ON THE ENDPOINT REGULARITY OF DISCRETE MAXIMAL OPERATORS

Transcription

1 ON THE ENDPOINT REGULARITY OF DISCRETE MAXIMAL OPERATORS EMANUEL CARNEIRO AND KEVIN HUGHES Abstract. Given a discrete function f : Z d R we consider the maximal operator X Mf n = sup f n m, r 0 Nr m Ω r where Ω r r 0 are dilations of a convex set Ω open, bounded and with Lipschitz boudary containing the origin and Nr is the number of lattice points inside Ω r. We prove here that the operator f Mf is bounded and continuous from l Z d to l Z d. We also prove the same result for the non-centered version of this discrete maximal operator.. Introduction.. Background. For a function f L loc Rd the centered Hardy-Littlewood maximal operator is defined as Mfx = sup fx y dy, r>0 mb r B r where B r is the ball of radius r centered at the origin and mb r is the d-dimensional Lebesgue measure of this ball. A basic result in harmonic analysis is that M : L p R d L p R d is a bounded operator for p >, and that it satisfies a weak-type estimate M : L R d L weak Rd at the endpoint p =. The same holds in the non-centered case, when we consider the supremum over balls that simply contain the point x. In both instances we may also replace the balls by dilations of a convex set with Lipschitz boundary since these have bounded eccentricity. Over the last years several works addressed the problem of understanding the behavior of differentiability under a maximal operator. This program began with Kinnunen [7] who investigated the action of the classical Hardy-Littlewood maximal operator in Sobolev spaces and showed that M : W,p R d W,p R d is bounded for p >. This paradigm that an L p -bound implies a W,p -bound was later extended to a local version of the maximal operator [8], to a fractional version [9] and to a multilinear version [5]. The continuity of M : W,p W,p for p > was established by Luiro in [] for the classical Hardy-Littlewood maximal operator and in [2] for its local version. Note that this is a non-trivial problem since we do not have sublinearity for the weak derivatives of the Hardy Littlewood maximal function. Date: June 30, Mathematics Subject Classification. Primary 42B25, 46E35. Key words and phrases. Discrete maximal operators; Hardy-Littlewood maximal operator; Sobolev spaces; bounded variation.

2 2 CARNEIRO AND HUGHES Understanding the regularity at the endpoint case seems to be a deeper issue. In this regard, one of the main questions was posed by Haj lasz and Onninen in [6, Question ]: is the operator f Mf bounded from W, R d to L R d? Observe that a bound of the type Mf L R d C f L R d f L R d. would imply, via a dilation invariance argument, the bound Mf L R d C f L R d,.2 and so the fundamental question would be to compare the variation of Mf with the variation of the original function f perhaps having the additional information that f is integrable. In the work [6], Tanaka obtained the bound.2 in dimension d = for the non-centered Hardy-Littlewood maximal operator with constant C = 2. This was later improved by Aldaz and Pérez Lázaro [] who obtained.2 with the sharp C = under the minimal assumption that f is of bounded variation still, only in dimension d = and for the non-centered maximal operator. None of these proofs extend to higher dimensions. The problem for the centered maximal operator remains untouched, even in dimension d =..2. The discrete analogue. We address here this problem in the discrete setting. We shall generally denote by n = n, n 2,..., n d a vector in Z d and for a function f : Z d R we define its l p -norm as usual: f lp Z d = f n /p p, if p <, and n Z d f l Z d = sup n Z d f n. The gradient f of a discrete function f will be the vector f f n = n, f n,..., f n, x x 2 x d where f n := f n e i f n, x i and e i = 0, 0,...,,..., 0 is the canonical i-th base vector. Now let Ω R d be a bounded open subset that is convex with Lipschitz boundary. Let us assume that 0 intω and normalize it so that e d Ω. We now define the set that will play the role of the ball of center x 0 and radius r in our maximal operators. For r > 0 we write and for r = 0 we put Ω r x 0 = x Z d ; r x x 0 Ω }, Ω 0 x 0 = x 0 }. Whenever x 0 = 0 we shall write Ω r = Ω r 0 for simplicity. For instance, to work with regular l p -balls one should consider Ω = x R d ; x p < }.

