Weak-type (1,1) inequality for discrete maximal functions and pointwise ergodic theorems along thin arithmetic sets

We establish weak-type $(1,1)$ bounds for the maximal function associated with ergodic averaging operators modeled on a wide class of thin deterministic sets $B$. As a corollary we obtain the corresponding pointwise convergence result on $L^1$. This contributes yet another counterexample for the conjecture of Rosenblatt and Wierdl from 1991 asserting the failure of pointwise convergence on $L^1$ of ergodic averages along arithmetic sets with zero Banach density. The second main result is a multiparameter pointwise ergodic theorem in the spirit of Dunford and Zygmund along $B$ on $L^p$, $p>1$, which is derived by establishing uniform oscillation estimates and certain vector-valued maximal estimates.

1. Introduction 1.1.Brief historical remarks.In 1991 Rosenblatt and Wierdl [1,Conjecture 4.1] formulated a famous conjecture asserting that for any arithmetical set A with zero Banach density and for any (X, B, µ, T ) aperiodic probability dynamical system, there exists a function f ∈ L 1 µ (X), such that does not converge almost everywhere, i.e. the set of x ∈ X such that lim N →∞ M A,N f (x) does not exist has positive measure.This was disproven in 2006 by Buczolich [2], where he provided a counterexample by constructing inductively an appropriate set A of zero Banach density for which one gets the pointwise convergence of the ergodic averages M A,N f for all f ∈ L 1 .Calderón's transference principle suggests that such questions are closely related to the study of weaktype (1, 1) estimates for the maximal function corresponding to those averages over the integer shift system, namely for the operator A year later it was shown [3] that for A = ⌊n c ⌋ : n ∈ N with c ∈ 1, 1001 1000 , the operator M A is of weak-type (1, 1), and as a corollary the authors proved pointwise convergence on L 1 for the corresponding ergodic averages along the set A, providing a counterexample of the aforementioned conjecture given by a concrete formula.This class of examples was extended in [4] where the author established the weaktype (1,1) bounds for M A and the corresponding pointwise ergodic theorem on L 1 for sets of the form ⌊n c ℓ(n)⌋ : n ∈ N , where c is close to 1 and ℓ is a certain kind of slowly varying function, for example any iterate of log, see Definitions 1.1, 1.2 below.
One of the main results of the present work is a natural extension of the result from [4] and in order to formulate it, we must introduce two families of functions that one may think of as slowly varying and regularly varying functions respectively.Definition 1.1.Fix x 0 ≥ 1 and let L denote the set of all functions ℓ : [x 0 , ∞) → [1, ∞) such that ℓ(x) = exp where ϑ ∈ C 2 ([x 0 , ∞)) is a real-valued function satisfying ϑ(x) → 0 , xϑ ′ (x) → 0 , x 2 ϑ ′′ (x) → 0 as x → ∞.
We note that L 0 ⊆ L.
Definition 1.3.Fix x 0 ≥ 1, c ∈ (1, ∞) and let R c be the set of all functions h : [x 0 , ∞) → [1, ∞) such that h is strictly increasing, convex and of the form h(x) = x c ℓ(x) for some ℓ ∈ L. We define R 1 analogously, but with the extra assumption that ℓ ∈ L 0 .
We are now ready to give the definitions of the arithmetic sets we are interested in.Let c 1 , c 2 ∈ [1, 2) and let us fix h 1 and h 2 in R c 1 and R c 2 respectively.Let ϕ 1 and ϕ 2 be the compositional inverses of h 1 and h 2 and for convenience, let γ 1 = 1/c 1 and γ 2 = 1/c 2 .Let us fix a function ψ Those sets have been introduced and studied in [5], where the authors proved that the Hardy-Littlewood majorant property holds for them, see Theorem 1 and 2 in [5], as a corollary of a restriction theorem.Recently, the author [6] proved an analogous result for P ∩ B ± , see Theorem 1.8 in [6], as well as Roth's theorem in these sets, namely, it was show that any subset of the primes of the form B ± with positive relative upper density contains infinitely many non-trivial three-term arithmetic progressions.
