DIMENSION-FREE ESTIMATES ON l 2 ( Z d ) FOR DISCRETE DYADIC MAXIMAL FUNCTION OVER l 1 BALLS: SMALL SCALES

. We give a dimension-free bound on l p ( Z d ) for discrete Hardy-Littlewood operator over l 1 balls in Z d with small dyadic radii, where p ∈ [2 , ∞ ] .


Introduction
Let G be a convex, bounded, closed symmetric subset of R d with non-empty interior, we will call such G a symmetric convex body.Natural examples are the l q balls: With each symmetric convex body one can associate the corresponding Hardy-Littlewood averaging operator.For any t > 0 and x ∈ R d we define for a locally integrable function f , where t • G = {tx : x ∈ G} and |t • G| denotes its Lebesgue measure.Now for any p let C p (d, G) > 0 be the smallest number such that the following inequality holds for every f ∈ L p (R d ).It is well known that C p (d, G) < ∞ for all p ∈ (1, ∞] and all symmetric convex bodies G.
In 1980s dependency of C p (d, G) on dimension d has begun to be studied.Various results were obtained, but as of now the major conjecture in this topic is still open, namely that C p (d, G) can be bounded from the above by a number independent of dimension d for each fixed p ∈ (1, ∞].We recommend the survey article [3] for a detailed exposition of the subject, which contains results that we skipped mentioning.
Similar questions can be considered for discrete analogue of the operator M G t .For every t > 0 and every x ∈ Z d we define the discrete Hardy-Littlewood averaging operator where |(t • G) ∩ Z d | is the number of elements of the set (t • G) ∩ Z d .Similarly as before we define C p (d, G) > 0 to be the smallest number such that the following inequality holds for every f ∈ l p (Z d ).Using similar methods as in the continuous case one can show that for any p ∈ (1, ∞] and any symmetric convex body we have C p (d, G) < ∞.
What about dependency of C p (d, G) on the dimension d?We can ask similar questions as in the continuous setup, yet it turns out that the situation is much more delicate.Indeed in [1] the authors constructed a family of ellipsoids E(d) ⊆ R d , with the property that for each p ∈ (1, ∞] there exists C p > 0 such that for every d ∈ N we have This means that if we want to establish dimension-free estimates for C p (d, G) we need to restrict ourselves to specific sets G, which contain more symmetries; one of the simpler options are B q balls.Literature in the discrete setting is not as fruitful, nevertheless there are some papers and positive results in this regard, for example: • In [1] it was proved that for every p ∈ ( 3 2 , ∞] there exists a constant C p > 0 such that for every d ∈ N and every f ∈ l p (Z d ) we have This result is as strong as the ones in continuous case.Unfortunately in the case of sets B q for q = ∞ authors of papers [1], [2], [6] could only obtain way weaker conclusions.
• In [2] it was proved that for every p ∈ [2, ∞) there exists C p > 0 such that for every d ∈ N and every f ∈ l p (Z d ) we have where D = {2 n : n ∈ N}.
• In [6] by extending methods of [2] it was proved that for any q ∈ (2, ∞) and any p ∈ [2, ∞) there exists constant C(p, q) > 0 such that for every d ∈ N and every f ∈ l p (Z d ) we have where D = {2 n : n ∈ N}.The paper [6] did not cover the range t < d 1/q nor q < 2.
In this paper we will prove the following result.
where D = {2 n : n ∈ N 0 } is the set of dyadic integers.
As far as the author is aware the result above is new.It is a small step beyond the results of [6] in the case q = 1.Additionally we give explicit values to various constants.In the proof of Theorem 1.1 we will rely on methods shown in the [2, Section 3].Right at the beginning we will replace t • B 1 by the simpler set S t = {x ∈ {−1, 0, 1} d : d i=1 |x i | = t} and then consider only S t .Later on we merely use tools from [2,Section 3].This replacement by a simpler set is easy due to the fact that there exists a formula for the number of lattice points in t • B 1 .This is not the case for other sets of the form t • B q , where q ∈ (1, ∞).We restrict ourselves to t ≤ √ d due to the fact that then (t • B 1 ) ∩ Z d consists mostly of the points from S t (see Lemma 2.4).This is not the case when t is sufficiently big in terms of d and we would need different argument in that range.Throughout the rest of this paper we will consider B q only for q = 1.
One can extend the discrete Fourier transform to f ∈ l 2 (Z d ).Then it turns out that f ∈ L 2 (T d ).
We have the following Parseval identity: Moreover, for any f, g ∈ l 2 (Z d ), F −1 will denote the inverse of the discrete Fourier transform, that is where G ∈ L 2 (T d ).(7) We let m t be the multiplier symbol (8) T d will denote the d-dimensional torus, which will be identified with the set [− 1 2 , 1 2 ) d .
Definition 1.2.The discrete Hardy-Littlewood averaging operator of the l 1 ball is defined for any function f : Z d → C by the formula Our goal is to prove the following theorem.
Using a complex interpolation argument one can show that Theorem 1.3 implies Theorem 1.1.Notice that the operator M t is given by convolution, that is Since we are considering l 2 norms, using Fourier theory we can reduce our problem to understanding pointwise bounds of the following function This will be explained in detail in the next sections.The goal of this section is to approximate function m t (ξ) by a simpler function; it will be obtained by approximating |B t ∩ Z d | by the number of lattice points in a simpler set.Definition 2.1.For t ∈ N we define next we introduce averaging operator S t related to the above multiplier by the formula Lemma 2.2.For any t ∈ N, t ≤ d the following formulas hold These formulas are well known.We give a proof for completeness.
Proof.For equality (2.1) we notice that there are d t options to choose t coordinates out of d coordinates, on which we have value −1 or 1.Next we can choose each sign in 2 ways, thus we obtain For equality (2.2) we consider the following disjoint decomposition where In the union above each l-th individual set has size Indeed, we have d l options of choosing nonzero coordinates and then 2 l options of choosing signs.The last equality above is classical "stars and bars" problem from combinatorics, see [4, p. 38].Hence for k ≥ 1 In the penultimate equality above we used the following fact: for any t ≥ l ≥ 1 we have One can prove this equality by segregating l-element subsets of the set {1, ..., t} based on biggest number which they contain.For each k there are k− In the penultimate equality above we changed summation index by substitution u = t − k and used the fact that t k = t t−k , in the last equality we denoted each term of the sum by a u .Then for each 1 ≤ u ≤ t − 1 we have that a u+1 ≤ t 2 d 1 u+1 a u .Indeed, Iterating the above estimate we get However, using the fact that 2t ≤ d we obtain that which implies Thus we proved the upper bound.
On the other hand the lower bound follows from estimating undermentioned sum by the term k = t − 1.
In the paper we will use only the upper bound of Lemma 2.
The simple bound above has consequences in terms of norm estimates for the corresponding maximal dyadic operator.Proof of the next lemma is based on Lemma 2.4 and a square function estimate.
Proof.Take any f ∈ l 2 (Z d ), then we have . The second inequality above comes from Corollary 2.5.In the above reasoning multiple times we made use of Parseval identity and the fact that