3 DISCRETE MAXIMAL OPERATORS 3 From now on we use the letter M to denote the centered discrete maximal operator associated to Ω given by Mf n = sup f n m,.3 r 0 Nr m Ω r where Nr is the number of lattice points in the set Ω r. We define the non-centered discrete maximal operator M associated to Ω in a similar way, by writing Mf n = sup f m,.4 r 0 N x 0, r m Ω r x 0 where the supremum is taken over all balls Ω r x 0 such that n Ω r x 0, and N x 0, r denotes the number of lattice points in the set Ω r x 0. These convex Ω-balls have roughly the same behavior as the regular balls, from the geometric and arithmetic points of view. For instance, we have the following asymptotics [0, Chapter VI 2, Theorem 2] for the number of lattice points N x 0, r = r d O r d.5 as r, where = mω is the d-dimensional volume of Ω, and the constant implicit in the big O notation depends only on the dimension d and on the set Ω e.g. if Ω is the l -ball we have the exact expression Nr = 2 r d. As in the continuous case, both M and M are of strong type p, p, if p >, and of weak type, see for instance [5, Chapter X]. It is then natural to ask how the regularity theory transfers from the continuous to the discrete setting. By the triangle inequality one sees that, in the discrete setting, the Sobolev norm f l p f l p is equivalent to the norm f l p, and thus the question of whether M and M are bounded in discrete Sobolev spaces is trivially true for p >. On the other hand, the regularity at the endpoint case p = is a very interesting topic and the main objective of this paper is to present the folllowing result. Theorem Endpoint regularity of discrete maximal operators. Let d and consider M and M as defined in.3 and.4. i Centered case The operator f Mf is bounded and continuous from l Z d to l Z d. ii Non-centered case The operator f Mf is bounded and continuous from l Z d to l Z d. The boundedness part in Theorem provides a positive answer to the question of Haj lasz and Onninen [6, Question ] in the discrete setting, in all dimensions and for this general family of centered or non-centered maximal operators with convex Ω-balls. The insight for this part was originated in a joint work of the authors with J. Bober and L. B. Pierce [2] where the case d = was treated, and it has two main ingredients: i a double counting argument to evaluate the maximal contribution of each point mass of f to Mf l ; ii a summability argument over the sequence of local maxima and local minima of Mf. The technique is now refined to contemplate the n-dimensional case and this general family of operators. The continuity result is a novelty in the endpoint regularity theory. Luiro s framework [] for the continuity of the classical Hardy-Littlewood maximal operator in the Sobolev space W,p R d, for p >, is not adaptable since it relies on

4 4 CARNEIRO AND HUGHES the L p -boundedness of this operator which we do not have here, and we will only be able to use a few ingredients of it. The heart of our proof lies instead on the two core ideas mentioned above for the boundedness part and a useful application of the Brezis-Lieb lemma [4]. Remark : One might ask if inequality.2 holds in the discrete case, which would be a stronger result than our Theorem. This has only been proved in dimension d = for the non-centered maximal operator see [2] with sharp constant C = i.e. the non-centered maximal function does not increase the variation of a function. Note that the dilation invariance argument to deduce.2 from. fails in the discrete setting. Remark 2: If we consider for instance the one-dimensional discrete centered Hardy- Littlewood maximal operator with regular balls applied to the delta function f0 = and fn = 0 for n 0, we obtain Mfn = /2 n and thus Mf n = O n 2. Examples like this may raise the question on whether Mf belongs to a better l p space i.e. p < when f l. It turns out that the general answer is negative, and Theorem is sharp in this sense. To see this consider a function f l Z such that f / l p Z for any p <, for example fn = / n log 2 n for n, and zero otherwise. Now choose a sequence = a < a 2 < a 3 < a 4 <... of natural numbers such that i a 2 4. ii a n a n > a n a n 2, for any n 2. f iii f > 2a. 2 a iv f f 3 > 2a. 2 a f v fn > 2a n a n vi fn f 3 > 2a n a n, for any n 2., for any n 2. Define the function g : Z R given by ga n = fn for n, and zero otherwise. Note that g l = f l. Conditions i-vi above guarantee that, for the one-dimensional discrete centered Hardy-Littlewood maximal operator M, we have Mga n = fn and Mga n = fn 3, for n. Thus Mg a n = 2fn 3, and thus Mg / l p Z for any p <. Remark 3: Another interesting variant would be to consider the spherical maximal operator [3, 4] and its discrete analogue [3]. The non-endpoint regularity of the continuous operator in Sobolev spaces was proved in [6] and it would be interesting to investigate what happens in the endpoint case, both in the continuous and in the discrete settings. 2. Proof of Theorem - Boundedness 2.. Centered case. We start with some arithmetic and geometric properties of the sets Ω r. From.5 we can find a constant c depending only on the dimension d and the set Ω such that and N x 0, r r c d, 2. N x 0, r max maxr c, 0} } d d, =: r c. 2.2