Following [6] we repeat and extend some comments on the sophisticated nature of the sets B ± , simultaneously we try to convince the reader about the richness of the family of the sets that we consider.More precisely, to motivate the definition of B ± , we note that For γ ∈ (0, 1), h 1 (x) = h 2 (x) = x 1/γ and ψ(x) = ϕ 1 (x + 1) − ϕ 1 (x), note that the last condition becomes m 1/γ ∈ [n, n + 1) or n = ⌊m 1/γ ⌋, thus B − = { ⌊m 1/γ ⌋ : m ∈ N }.It is not difficult to see that any set (1.4) can also be brought in the form B − by similar appropriate choices.Thus the family of sets we consider includes the fractional powers with exponent close to 1 and even the more general sets considered in [4].
1.2.One-parameter ergodic theorem on L 1 .We are now ready to state one of the main results of our paper.Due to some technical complications, we demand further that ϕ 1 ≃ ϕ 2 , and note that this implies that c 1 = c 2 .
Theorem 1.5 (Weak-type (1,1) inequality for M B ± ).Assume c 1 ∈ (1, 30/29) and ϕ 1 ≃ ϕ 2 .Then the maximal function is of weak-type (1,1), i.e.: |{x ∈ Z : By interpolation, this implies that for all p ∈ (1, ∞], there exists a constant C p > 0 such that We use this, together with 2-oscillation estimates, see Theorem 1.9, to obtain the following pointwise convergence result. Theorem 1.6 (Pointwise ergodic theorem).Assume c 1 ∈ (1, 30/29), ϕ 1 ≃ ϕ 2 and let (X, B, µ, T ) be an invertible σ-finite measure preserving system.For any p ∈ [1, ∞) and any f ∈ L p µ (X), we have that Before discussing the strategy of our proofs, we would like to further examine the sets B + and B − .Let us restrict our attention to the sets B + , which we call B from now on, as the results for B − are of equal difficulty.Note that Even in simple examples, we should expect that B will contain a lot of consecutive integers after n 0 .For example, for any ϕ which implies that B contains 99 consecutive integers after n 0 .Clearly, m ≤ ϕ 1 (n 0 + l) for all l ∈ N 0 .If we assume for the sake of a contradiction that for some l ∈ {0, . . ., 99} we have that m < ϕ 1 (n 0 + l) − ψ 1 (n 0 + l), then and by the Mean Value Theorem, there exists a ξ n 0 ,l ∈ (n 0 − 1, n 0 + l) such that Thus 100 < l + 1 which is a contradiction.This shows that the set B we considered here contains infinitely many full blocks of 100 consecutive integers.Such a set B stands in sharp contrast to the sets of the form N h = { ⌊h(m)⌋ : m ∈ N }, h ∈ R c , as the gaps between members of such sets tend to infinity.
In general, the constant determines an important qualitative aspect of the form of the sets B, see Lemma 2.4.Loosely speaking, for big intervals of integers where the ratio is bigger than L, we expect that B will contain blocks of length at least L/C, where C is the doubling constant of ϕ ′ 1 .The technical issues that arose when trying to handle the case where B contains arbitrarily long intervals of integers (specifically in the counting Lemma 5.1) forced the author to impose the restriction ϕ ′ 1 ≃ ϕ ′ 2 , which is equivalent to ϕ 1 ≃ ϕ 2 .Even in this simpler case, we note that ϕ ′ 2 (x) ϕ ′ 1 (x) could oscillate and thus B could contain blocks of various oscillating lengths.We are now ready to make the following remarks.
Remark 1.7 (Smooth dyadic maximal operator).To establish Theorem 1.5, it is convenient to work with a smooth dyadic variant of the maximal operator.More precisely, let's fix η ∈ C ∞ (R) such that 0 ≤ η(x) ≤ 1 for all x ∈ R, supp(η) ⊆ (1/2, 4) and η(x) = 1 for all x ∈ [1,2].We define and note that The first inequality is straightforward, and for the second one, note that Lemma 2.1 implies that Thus it suffices to establish the weak-type (1,1) bound for the smooth dyadic maximal function, so let us denote M (sd) B by M and let We give a brief overview of the main ideas of the proof of Theorem 1.5.We use a subtle variation of the Calderón-Zygmund decomposition that was introduced by Fefferman [7], see also [8], in a similar manner to [3] and [4].More specifically, after approximating K n * K n by suitable well-behaving functions, see Lemma 5.2, we employ a refined Calderón-Zygmund decomposition which allows us to use ℓ 2 -estimates for the "very bad" part of the decomposition, see subsection 5.2.The aforementioned approximation is analogous to the one presented in section 5 of [4] and similar techniques are used here.The novelty lies in the sophisticated nature of the sets B which complicates the situation substancially.For example, bounding |K n * K n (x)| for small values of x, see Lemma 5.1, is precisely what forced the author to impose the extra assumption ϕ 1 ≃ ϕ 2 .To carry out the approximation one needs to estimate certain exponential sums and the main tool is Van der Corput's inequality.Some of the necessary exponential sum estimates can be readily found in section 3 of [4] and suitable extensions are already established by the author in [6].Finally, we formulate an abstract result, see Theorem 5.5, which is a generalization of Theorem 6.1 of [4], adapted to our approximation for K n * K n .We give the full proof of Theorem 1.5 in section 5, see section 3 of [3], and sections 5 and 6 of [4].