3.
Estimates for the multiplier s t .
In the proof of Lemma 2.6 we have seen how a pointwise bound of |m t (ξ) − s t (ξ)| impacted sup t∈D,t≤ The bound in Corollary 2.5 is fairly strong and uniform in ξ ∈ T d .Unfortunately we will not be that lucky in the future.Our goal for now is to obtain bounds on |s t (ξ)| for big ξ (then we expect some decay in ξ) and |s t (ξ) − 1| for small ξ.Proof of the next lemma exploits in a simple way invariance of the set S t under both permutation of coordinates and sign changes of each coordinate.Proof.Notice that for any ǫ ∈ {−1, 1} d we have that where ǫx = (ǫ 1 x 1 , ..., ǫ d x d ).This implies that Note that for any sequences of complex numbers {a j } d j=1 , {b j } d j=1 such that max 1≤j≤d |a j | ≤ 1 and max 1≤j≤d |b j | ≤ 1 we have This follows from an easy induction argument.Using (3.1) and the formula cos(2x) = 1 − 2 sin 2 (x) we obtain In the last inequality we used the fact that | sin(xy)| ≤ |x|| sin(y)| for any x ∈ Z, y ∈ R (this can be proved by only considering x ∈ N and induction with respect to x using formula for sin(a + b)).Now to understand the sum x∈St x 2 j we make use of the fact that S t is closed under permutations of coordinates, that is for any σ ∈ Sym(d) we have where σ(x) = (x σ(1) , x σ(2) , ..., x σ(d) ).Using similar argument as in the beginning of the proof one can show that for any j ∈ {1, ..., d} we have sin 2 (πξ j ).
Above we used the fact that any for any x ∈ S t we have x 2 l 2 = x l 1 (since x has coordinates in the set {−1, 0, 1}) and that x l 1 = t. .
Next theorem describes important facts regarding Krawtchouk polynomials.
Theorem 3.3.For every n ∈ N 0 and integers x, k ∈ [0, n] we have (1) Symmetry: k (3) Uniform bound: For any x, k ≤ n/2 we have , where c = −2 log(0.93)= 0.14514...The proof of the first two points is contained in [7], the last point is [5,Lemma 2.2].The value of the constant c is not explicitly given in [5], however one can deduce it from its proof.From now on we define c = −2 log(0.93)= 0.14514... Lemma 3.4.For any t, d ∈ N 0 such that t ≤ d/2 we have Proof.We will proceed as in the later part of the proof of [2,Proposition 3.3].Using the same symmetry invariance arguments as in the proof of Lemma 3.1 we have the following equalities It is sufficient to prove that for any y ∈ S t we have that Fix any y ∈ S t , notice that we have following disjoint decomposition.
Notice that each set under the union has size (d − t)! • t!.Since cosine is even, from (3.3) we obtain For any S ⊆ N d we define For any set J ⊆ N d such that |J| = t we define ε(J) ∈ {−1, 1} d to be vector, such that ε(J) j = −1 exactly when j ∈ J. Then we have where w S : {−1, 1} d → {−1, 1} is defined by the formula w S (ε) = j∈S ε j .Now by changing the order of summation we obtain Notice that for fixed S ⊆ N d the following equalities hold Using the above we get If |S| ≤ d/2 then by our assumption that t ≤ d/2 and point 3 of Theorem 3.3 we get otherwise if |S| > d/2 then using reflection symmetry and point 3 of Theorem 3.3 we obtain This gives us In the penultimate inequality we used inequality 1 − x ≤ e −x for x ≥ 0. In the ultimate inequality we used x/2 ≤ 1 − e −x for 0 ≤ x ≤ 1/2.This concludes the proof of Lemma 3.4 .

