3k+i+m x is either disjoint from F or contained in F. If P 3k+i+m x F, then for every r Rk+ i, Bx, r B x, 3k+i+m B x, β 3k+i+m P 3k+i+m x F, 5 where we used the fact that r 3k+ +im and m β. The inclusion 5 implies that both sides of the equation 50 are equal to M R i k+ gx. On the other hand, if P 3k+i+m x is disjoint from F, then Bx, r is disjoint from F for all r Rk+ i, implying that both sides of the equation 50 vanish. This concludes the proof of 49. Fix g L E, and extend g to a function on X whose value is 0 outside E. Assume that E g F3k+i+m L E =. This implies that for all F F 3k+i+m we have g dµ = F g dµ µ F E µf. F E 5 Fix r R i k and x E. Denote F def = { C P3k+i+m : C Bx, r } F 3k+i+m. Note that Bx, r E, which implies that F Bx, r. 53 8

19 Moreover, F B x, r + sup C P 3k+i+m diamc B x, r + 3k+i+m B x, + m r, 54 where in the last inclusion in 54 we used the fact that r Rk i implies that r 3k+im. Hence, g dµ 53 g dµ 5 µf µbx, r Bx,r µbx, r F µbx, r 54 µ B x, + m r µbx, r µ B x, + n µbx, r r 8 K, 55 We are now in position to apply Theorem. to the increasing sequence of σ-algebras { F3k+i+m } k=0 and the sublinear operators { M R i k } k=0, with A = v /p Q, due to 48, and B = K, due to 55: µ sup M R i k f > k N {0} p vq p + p K p X f p dµ p + Using 47 and 44, we therefore deduce that /p [µ M R f > ] /p log log K + Q + K + log n as required. 4. An argument of E. Lindenstrauss log log K Q p + p K p f p dµ. + log n X f LpX, We now present an alternative approach to Corollary., following an argument of E. Lindenstrauss [3]. Let us first make some definitions. We fix a metric measure space X, d, µ. Given any two radii r, r > 0 and a center x X, we define the enlarged ball Bx, r, r by Bx, r, r def = By, r = {z X : dx, y r dy, z r for some y X}. Thus, for instance, y Bx,r Bx, r Bx, r, r Bx, r + r. 56 In analogy to [3], we say that a finite sequence of radii 0 < r < r < < r k is tempered with constant K if we have the bound j {,..., k} x X y Bx, r j, µ Bx, r j j B x, r j, r i Kµ B y, r j i=

20 Theorem 4. Lindenstrauss maximal inequality. Let X, d, µ be a metric measure space, and let 0 < r < r <... < r k be a sequence of radii which is tempered with constant K. Then we have the weak, maximal inequality µ x X : max j k for all f L X and λ > 0. f dµ > λ Bx, r j Bx,r j e K e λ f L X Proof of Corollary. assuming Theorem 4.. Assume that X, d, µ obeys the strong microdoubling condition 4. It is immediate to check that any sequence 0 < r < r <... < r k obeying the lacunarity condition r j nr j will be tempered with constant K, and hence by Theorem 4., µ x X : max j k f dµ > λ Bx, r j Bx,r j e K e λ f L X. If instead we have the lacunarity condition r j r j, then we can sparsify this sequence into Olog n subsequences obeying the prior lacunarity condition, and hence, by subadditivity, µ x X : max f dµ > λ K log n f L X. j k Bx, r j λ Bx,r j From monotone convergence we then conclude 6. Similarly, any sequence obeying the lacunarity condition r j + r n j can be sparsified into On log n sequences which have a lacunarity ratio of n. By monotone convergence this implies that µ x X : sup r + Bx, r n Z Bx,r f dµ > λ Kn log n f L X, λ where + n Z denotes the integer powers of +. Now note from 4 that every ball is n contained in a ball whose radius is an integer power of +, and whose measure is at most n K times larger. Thus Mfx K sup f dµ, r + Bx, r n Z Bx,r and 5 follows. Proof of Theorem 4.. As in [3], this is achieved by a randomized variant of the Vitali covering argument. We may take f to be non-negative, and normalize λ =. For each j {,..., k}, let E j be a compact subset of X on which we have x E j = Bx, r j Bx,r j By inner regularity it will suffice to show that k µ E j e e K j= 0 X f dµ >. 58 f dµ. 59