Combining Theorem 1.5 with the trivial estimate M B ℓ ∞ (Z)→ℓ ∞ (Z) 1, we obtain by interpolation that M B ℓ p (Z)→ℓ p (Z) p 1, for all p ∈ (1, ∞].Calderón's transference principle implies that for any invertible σ-finite measure preserving system (X, B, µ, T ) we have ), and thus to establish Theorem 1.6 it suffices to exhibit an L p µ -dense class of functions D p contained in L p B .The exponential sum estimates of Lemma 2.1 together with a straightforward adaptation of the argument presented in section 3 of [4], which uses ideas from [11], shows that one may take D p = L p µ (X) ∩ L 2 µ (X) and conclude.Instead of this, one can derive the pointwise convergence theorem immediately by the much stronger uniform 2-oscillation L p µ -estimates of Theorem 1.9, which are also exploited in the sequel.
1.3.Multi-parameter ergodic theorem.The second main result of our paper is a multi-parameter variant of Theorem 1.6.Here we discard the assumption ϕ 1 ≃ ϕ 2 and the acceptable range of c 1 and c 2 is considerably larger.In contrast to the one-parameter situation, weak-type (1, 1) estimates do not hold here.
To make the exposition slightly cleaner, let's fix k ∈ N and B 1 , . . ., B k as in the introduction where if We are ready to state the second main result.
Theorem 1.8.Assume (X, B, µ) is a σ-finite measure space and {S i : i ∈ [k]} is a family of invertible µ-invariant commuting transformations.Then for any p ∈ (1, ∞) and any f ∈ L p µ (X) we have that where We make some brief historical remarks.In 1951 Dunford [12] and Zygmund [13] independently showed that given a σ-finite measure space (X, B, µ) and a family of µ-invariant transformations {T i : i ∈ [k]}, for any p ∈ (1, ∞) and any f ∈ L p µ (X), we have For k ≥ 2, pointwise convergence fails on L 1 .Motivated by that observation and after his seminal work on pointwise ergodic theory [9,10,11], Bourgain showed that for any p ∈ (1, ∞) and any f ∈ L p µ (X), we have } is a family of commuting and invertible µ-invariant transformations.In contrast to Dunford and Zygmund's result, the commutativity assumption turns out to be indispensable for the polynomial case.For a more thorough exposition on the matter we refer the reader to Section 1.2 in [14] as well as the introduction from [15], see page 3.In the spirit of the above, Theorem 1.8 establishes the multi-parameter result for orbits along sets of the form B. For example, for appropriate choices of parameters, Theorem 1.8 implies that for any p ∈ (1, ∞) and any f ∈ L p µ (X), we have where c 1 , . . ., c k ∈ (1, 6/5) and {T i : i ∈ [k]} is a family of commuting and invertible µ-invariant transformations.
Using an abstract multi-parameter oscillation result from [14], we reduce the task of proving the above theorem to showing the following useful quantitative uniform estimates, which may be of independent interest.Theorem 1.9 (Uniform 2-oscillation and vector-valued maximal estimates).Assume ) and B as in the introduction.Assume (X, B, µ) is a σ-finite measure space and T is an invertible µ-invariant transformation.Let Then for any p ∈ (1, ∞), there exists a constant C p such that (1.10) sup and such that for any (f j ) j∈Z ∈ L p µ Z; ℓ 2 (Z) .
We briefly note that for any Y ⊆ X ⊆ R, with |Y | > 2, we have that For the definition of multi-parameter oscillations as well as the basic properties of oscillations we refer the reader to section 2 from [14].