5 DISCRETE MAXIMAL OPERATORS 5 Over 2.2 it should be clear that if x 0 Z d we can take r 0, and if x 0 / Z d we shall only be taking radii r so that the corresponding ball contains at least one lattice point to calculate the average. We define c 2 > c as the constant such that c 2 c d =. Since Ω is bounded, there exists λ > 0 depending only on Ω such that Ω B λ note that λ since e d Ω. This means that if p Ω r x 0 then p x0 λr. 2.3 These constants c, c 2 and λ will be fixed throughout the rest of the paper Set up. We want to show that Mf l Z d C f l Z d 2.4 for a suitable C that might depend on d and Ω in principle. We assume without loss of generality that f 0. It suffices to prove that Mf x i C f l Z, d l Z d for any i =, 2,..., d. We will work with i = d the other cases are analogous. Let us write each n = n, n 2,..., n d Z d as n = n, n d, where n = n, n 2,..., n d Z d. For each n Z d we will consider the sum over the line perpendicular to Z d passing through n, i.e. Mf n, l x d = Mf n, l Mf n, l. l= l= For a discrete function g : Z R we say that a point a is a local maximum of g if ga ga and ga < ga. Analogously, we say that a point b is a local minimum of g if gb gb and gb > gb. We let a i } i Z and b i } i Z be the sequences of local maxima and local minima of Mf n, ordered as follows:... < b < a < b 0 < a 0 < b < a <... Observe that this sequence that depends on n might be finite either on one side or both. In this case, since Mf lweak Zd, it would terminate in a local maximum and minor modifications would have to be done in the argument we present below. For simplicity let us proceed with the case where the sequence of local extrema is infinite on both sides. In this case we have Mf n, l x d = 2 Mf n, a j Mf n }, b j. 2.5 l= j= The double counting argument. Let r j be the minimum radius such that the supremum in.3 is attained for the point n, a j, i.e. Mf n, a j = Arj f n, a j := n, a j m. 2.6 Nr j m Ω rj f If we consider the radius s j = r j a j b j centered at the point n, b j we obtain Mf n, b j Asj f n, b j = n, b j m. 2.7 Nr j a j b j m Ω sj f

6 6 CARNEIRO AND HUGHES The observation that motivates this particular choice of the radius s j is that Ω rj n, a j Ωsj n, b j, which follows from the convexity of Ω and the fact that e d Ω. From 2.5, 2.6 and 2.7 we obtain Mf = x d Mf n, l x d l Z d n Z d l= n Z d 2 j= Arj f n, a j Asj f n, b j }, 2.8 where a j = a j n and b j = b j n. We now consider a general point p = p, p 2,..., p d Z d, also represented as p = p, p d with p Z d. We want to evaluate the maximum contribution that f p, p d might have to the right-hand side of 2.8. For given n and j, this contribution will only be positive if the point p, p d belongs to both sets Ω rj n, a j and Ωsj n, b j in case the point p, p d belongs only to Ω sj n, b j or does not belong to any of these Ω-balls, the contribution is negative or zero and we disregard it. Since p, p d Ωrj n, a j, from 2.3 we have p, p d n, a j λrj. 2.9 Using 2., 2.2 and 2.9, we can estimate the maximum contribution of f p, p d, for given n and j, on the associated summand on right-hand side of 2.8 as f p, p d Nr j Nr j a j b j f p, p d Nr j Nr j a j a j f p, p d r j c d r j a j a j c d f p, p d λ p n 2 p d a j 2 /2 d c c 2 a j a j c d, λ p n 2 p d a j 2 /2 d, aj a j c 2.0 In the last inequality of 2.0 we have used 2.9 and the fact that the function gx = x c d x a j a j c d is decreasing as x, for x c 2. If we sum 2.0 over all j and then over all n Z d we find an upper bound for the contribution of f p, p d to the right-hand