21 We establish 59 by induction on k. The case k = 0 is vacuously true, so suppose k and the claim has already been proven for k i.e, that 59 holds true for all non-negative f L X and all sets {E j } k j= satisfying 58. By compactness, we see that there exists an ε > 0 such that x E k = µbx, r k > ε. We then define the extended ball B x def = Bx, r k k Bx, r k, r j. Thus, since the sequence of radii {r j } k j= is tempered, for all y Bx, r k, If we then define the intensity function j= ε < µ B y KµBx, r k. 60 px def = inf y Bx,r k µb y, then p is a measurable function on E k which is bounded both above and below: Kµ Bx, r k px < ε. 6 We now introduce a Poisson process Σ on E k with intensity px. Thus Σ is a random subset 6 of E k which will be almost surely finite, and more precisely, for any non-negative measurable weight w : E k R +, the quantity x Σ wx is a Poisson random variable with expectation [ ] def α w = E wx = wp dµ, 6 E k i.e., for any integer k 0 Now we define the random sets x Σ Pr wx = k x Σ E def = B x and F def = Bx, r k. x Σ x Σ = e αw αw k. 63 k! Then, k k µ E j µe k + µe + µ E j \ E. 64 j= j= 6 If E k contains atoms, then Σ may contain multiplicity, thus it is really a multiset rather than a set in this case. One way to create Σ is to let N be a Poisson random variable with expectation P def = E k pdµ and then let Σ = {x,..., x N } where x,..., x N are iid elements of E chosen using the probability distribution pdµ/p.

22 Let us investigate the third term in 64. Fix j {,..., k }. If x E j \ E, then f dµ >. Bx, r j Bx,r j But, since x / E it follows from our definitions that Bx, r j is disjoint from F. Thus we have f X\F dµ >. Bx, r j Bx,r j We can therefore apply the induction hypothesis to the sets {E j \ E } k j= f X\F, and conclude that k µ E j \ E e e K f dµ. X\F j= It follows from 64 that it suffices to show that µe k + E [µe ] E [µe k + µe ] Now, applying 6 and 63 with w def = /p, we have [ ] µe k = E, px while from definition of E we have µe x Σ x Σ x Σ µ B x = x Σ Thus, in order to prove 65 it suffices to show that [ ] E e [ px e KE From 58 we know that for all x Σ, and hence E x Σ [ x Σ px < pxµbx, r k X and the function e [ ] e KE f dµ. 65 F F px. ] f dµ. 66 Bx,rk f dµ, ] [ ] E px X pxµbx, r k Bx,r k f dµ. 67 x Σ Bx,r k y Fix y X. From 6 with wx = pxµbx,r k, we see that [ ] E pxµbx, r k Bx,r k y = µbx, r k E k By,r k dµx. 68

23 By substituting 68 into 67, we see that in order to prove 66 it will suffice to prove the pointwise estimate E k By,r k µbx, r k dµx ek e E [ F y], 69 for all y X. Now observe that the definition of F implies that F y = if and only if Σ By, r k. But, recall from 6 using wx = By,rk x that Σ By, r k is a Poisson random variable with expectation αy def = By,rk p dµ = E k E k By,r k px dµx, 70 and thus E [ F y] = e αy. 7 A combination of 6 and 70 yields the bound dµx Kαy. 7 µbx, r k E k By,r k The definition of px implies that if y Bx, r k then px, since µb y µby,r k B y Bx, r k. In combination with 70, we deduce that αy. But, the function α e α is decreasing on [0,, and therefore e αy e αy. This, in α combination with 7 and 7, implies 69, and completes the proof of Theorem 4.. As observed in [3], the above argument allows us to extract a good maximal inequality for sufficiently sparse subsequences of radii if the situation is sufficiently amenable. In our current context, the analogue for amenability is in fact subexponential growth: Corollary 4.. Let X, d, µ be a metric measure space such that µbx, r is independent of x X for all r > 0. Suppose also that we have the sub-exponential growth condition log µbx, r lim = 0 73 r r for any x X note that our assumption implies that the choice of x is in fact irrelevant. Then there exists a sequence of radii 0 < r < r <... tending to infinity such that we have the maximal inequality f L X = sup k A rk f L, X where the averaging operators A r are given by A r g def = Bx,r 4 f L X, Bx,r g dµ. Proof. We construct the radii recursively as follows. We set r def =. If r,..., r k have already been chosen, we choose r k+ > max {r k, k} so that log µ B x, r k+ + r k µ B x, r k for any x X. Such a radius must exist, since otherwise one would easily contradict 73. The sequence of radii is tempered with constant K = e 0.00, and the claim follows since K e < 4. 3