We now comment on the proof of Theorem 1.9.Again, Calderón's transference principle suggests that it suffices to establish these estimates for the integer shift system.Ultimately, those estimates are derived from the analogous ones for the standard discrete Hardy-Littlewood averaging operator.For the vectorvalued maximal inequality we use the exponential sum estimates of Lemma 2.1 together with the fact that ψ behaves "like a constant" in dyadic blocks in order to eventually be able to use the corresponding estimates for the Hardy-Littlewood averaging operator (for example see Theorem 1 in [16] or Theorem C in [17]).The situation is much more complicated for the oscillations.We follow the strategy from [18] and [19], and we break our analysis into short and long oscillations, and instead of opting to handle as our "long oscillations" the rather natural choice {2 n : n ∈ N 0 }, we choose a much denser set, namely on {⌊2 n τ ⌋ : n ∈ N 0 }, for τ small.This affords us to bound the short oscillations straightforwardly.Loosely speaking, the long oscillations are treated in a similar manner to the vector-valued maximal inequality, but the fact that the 2-oscillations are not a positive operator makes the use of the fact that ψ behaves nicely in dyadic blocks difficult.Here, we adapt the argument from section 5 in [20] to our oscillation setting in order to compare averages with different weights.Again, we use the uniform oscillation estimates for the discrete Hardy-Littlewood averaging operator to conclude (which one may find for example in [21] or [22]).Finally, we mention that the exponential sum estimates help us understand some error terms on ℓ 2 , and Riesz-Thorin interpolation together with trivial bounds coming from the fact that we deal with averaging operators help us establish the corresponding ℓ p bounds.1.4.Notation.We denote by C a positive constant that may change from occurrence to occurrence.If A, B are two non-negative quantities, we write A B or B A to denote that there exists a positive constant C such that A ≤ CB.Whenever A B and A B we write A ≃ B. For two complexvalued functions f, g we write f ∼ g to denote that lim x→∞ f (x) g(x) = 1.We denote the average value of a function f : For any natural number N , we let [N ] = {1, 2, . . ., N }.

Basic Properties of the sets B
In this section we collect some useful properties of the sets B. We begin by stating an exponential sum estimate proven in [5].Here we fix two constants as in the introduction and all the implied constants may depend on them.We note that we use the basic properties of those functions as described in Lemma 2.6 and Lemma 2.14 from [4] without further mention.
as well as where the implied constant does not depend on ξ or N .
Proof.We note that one is derived from the other using summation by parts.The proof can be found in page 6 as well as in Lemma 3.2 in [5].
Proof.Let us assume that ϕ 2 ϕ 1 and {n, n + 1, . . ., n + l − 1} ⊆ B. We wish to bound l.Let us notice that B may be partitioned as follows where For sufficiently large m, k with m < k, we have that dist(B m , B k ) ≥ 2 since if we assume for the sake of a contradiction that dist(B m , B k ) = 1 then there exists n ∈ B m such that and n + 1 ∈ B k .But then and thus and by the Mean Value Theorem there exists ξ n ∈ (n, n + 1) such that ϕ ′ 1 (ξ n ) > 1/2, which will be a contradiction for sufficiently large n, since ϕ ′ 1 (x) → 0, as x → ∞.Thus, if we ignore some first few terms of the set B, then the fact that {n, n + 1, . . ., n + l − 1} ⊆ B, together with our previous observation, imply that there exists an m ∈ N such that {n, n + 1, . . ., n and thus (l − 1) 1, since ϕ 2 ϕ 1 .Thus the set B does not contain arbitrarily long intervals.

Uniform 2-oscillation Estimates
Here we wish to prove the first half of Proposition 1.9, namely to establish the estimate (1.10).Similarly to the previous section, we fix B as in the introduction with ) and all the implied constants may depend on them.By the Calderón Transference Principle, in order to establish that estimate for any σ-finite measure preserving system, it suffices to establish it for the integer shift system, namely for (Z, P(Z), | • |, S) where S is the shift map S(x) = x − 1 and the | • | is the counting measure.To simplify the notation, we let B t = B ∩ [1, t] and write We therefore wish to show that for any p ∈ (1, ∞), there exists a constant C p such that To establish this result, we break the 2-oscillations into short and long ones.To do that, we need to carefully choose some parameters first.Let where , see [14] page 17.One may adapt the argument appearing in Lemma 1.3 in [23] to establish this.We deal with the second term of (3.2).Note that where we have used the fact that Then we use the triangular inequality to bound the last expression by , where δ n (x) = 1 {n} (x), and note that M t f (x) = K t * f (x).Thus we rewrite the expression above as We firstly consider the case p ∈ (2, ∞), we get where we have used Minkowski's inequality for p/2 > 1 and p > 1, and then Young's convolution inequality.