7 DISCRETE MAXIMAL OPERATORS 7 side of 2.8. This is given by 2f p, p d n Z d j= λ p n 2 p d a j 2 /2 d c c 2 a j a j c d, λ p n 2 p d a j 2 /2 d. aj a j c The summability argument. We now prove that the double sum in 2. is bounded independently of the the point p, p d and the increasing sequence aj }. For this we may assume p = 0 since the sum is over all n Z d we can just change variables here to m = n p. We also assume p d = 0, since we may consider the increasing sequence a j = a j p d. The problem becomes then to bound Sa j } = n Z d j= λ n 2 a 2 /2 d j c c 2 a j a j c d, λ n 2 a 2 /2aj d j a j c 2.2 independently of the increasing sequence a j } of integers. The key tool is the lemma below. Lemma 2 Summability lemma. For any increasing sequence a j } j Z of integers consider the sum Sa j } given by 2.2. The sum Sa j } is maximized for the sequence a j = j, and in this case the sum is finite. Proof. Suppose we have two terms in the sequence, say a 0 and a that are not consecutive. Let us prove that if we introduce a term ã 0 in the sequence, with

8 8 CARNEIRO AND HUGHES a 0 < ã 0 < a, the overall sum does not decrease. For this it is sufficient to see that λ n 2 a 2 /2 d c c 2 a a 0 c d, λ n 2 a 2 /2a d a 0 c λ n 2 a 2 /2 d c c 2 a ã 0 c d, λ n 2 a 2 /2a d ã 0 c λ n 2 ã 2 /2 d 0 c c 2 ã 0 a 0 c d, λ n 2 ã 2 /2ã0 d, 0 a 0 c and this is true if and only if min c 2 a ã 0 c d, λ n 2 a 2 /2a d ã 0 c c 2 a a 0 c d, λ n 2 a 2 /2a d a 0 c λ n 2 ã 2 /2 d 0 c c 2 ã 0 a 0 c d, λ n 2 ã 2 /2ã0 d. 0 a 0 c The last inequality can be verified from the fact that gx = x d x ã 0 a 0 d is decreasing as x, for x 0, and the fact that λ n 2 a 2 /2 a ã 0 λ n 2 ã 2 0 /2.

9 DISCRETE MAXIMAL OPERATORS 9 The latter follows by calling a = ã 0 t note that t 0, and then differentiating the expression with respect to the variable t to check the sign here we make use of the fact that λ, since we might have ã 0 > a. Therefore the required sum 2.2 is bounded by above by the sum considering the particular sequence a j = j. This gives us S = n Z d j= λ n 2 j 2 /2 d c c 2 c d, λ n 2 j 2 /2 d c = } 2.3 n Z C d Ω λ d n c c 2 c d, C Ω λ d n c λ n c 2 = Cd, Ω <. λ n >c 2 λ n c d λ n c d Conclusion. We have proved that the contribution of a generic point fp, p 2,..., p d to the right-hand side of 2.8 is at most a constant 2 C = 2 Cd, Ω and therefore, when we sum over all points, we get Mf 2 x C f l d l Z d Z. d Since the same holds for any direction we obtain the desired inequality Non-centered case. We will indicate here the basic modifications that have to be made in comparison with the proof for the centered case. The set up is the same up to the beginning of the double counting argument. For a given point n, a j we can pick a point xj and a radius r j such that n, a j Ωrj x j and the average over the set Ω rj x j realizes the supremum in the maximal function, i.e., Mf n, a j = A xj,r jf n, a j := N x j, r j m Ω rj x j f m. 2.4 This is guaranteed since any maximizing sequence x k j, rk j of the right-hand side of 2.4 must be stationary. In fact, we should have the sequence x k j, rk j trapped in a bounded subset x k j R and r k j R, for some R > 0 since f l Z d, and then we would have only a finite number of subsets of Z d to choose from for the sum in 2.4. We now consider the Ω-ball of radius s j = r j a j b j centered at y j = x j a j b j e d. Note that n, b j Ω rj y j Ω sj y j. From the convexity of Ω and