24 5. The infinite tree Fix k and let T be the infinite rooted k-ary tree with the usual graph metric and the counting measure µ. In this section we prove Theorem.5. The first standard step is to replace the Hardy-Littlewood maximal function with the spherical maximal function where Sx, r is the sphere M fx def = sup r 0 Sx, r y Sx,r fy, Sx, r def = {y T : dx, y = r}. Since every ball can be written as the disjoint union of spheres, we have the pointwise estimate and so it suffices to show that Mfx M fx, µ x T : M fx λ λ f L T, 74 for all f L T and λ > 0. Our arguments rely on the following expander-type estimate. We use E = µe to denote the cardinality of a finite set E T. Lemma 5.. Let E, F be finite subsets of T and let r 0 be an integer. Then {x, y E F : dx, y = r} E / F / k r/. This bound should be compared against the trivial bounds of E k r and F k r. It is superior when E / F lies between k r and k r. By setting E and F equal to concentric spheres one can verify that the bound is essentially sharp in this case. Proof. Let us subdivide T = j=0 T j, where T j is the generation of the tree at depth j thus for instance T j = k j def def. We then define E j = E T j and F j = F T j. Observe that in order for an element in E j and an element in F i to have distance exactly r, we must have i = j + r m for some m {0,..., r}. Thus we can write {x, y E F : dx, y = r} = r m=0 i,j N {0} i=j+r m {x, y E j F i : dx, y = r}. 75 Fix m {0,..., r} and i, j N {0} such that i = j + r m. Observe that if x T j and y T i are at distance r in T, then the m th parent of x equals the r m th parent of y. From this we conclude that for each x T j there are at most k r m elements of y T i with dx, y = r, and conversely for each y T i there are at most k m elements of x T j with dx, y = r. Thus {x, y E j F i : dx, y = r} min { k r m E j, k m F i }. 76 4

25 A combination of 75 and 76 implies that our task is therefore to show that r min { k r m E j, k m F i } E / F / k r/. 77 If we write c j def = E j k j and we have r m=0 i,j N {0} i=j+r m m=0 i,j N {0} i=j+r m def and d j = F j k j j=0 for j 0 and c j def = d j def k j c j = E min { k r m E j, k m F i } = r and = 0 for j < 0 then we have k j d j = F, 78 j=0 m=0 i,j N {0} i=j+r m k i+j+r/ min {c j, d i } k r/ i,j=0 k i+j/ min {c j, d i }. 79 A combination of 78 and 79 shows that in order to prove 77 it will suffice to show that / / k i+j/ min {c j, d i } k j c j k i d i. i,j=0 j 0 i 0 To prove this inequality, let α be a real parameter to be chosen later, and estimate k i+j/ min {c j, d i } k i+j/ c j + k i+j/ d i k j+ α cj + k i α di. i,j=0 i,j N {0} i<j+α Optimising in α we obtain the required result. i,j N {0} i j+α For each r 0, let A r denote the spherical averaging operator A rfx def = fy. µsx, r y Sx,r Thus M fx = sup r 0 A rfx. We can use Lemma 5. to obtain a distributional estimate on A r. Lemma 5.. Let f L T, r > 0 and λ > 0. Then µ A rf λ n k r n µ f n λ. n N {0} n k r Proof. We may take f to be non-negative. By dividing f by λ we may normalize λ =. We bound f + n En + f {f kr }, 80 n N {0} n k r 5 j=0 i=0