In the case where p ∈ (1, 2] we note where we have used Minkowski's inequality and Young's convolution inequality.Combining the two cases gives where q = min{2, p}.We focus on We know that there exists ε > 0 such that [5], page 5), thus we get We also note that by the Mean Value Theorem for f In either case, the first series is summable.The second series is also summable, since for example 2 −qεn τ p,τ n −2 , and thus To establish the desired estimate, it remains to bound the first term of (3.2).Note that We wish to show Proof of the estimate (3.4).We introduce some auxiliary averaging operators.Let where L t (x) = 1 |Bt| 1≤s≤t ψ(s)δ s (x).We may compare M t with A t as follows The first term of the expression (3.5) will be bounded using the Lemma 2.1 and interpolation.More specifically, we start with p = 2, and we note and for each n ∈ N 0 , we have and note that there exists χ > 0 such that for any ξ ∈ T we get which gives For the case of p = 2, firstly, let us assume that p ∈ (2, ∞).Note that there exists a positive constant where we have used Lemma 2.1 for ξ = 0. We may choose θ ∈ (0, 1) such that and use Riesz-Thorin interpolation theorem.Since for any n ∈ N 0 we have we interpolate to obtain We now note that since p/2 > 1 we have The case of p ∈ (1, 2) is similar; we choose θ ∈ (0, 1) such that 1 p = θ 2 + 1−θ p 0 .Riesz-Thorin interpolation theorem yields the same estimate as before and we note that We have appropriately bounded the first term of equation (3.5).We now focus on the second term.We will reduce the 2-oscillation ℓ p estimates for A 2 n τ to the corresponding ones for the C 2 n τ .Firstly, the analysis of the 2-oscillations will be made easier if we adjust A t to the following very similar operator One may use a similar argument to the one presented earlier to compare A 2 n τ and D 2 n τ .More precisely we have sup , we can bound the first term of equation (3.6) by C p,τ f ℓ p (Z) and our task has been reduced to estimating the 2-oscillations of D 2 n τ .In fact, we will be able to estimate the 2-oscillations of D n by comparing it with H n by adapting the strategy of [20], see section 5.For convenience, let Ψ(k) = 1≤s≤k ψ(s).We perform summation by parts where Without loss of generality, ψ(s) is decreasing.Thus (λ k s ) s,k∈N is a family of non-negative real numbers (in the spirit of Lemma 2 from [20]) and we note that for any k ∈ N we have that and we also have that for any fixed N ∈ N the sequence N s=1 λ k s is decreasing in k, since We note that for all k ∈ N, N k is increasing in t.Also, since for any fixed N ∈ N the sequence N s=1 λ k s is decreasing in k, we have that for any fixed t ∈ [0, 1), the sequence N k (t) is increasing in k.We also note that Finally, for any J ∈ N 0 and any I = {I 0 , . . ., I J } ∈ S J (N 0 ), we have that and now we finish the argument by noting that where we have used the fact that uniform 2-oscillation ℓ p -estimates do hold for the standard averaging operator, see [21] or [22], for any p ∈ (1, ∞), or more precisely, We note that This establishes the estimate (1.10).

4.
Vector-Valued maximal estimates and concluding the proof of Theorem 1.8 In this section we establish vector-valued estimates for the maximal function corresponding to M t .We fix a set B as in the introduction with c 1 ∈ [1, 2), c 2 ∈ [1, 6/5).By the Calderón Transference Principle, in order to establish estimate (1.11), it suffices to show the following.Proposition 4.1.For any p ∈ (1, ∞), there exists a constant C p such that for any (f j ) j∈Z ∈ ℓ p Z; ℓ 2 (Z) we have We focus on the first term.For p = 2 we note that We note that Plancherel theorem combined with Lemma 2.1 yield the following and thus For the case of p = 2, we proceed in a manner identical to the one of the previous section.Firstly, let us assume that p ∈ (2, ∞).We fix a p 0 > p and we note that there exists a positive constant C such that We may choose θ ∈ (0, 1) such that 1 p = θ 2 + 1−θ p 0 and use Riesz-Thorin interpolation theorem.Since for any n ∈ N 0 we have we interpolate to obtain Thus if we let T n : ℓ p (Z) → ℓ p (Z) such that T n f = M 2 n f − A 2 n f , then we know that T n is a bounded linear operator with T n ℓ p (Z)→ℓ p (Z) ≤ C(2 χθ ) −n .Thus, we know that T n has an ℓ 2 -valued extension (see [24], page 386) with the same norm, that is Finally, since p/2 > 1, we get For p ∈ (1, 2) the situation is similar, we choose p 0 ∈ (1, p), and θ ∈ (0, 1) such that 1 p = θ 2 + 1−θ p 0 and Riesz-Thorin interpolation theorem yields the estimate of (4.3), which in turn implies the estimate (4.4).Since p < 2, we have We have bounded appropriately the first term of (4.2) and all is left is to bound the second term.We firstly observe that for any n ∈ N 0 we get Since n ∈ N 0 was arbitrary, we have shown that , and thus the second term of 4.2 may dominated by This completes the proof.