10 0 CARNEIRO AND HUGHES the fact that e d Ω we also have Ω rj x j Ω sj y j. Therefore Mf n, b j A yj,s jf n, b j = f m, 2.5 N y j, s j and Mf x d l Z d = n Z d l= n Z d 2 j= Mf n, l x d m Ω sj y j A xj,r jf n, a j A yj,s jf n, b j }. 2.6 Consider a point p = p, p d Z d. The term f p, p d will only contribute positively to a summand on the right-hand side of 2.6 if p, p d Ωrj x j. In this case, since n, a j Ωrj x j, using 2.3 we have p, p d n, a j 2 λ rj. 2.7 The rest of the proof is the same. 3. Proof of Theorem - Continuity 3.. Centered case. We want to show that if f k f in l Z d then Mf k Mf in l Z d Set up. Since f k f f k f and the maximal operator only sees the absolute value we may assume without loss of generality that f k 0 for all k, and that f 0. It suffices to prove the result for each partial derivative, i.e. that Mf k Mf x i x i 0 3. l Z d as k, for each i =, 2,..., d. We shall prove it for i = d and the other cases are analogous A discrete version of Luiro s lemma. For a function g l Z d and a point n Z d let us define Rg n as the set of all radii that realize the supremum in the maximal function at the point n, i.e. Rg n = r [0, ; Mg n = A r g n = Nr m Ω r g n m. The next lemma gives us information about the convergence of these sets of radii. It can be seen as the discrete analogue of [, Lemma 2.2]. Lemma 3. Let f k f in l Z d. Given R > 0 there exists k 0 = k 0 R such that, for k k 0, we have Rf k n Rf n for each n B R. Proof. Fix n B R and consider the application r A r f n for r 0. From the fact that f l Z d together with 2.2 we can see that A r f n 0 as r. Therefore the set of values in the image A r f n; r 0} such that A r f n 2 Mf n is a finite set. There exists then a second larger value which falls short

11 DISCRETE MAXIMAL OPERATORS of the maximum by a quantity we define as ɛ n, i.e. if A r f n > Mf n ɛ n then A r f n = Mf n and r Rf n. Define ɛ = 3 min ɛ n; n B R }. Since f k f in l Z d, we have f k f in l Z d. Pick k 0 such that for k k 0 we have f k f l ɛ. For any n B R if we take s Rf n we have Mf n = A s f n = A s f k n A s f f k n Mf k n ɛ. 3.2 Now given r k Rf k n we can use 3.2 to obtain A rk f n = A rk f k n A rk f f k n = Mf k n A rk f f k n Mf k n ɛ Mf n 2ɛ, and from the definition of ɛ and ɛ n we conclude that r k Rf n Reduction via the Brezis Lieb lemma. Given ɛ > 0, we can find k 0 such that f k f l ɛ, and using Lemma 3 for a fixed n Z d, we can choose k k 0 so that we also have Rf k n Rf n for k k. Taking any r k Rf k n we have Mf n Mfk n = Ark f n A rk f k n ɛ, 3.3 for k k and thus Mf k n Mf n as k. The same can be said replacing n by n e d and thus we find that Mf k n Mf n 3.4 x d x d pointwise as k. Since Mf k n Mf k n Mf n x d x d x d Mf n x d and the latter is in l Z d from the boundedness part of the theorem, an application of the dominated convergence theorem with 3.4 gives us lim k Mf k x d Mf k } Mf = l Z d x d x d Mf. x d l Z d Therefore, to prove 3. it suffices to show that lim k Mf k x d = Mf l Z d x d l Z d l Z d. 3.5 The reduction to 3.5 is the content of the Brezis-Lieb lemma [4] in the case p =. We henceforth focus our efforts in proving Lower bound. From Fatou s lemma and 3.4 we have Mf x d lim inf l Z d k Mf k x d. 3.6 l Z d