Introduction of new multipliers
Now we will define a pair of Fourier multipliers, whose corresponding maximal operators are bounded independently of the dimension by a general theory.Then we will try to approximate s t pointwise by these multipliers.

Conclusions and proof of the main theorem
We are almost ready to prove the main theorem.The next lemma is simple, however we give a proof to calculate the explicit constant 5  3 .Lemma 5.1.For any x ∈ R + we have above we used the fact that function ( 1 4 , 1] ∋ s → s + 1 4s is bounded by 5  4 .We are finally ready to prove Theorem 1.3.

Proof. Take any
By Lemma 2.6 the first term above is bounded by 6 f l 2 .For the second term let By Theorem 4.2 the first sum is bounded by To finish the proof it remains to estimate for i = 1, 2 the following term We will only bound the above for i = 1, the proof for i = 2 is basically the same.From Lemma 4. Finally the following is true Similarly we obtain Which gives us This completes the proof of Theorem 1.3.

( 3 )
We define e(x) = e 2πix for any x ∈ R d .(4) We have standard scalar product on R dx • y = d k=1 x k y k , where x, y ∈ R d .(5) For f, g ∈ l 2 (Z d ) we define f * g ∈ l 2 (Z d ) by the series f * g(x) = y∈Z d f (y)g(x − y) = y∈Z d f (x − y)g(y),which is absolutely convergent for each x ∈ Z d .(6) If f ∈ l 1 (Z d ) we define the discrete Fourier transform by the formula

Theorem 1 . 3 .
For any d ∈ N, d ≥ 4 and any f ∈ l 2 (Z d ) we have sup t∈D, t≤ √ d

2 .
Estimate for |B t ∩ Z d | and its consequences.

Lemma 2 . 6 .
For d ∈ N, d ≥ 4 and any f ∈ l 2 (Z d ) the following inequality holds sup t∈D,t≤ √ d

Definition 4 . 1 . 2 )
For any ξ ∈ T d and any t ∈ N 0 , t ≤ d we define Proof of the following theorem is a consequence of the theory of symmetric diffusion semigroups, see[8, p. 73]  and [2, Theorem 2.2].Theorem 4.2.For any f ∈ l 2 (Z d ) the following inequality holds Remark 2.3.Notice that |S t | is equal to term corresponding to k = t in the sum (2.2).When t is small in terms of d then d k+1 is roughly d times bigger than d k for each k ≤ t; because of that we expect |B t ∩ Z d | to be similar in size to |S t | when t is sufficiently small in terms of d.Lemma 2.4 quantifies our expectations.