The work of Section 3 and 4 proves Proposition 1.9.We immediately describe how Proposition 1.9 implies Theorem 1.8.
Proof of Theorem 1.8 assuming Proposition 1.9.We simply apply Proposition 4.1 from [14] to establish a multi-parameter uniform 2-oscillation estimate, which according to Remark 2.4 together with Proposition 2.8, page 15, from the same paper yield the desired result.

Proof of the weak-type (1,1) inequality
Throughout this section we have fixed a set B with ϕ 1 ≃ ϕ 2 and c 1 ∈ (1, 30/29).All constants may depend on ϕ 1 , ϕ 2 and ψ but on nothing else, unless stated otherwise.We remind the reader that it suffices to establish the weak type (1,1) bound for the smooth dyadic maximal function The next two Lemmas are devoted to studying the properties of K N * K N and they will be key ingredients for establishing the weak-type (1,1) bound.Proof.We have assumed that ϕ 2 ≃ ϕ 1 and thus, by Lemma 2.4, there exists a uniform bound C B for the length of intervals contained in B. For any |x| ∈ Z such that x ≥ C B + 1, we get and with a change of variables we see that , and thus, without loss of generality, let us assume that x ≥ C B + 1.Since the supp(η) ⊆ (1/2, 4) and 0 ≤ η(x) ≤ 1, we have and all is left to do is estimate the cardinality of that set.Let ) }, and notice that for any n ∈ A x N , we have that there exists a unique s, m Notice that since x > C B , we have that s ≥ 1, since n and n + x cannot correspond to the same m.By combining the previous set of inequalities we obtain For constants depending only on ϕ 1 , ϕ 2 and ψ, we have Note that for all l we get that l − h 1 (ϕ . Since ϕ 2 ≃ ϕ 1 , we get that there exists an absolute constant T , such that l − h 1 (ϕ 1 (l) − ψ(l)) ≤ T , and thus 1 N and the proof is complete.

Lemma 5.2. There exists a real number
We also have that Proof.We note that for all n ∈ N we have that 1 We can therefore split our kernel to several manageable pieces.
We will exploit a famous truncated Fourier Series.More precisely, we know that if Φ(x) = {x} − 1/2 then for all M ∈ N we get (see section 2 from [25]).
Importantly, we also have where x = min{|x − n| : n ∈ Z}, and Finally, we can rewrite Let's use the truncated Fourier Series and define and thus Let's firstly estimate I 1 , we have and for any h ∈ Z we have ] since the integrand is zero outside that interval.Thus we get This shows the properties of G N claimed in the Lemma.Now we bound E N .Let's start with I 2 .We can rewrite I 2 as According to Corollary 3.12 in [4], if we let Let us follow the notation of [4] and write where we used that for all real numbers x we have |e ix − 1| ≤ |x|.Similarly using the mean value theorem and the estimate Since γ ∈ (29/30, 1), we get that for M = N 1+2χ+ε ϕ 2 (N ) −1 , χ = 1 − γ and ε < χ/10 we get We use that for all ε 1 > 0 we have σ(x) For a fixed ε 1 = ε ∈ (0, χ/10) we get which is true since γ > 29/30.Therefore we have show that |I 2 (x)| N −1−χ , as desired.The term I 2 is treated similarly

we obtain in an almost
identical fashion the bound |I 4 (x)| N −1−χ .We will now deal with I 3 , I 5 , I 7 , I 8 , I 9 .We are going to follow the recipe of [4], and we are going to use Lemma 3.18 [4].Let's state it here. where . Therefore we have that and also by the mean value theorem together with the following calculation We let m = max{m 1 , m 2 } and we use Corollary 3.12 from [4] for α = 0 and κ = 1 to obtain We can now finish our estimates for I 6 We wish to have that 8/3 + 16/3χ + 8/3ε − 10/3γ + 2ε 1 − 2/3 ≤ −1 − χ but we have that And we can choose ε 1 > 0 to make this true.