12 2 CARNEIRO AND HUGHES Upper bound. Given ɛ > 0 we shall prove that there exists k 0 = k 0 ɛ such that for k k 0 we have Mf k x d Mf l Z d x d ɛ. 3.7 l Z d This would imply that lim sup k Mf k x d Mf l Z d x d l Z d which together with 3.6 would prove that the limit exists and 3.5 holds. Let us start with a sufficiently large integer radius R to be properly chosen later and consider the cube x R d ; x 2R }. Let us continue writing n Z d as n = n, n d with n Z d. We write the required sum in the following way Mf k = x d Mf k n, n d x d Mf k n, n d x d l Z d n 2R n d 2R := S S 2 S 3. We shall bound S, S 2 and S 3 separately. n 2R n d >2R n >2R n d Z Mf k n, n d x d, Bound for S. Let us pick ɛ > 0 to be properly chosen later. With the aid of Lemma 3 we find k = k ɛ, R such that Rf k n Rf n for each n with n 2R and f k f l Z d ɛ, 3.9 for k k. Using 3.3 we have that Mf k n Mf n x d x d 2ɛ, for any n with n 2R. Thus S = Mf k n, n d x d n 2R n d 2R n 2R n d 2R Mf x d 2 ɛ 4R d. l Z d Mf n, n d x d 2 ɛ 4R d Bound for S 2. Here we start with the same idea and notation for the local maxima and local minima over vertical lines as in 2.8 S 2 = Mf k n, l x d n >2R n >2R 2 l= 3. Arj f k n, a j Asj f k n }, b j. j=

13 DISCRETE MAXIMAL OPERATORS 3 We find an upper bound for the contribution of a generic point f k p, p d to the right-hand side of 3. as previously done in 2.. This is given by 2f k p, p d n >2R j= λ p n 2 p d a j 2 /2 d c c 2 a j a j c d, 3.2 λ p n 2 p d a j 2 /2 d. aj a j c Using Lemma 2 we see that the sum on the right-hand side of 3.2 is majorized by the sum with the sequence a j = j. This gives us 2f k p, p d n >2R j= λ p n 2 j 2 /2 d c c 2 c d, 3.3 λ p n 2 j 2 /2 d. c We now evaluate this contribution in two distinct sets. Firstly, we consider the case when p, p d BR, for which we have p n R. Imposing the condition that λ R > c we can ensure that the contribution of f k p, p d is majorized by 2f k p, p d C n R Ω λ n c d λ n c d 3.5 := 2 f k p, p d hr. The fact that hr 0 as R is a crucial point in this proof and shall be used when we choose R at the end. Secondly, when p, p d / BR the contribution will simply be bounded by 2 Cf k p, p d as we found in 2.3. If we then sum up these contributions and plug them in on the right-hand side of 3. we find S 2 2 hr χ BR f k l Z d 2 C χ BR cf k l Z d Bound for S 3. We start by noting that S 3 = Mf k n, l x d n 2R := S 3 S 3. l=2r 2R n 2R l= Mf k n, l x d

14 4 CARNEIRO AND HUGHES Let us provide an upper bound for S 3. The upper bound for S 3 is analogous. We consider the sequence of local maxima a j } and local minima b j } for Mf k n, l when l 2R. In this situation we do have a first local maximum a which might be the endpoint 2R and we order this sequence as follows: 2R a < b 2 < a 2 < b 3 < a 3... If the sequence terminates, it will be in a local maximum since Mf k l weak Zd, and we can just truncate the sum in the argument below. Keeping the notation as before and including for convenience a 0 = b = we have S 3 n 2R 2 Arj f k n, a j Asj f k n }, b j. 3.7 j= The contribution of a generic point f k p, p d to the right-hand side of 3.7 following the calculation 2., has an upper bound of 2f k p, p d n 2R j= λ p n 2 p d a j 2 /2 d c c 2 a j a j c d, 3.8 λ p n 2 p d a j 2 /2 d. aj a j c Following the ideas of Lemma 2, keeping the constraint that a 0 =, the sum on the right-hand side of 3.8 is maximized when a j = 2R j for j. We would then have the upper bound 2f k p, p d n 2R λ p n 2 p d 2R 2 /2 d c 2f k p, p d n 2R j=2 λ p n 2 p d 2R j 2 /2 d c c 2 c d, λ p n 2 p d 2R j 2 /2 d. c 3.9 Again, we evaluate this contribution separately for p, p d in the sets BR and B R c. In the first case, if p, p d BR we have p d 2R R, and if we choose R