We use Lemma 5.1 and Lemma 5.2 to prove the weak-type (1,1) estimates of Theorem 1.5.We state and prove a general Theorem that allows us to conclude.It is a natural extension of Theorem 6.1 in [4], and the novelty lies in our handling of the problematic initial part of and let F n n∈Z be a family of nonnegative functions.Assume that there are sequences n for some ε 0 ∈ (0, 1) and assume there exists a finite constant M > 1 such that M d n ≤ d n+1 and M D n ≤ D n+1 ≤ 2 n+1 for all n ∈ N. Also, assume that exists a real number ε 1 > 0 such that for all n ∈ N and x ∈ Z we have Then we have that there exists a constant C > 0 such that Before proving the Theorem let us briefly show how it implies the weak-type (1,1) bound.
Proof of Theorem 1.5.By letting we can apply the theorem for ) and for smaller values of x the estimate is trivially established from the definition of F n .We also have and thus if C is the constant appearing in Lemma 5.1, for all integers x such that |x| ≤ C, we get that We can conclude by letting ε 2 = 1, and using the estimates from Lemma 5.1 and Lemma 5.2.This completes the proof.
Proof of Theorem 5.5.Let f ∈ ℓ 1 (Z) such that f ≥ 0 and let α > 0. We will perform a subtle variation of the Calderón-Zygmund decomposition.There exists a family of disjoint dyadic cubes (Q s,j ) (s,j)∈B , where B ⊆ N 0 × Z and Q s,j = [j2 s , (j + 1)2 s ) ∩ Z and functions g, b such that and for every n ∈ N 0 we decompose further and our treatment will be different for each summand.The following subsections are devoted to this task, and the most difficult part will be to bound the final one, where we will exploit the cancellation properties of B n s together with the properties of F n .
5.1.Estimates for the first three summands.For the good part we will use ℓ ∞ -bounds together with the fact that and g(x) ≤ 2α for all x ∈ Z, and thus g + s≥0 g n s ℓ ∞ (Z) ≤ 6α and thus K n * g + s≥0 g n s (x) ≤ T 6α for all x ∈ Z and n ∈ N. Thus for any C > 24T we get since it is the empty set.
For the second summand we use the lacunary nature of (d n ) n∈N as well as the bounds for the cardinality of the support of K n .Specifically, we have For the third summand we simply use the fact for any n ∈ N and any s ≥ s(n) we get that 2 s ≥ D n and thus where 3Q denotes the interval with the same center as Q and three times its radius.Therefore 5.2.Estimates for the fourth summand.The fourth summand is the most difficult to estimate and here we will use the regularity of K n * K n .We have We need the following result to conclude.
Claim.There exists 0 < λ < 1 such that for all n ∈ N and 0 Assuming that (5.8) holds, let us see how we can deduce the desired estimate.We have For the first and the third term note that where we have used the estimates from the Calderón-Zygmund decomposition in the beginning of the proof.
The fourth term is bounded as follows.
Fix |j| ≤ A and x ∈ Z such that x−j ∈ supp(b s 0 ) for some integer s 0 .Since the supports of b s 's are disjoint we have that there can be at most one integer s ′ 0 such that x ∈ supp(b s ′ 0 ).Note also that [|h n s |] Q s,j α, as we observed earlier.We have (5.9) where we have used the existence of a finite constant M > 1 such that M d n ≤ d n+1 .We get that A similar argument may be used to bound the second term.For the sake of completeness we note that Fix |j| ≤ A and x ∈ Z such that x − j ∈ supp(b s 0 ) for some integer s 0 .Since the supports of b s 's are disjoint we have that there can be at most one integer s ′ 0 such that x ∈ supp(b s ′ 0 ).We have In fact if s ′ 0 = s 0 , then the left-hand side equals 0. Nevertheless, the inequality above holds for some s 0 , s ′ 0 that depend on x, j and by comparing it with (5.9), we see that an argument identical to the one used previously may be used here.The proof will be completed once we established the estimate (5.8) of the claim.We do this in the following subsection.
5.3.Proof of the estimate (5.8).Let n ∈ N and 0 ≤ s 1 ≤ s 2 ≤ s(n) − 1 and let us note that on the one hand ≤ Now decompose F n (x) = F n (x)1 |x|≤A +F n (x)1 |x|>A = −A≤j≤A F n (j)δ j (x)+F n (x)1 |x|>A and let G n (x) = F n (x)1 |x|>A .We obtain In the right hand side of our inequality one of the terms of the desired estimate already appeared and thus we can now focus on the other two summands.Let Z m,n = m2 s(n) , (m + 1)2 s(n) ∩ Z, Z m,n = (m − 1)2 s(n) , (m + 2)2 s(n) ∩ Z and We have that Now we note that for any m ∈ Z we have (5.10) On the one hand (5.11) and on the other hand |{k ∈ Z : (s 1 , k) ∈ B & Q s 1 ,k ∩ Z m,n = ∅}| 2 s(n)−s 1 since Z m,n can be partitioned into 3 dyadic intervals, Q s 1 ,k is a dyadic interval and s 1 ≤ s(n).Therefore (5.12) B n s 1 1 Zm,n ℓ 1 (Z) 2 s(n)−s 1 4α2 s 1 = 8α2 s(n)−1 ≤ 8αD n and finally Let us define B n s,j = B n s 1 Q s,j , and note that for any m ∈ Z, x ∈ Z m,n , we have We also note that x∈Z B n s 1 ,j (x)1 Zm,n (x) = 0. To see this note that if supp(B n s 1 ,j ) ∩ Z m,n = ∅, then it is trivial, and if they intersect, we must have that supp(B n s 1 ,j ) ⊆ Z m,n , since supp(B n s 1 ,j ) ⊆ Q s 1 ,j which is a dyadic interval of length 2 s 1 and Z m,n is the union of three dyadic intervals of larger length.In the second case we get x∈Z B n s 1 ,j (x)1 Zm,n (x) = x∈Z B n s 1 ,j (x) = 0 from the definition of B n s 1 ,j (x).Fix m ∈ Z and j ∈ Z such that (s 1 , j) ∈ B and let x s 1 ,j be the center of the cube Q s 1 ,j .Assume x ∈ Z m,n is such that |x − x s 1 ,j | ≥ Cd ε 2 n + C2 s 1 .Using the cancellation property we have established together with the regularity assumptions for F n , we get  The calculations from (5.10), (5.12) show that sup m∈Z B n s 1 1 Zm,n ℓ 1 (Z) αD n and thus the first summand is bounded by a constant multiple of For the second summand we consider two cases.In the first case we assume that 2 s 1 ≤ d ε 2 n .In that case, any interval of radius d where we used the estimate (5.11), the fact that d n ≤ D ε 0 n and the fact that 2 s(n)−1 < D n ≤ 2 s(n) .We have established the appropriate bound for the first case.
Department of Mathematics, Rutgers University, Leonidas Daskalakis is supported by the NSF grant DMS-2154712.

Lemma 5 . 1 .
There exists a positive constant C such that for all N ∈ N and all x ∈ Z with C ≤ |x| ≤ ϕ 1 (N ), we have that |K N * KN (x)| ≤ CN −1 .
according to the previous calculation, for any fixed s ≃ xϕ 1 (N ) N , we have that there are at most 1 + C T ϕ 1 (N ) x ϕ 1 (N ) x , where in the last estimate we used that x ≤ ϕ 1 (N ).The number of s's in [1, C xϕ 1 (N ) N ], where C is the implied fixed constant appearing in s ≃ xϕ 1 (N ) N , are bounded by C xϕ 1 (N ) N , and therefore

n
and assume that there exists a constant A > 0 such that (5.6) F n (x) d −1 n for all x with |x| ≤ A and |F n (x)| D −1 n for all x with |x| > A Finally, assume that there exists an ε 2 ∈ (0, 1] such that (5.7) |F n (x + y) − F n (x)| D −2 n |y| whenever |x|, |x + y| d ε 2 n

s 1 ,n 2 s 1 B n s 1 ,j 1
j (y)1 Zm,n (y) y∈Z D −2 n |y − x s 1 ,j ||B n s 1 ,j (y)1 Zm,n (y)| D −2 Zm,n ℓ 1 (Z)Here we have used the fact that |x − y|≥ |x − x s 1 ,j | − |x s 1 ,j − y| ≥ Cd ε 2 n + C2 s 1 − 2 s 1 d ε 2n and thus we may use (5.7).Taking into account(5.6)and the definition of G n , we get that for any x ∈ Z G n * B n s 1 ,j