15 DISCRETE MAXIMAL OPERATORS 5 satisfying 3.4 the contribution of f k p, p d will be less than or equal to 2f k p 4R d, p d λ R c d C n R Ω λ n c d λ n c d = 2f k p } 4R d, p d λ R c d hr In the second case, if p c, p d BR, we just bound the contribution of fk p, p d by 2 C f k p, p d as in 2.3. Plugging these upper bounds in 3.7 we find } S 3 2 4R d λ R c d hr χ BR f k l Z d 2 C χ BR cf k l Z. 3.2 d By symmetry the same bound holds for S Conclusion. Putting together 3.8, 3.0, 3.6 and 3.2 we obtain Mf k x d Mf l Z d x d 2 ɛ 4R d l Z d } R d 4 λ R c d 6 hr χ BR f k l Z d 6 C χ BR cf k l Z. d We choose in this order R large enough so that it satisfies 3.4, } 4R d 4 λ R c d 6 hr ɛ 3, 3.23 f l Z d and Then we choose ɛ such that χ BR cf l Z d ɛ ɛ 36 C. ɛ 64R d, 3.24 and this generates a k as described in 3.9. We now choose k 0 k such that for all k k 0 we have } ɛ f k f l Z d min 36 C,, which then implies that χ BR f k l Z d f k l Z d f l Z d 3.25 and χ BR cf k l Z d χ BR cf l Z d χ BR cf k f l Z d ɛ 8 C Plugging 3.23, 3.24, 3.25 and 3.26 into 3.22 gives us Mf k x d Mf l Z d x d ɛ l Z d 3 ɛ 3 ɛ 3,

16 6 CARNEIRO AND HUGHES for all k k 0, and the proof is now complete Non-centered case. We will indicate here the basic changes that have to be made in comparison with the centered case argument. For a function g l Z d and a point n Z d, let us define the set Rg n as the set of all pairs x, r R d R such that n Ω r x and the supremum in the non-centered maximal function at n is attained for Ω r x, i.e. Rg n = x, r Rd R ; Mg n = A x,r g n = N x, r m Ω r x The proof of the following result is essentially the same as in Lemma 3. g m. Lemma 4. Let f k f in l Z d. Given R > 0 there exists k 0 = k 0 R such that, for k k 0, we have Rf k n Rf n for each n B R. The rest of the proof is also similar, using 2.5, 2.6 and 2.7 in the appropriate places. 4. Acknowledgements The first author acknowledges support from CNPq-Brazil grants 47352/20 8 and /20 2. The second author acknowldges support from NSF grant DMS We would like to thank Carlos Cabrelli and Ursula Molter for discussions related to Remark 2. References. J. M. Aldaz and J. Pérez Lázaro, Functions of bounded variation, the derivative of the one dimensional maximal function, and applications to inequalities, Trans. Amer. Math. Soc , no. 5, J. Bober, E. Carneiro, K. Hughes and L. B. Pierce, On a discrete version of Tanaka s theorem for maximal functions, Proc. Amer. Math. Soc , J. Bourgain, Averages in the plane over convex curves and maximal operators, J. Anal. Math , H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc , E. Carneiro and D. Moreira, On the regularity of maximal operators, Proc. Amer. Math. Soc , no. 2, P. Haj lasz and J. Onninen, On boundedness of maximal functions in Sobolev spaces, Ann. Acad. Sci. Fenn. Math , no., J. Kinnunen, The Hardy-Littlewood maximal function of a Sobolev function, Israel J. Math , J. Kinnunen and P. Lindqvist, The derivative of the maximal function, J. Reine Angew. Math , J. Kinnunen and E. Saksman, Regularity of the fractional maximal function, Bull. London Math. Soc , no. 4, S. Lang, Algebraic number theory, Addison-Wesley Publishing Co., Inc H. Luiro, Continuity of the maximal operator in Sobolev spaces, Proc. Amer. Math. Soc , no., H. Luiro, On the regularity of the Hardy-Littlewood maximal operator on subdomains of R n, Proc. Edinburgh Math. Soc , no, A. Magyar, E. M. Stein and S. Wainger, Discrete analogues in harmonic analysis: Spherical averages, Ann. Math , E. M. Stein, Maximal functions. I. Spherical means, Proc. Nat. Acad. Sci. U.S.A ,

17 DISCRETE MAXIMAL OPERATORS 7 5. E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, Vol H. Tanaka, A remark on the derivative of the one-dimensional Hardy-Littlewood maximal function, Bull. Austral. Math. Soc , no. 2, IMPA - Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina, 0, Rio de Janeiro, Brazil address: carneiro@impa.br Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ, address: kjhughes@math.princeton.edu