ON THE ENDPOINT REGULARITY OF DISCRETE MAXIMAL OPERATORS. Emanuel Carneiro and Kevin Hughes

Transcription

1 Math Res Lett 9 202, no 06, c International Press 202 ON THE ENDPOINT REGULARITY OF DISCRETE MAXIMAL OPERATORS Emanuel Carneiro and Kevin Hughes Dedicated to Professor William Beckner on the occasion of his 70th birthday Abstract Given a discrete function f : Z d R, we consider the maximal operator Mf n =sup f n m, r 0 Nr m Ω r where { Ω r are dilations of a convex set Ω open, bounded and with Lipschitz }r 0 boundary 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 isthed-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 weaktype 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 [8], 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 [9], to a fractional version [0] and to a multilinear version [5] The continuity of M : W,p W,p for p> was established by Luiro in [3] for the classical Hardy Littlewood maximal operator and in [4] for its Received by the editors July 9, 202 Key words and phrases Discrete maximal operators; Hardy Littlewood maximal operator; Sobolev spaces; bounded variation 245

2 246 EMANUEL CARNEIRO AND KEVIN HUGHES 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 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 [7, 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 2 Mf L R d C f L R, d 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 [8], Tanaka obtained the bound 2 in dimension d = for the non-centered Hardy Littlewood maximal operator with constant C = 2This 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 The progress in the centered case is very recent and also only in dimension d = In [] O Kurka showed that if f is of bounded variation on R then VarMf C Varf for some constant C>, where Varf denotes the total variation of the function f This result was later adapted to the one-dimensional discrete setting by Temur [9] It is likely that the sharp constant in Kurka s inequality should be C =, but this remains an open problem Regularity results of similar flavour for the heat flow maximal operator and the Poisson maximal operator were obtained in [6] 2 The discrete analog 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 = /p f n p, n Z d if p<, and f l Z d =sup f n n Z d 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 ith 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

3 DISCRETE MAXIMAL OPERATORS 247 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 R 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 < },where x p = x p x 2 p x d p /p for x =x,x 2,,x d R d From now on we use the letter M to denote the centered discrete maximal operator associated to Ω given by 3 Mf n =sup r 0 Nr m Ω r f n m, 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 4 Mf n = sup f m, n Ω r x 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 [2, Chapter VI, Section 2, Theorem 2] for the number of lattice points 5 N x 0,r= r d O r d 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 eg, 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 [7, 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 following 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

4 248 EMANUEL CARNEIRO AND KEVIN HUGHES The boundedness part in Theorem provides a positive answer to the question of Haj lasz and Onninen [7, 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 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 d-dimensional case and this general family of operators The continuity result is a novelty in the endpoint regularity theory Luiro s framework [3] 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 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 = ie, the non-centered maximal function does not increase the variation of a function and, more recently, for the centered maximal operator with C> see [9] 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 =0forn 0,weobtainMfn =/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 ie, 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 < 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, wehave Mga n =fn andmga n = fn 3, for n Thus Mg a n = 2fn 3,and thus Mg / l p Z for any p<

5 DISCRETE MAXIMAL OPERATORS 249 Remark 3 Another interesting variant would be to consider the spherical maximal operator [3, 6] and its discrete analog [5] The non-endpoint regularity of the continuous operator in Sobolev spaces was proved in [7] and it would be interesting to investigate what happens in the endpoint case, both in the continuous and in the discrete settings The techniques we use here do not seem to apply to the discrete spherical maximal operator 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 2 N x 0,r r c d, and 22 N x 0,r max { max{r c, 0} } d d, =: r c Regarding 22 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 23 p x0 λr These constants c, c 2 and λ will be fixed throughout the rest of the paper 2 Setup We want to show that 24 Mf l Z d C f l Z d 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 each i =, 2,,d We will work with i = d this is the reason we normalized at the beginning to have e d Ω The other cases are analogous, via a renormalization to put e i Ω 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 Foreachn Z d we will consider the sum over the line perpendicular to Z d passing through n, ie, l= Mf n,l x d = l= Mf n,l Mf n,l For a discrete function g : Z R we say that a point a is a local maximum of g if ga ga andga <ga Analogously, we say that a point b is a local

6 250 EMANUEL CARNEIRO AND KEVIN HUGHES minimum of g if gb gb andgb >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 25 Mf n,l x d =2 { Mf n,a j Mf n },b j l= j= 22 The double counting argument Let r j be the minimal radius such that the supremum in 3 is attained for the point n,a j such a radius exists since f l Z d We write 26 Mf n,a j = Arj f n,a j := Nr j m Ω rj f n,a j m If we consider the radius s j = r j a j b j centered at the point n,b j we obtain 27 Mf n,b j Asj f n,b j = Nr j a j b j m Ω sj f n,b j m 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 25 to 27 we obtain Mf = x d Mf n,l 28 x d l Z d n Z d l= n Z d 2 j= { Arj f n,a j Asj f n,b j }, where a j = a j n andb 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,withp Z d We want to evaluate the maximum contribution that f p,p d might have to the right-hand side of 28 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 23 we have 29 p,p d n,a j λrj

7 DISCRETE MAXIMAL OPERATORS 25 Using 2, 22 and 29, 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 28 as f p,p d Nr j 20 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 In the last inequality of 20, we have used 29 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 20 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 side of 28 This is given by 2 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 23 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

8 252 EMANUEL CARNEIRO AND KEVIN HUGHES increasing sequence a j = a j p d The problem becomes then to bound 22 S{a 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 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 S{a j } given by 22 ThesumS{a 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 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

9 DISCRETE MAXIMAL OPERATORS 253 andthisistrueifandonlyif 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 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,sincewemighthave ã 0 > a Therefore, the required sum 22 is bounded above by the sum considering the particular sequence a j = j This gives us 23 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 = { } n Z C d Ω λ d n c c 2 c d, C Ω λ d n c λ n c 2 λ n >c 2 λ d n c λ d n c = Cd, Ω <

10 254 EMANUEL CARNEIRO AND KEVIN HUGHES 24 Conclusion We have proved that the contribution of a generic point fp,p 2,,p d to the right-hand side of 28 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 and this completes the proof 22 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 setup is the same up to the beginning of the double counting argument For a given point n,a j,we can pick a point x j 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, ie, 24 Mf n,a j = A xj,r j f n,a j := f m N x j,r j m Ω rj x j The existence of such a pair x j,r j for which the supremum is attained is guaranteed, since for any maximizing sequence x k j,rk j, the right-hand side of 24 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>0sincef l Z d,andthenwewouldhave only a finite number of subsets of Z d to choose from for the sum in 24 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 Notethatn,b j Ω rj y j Ω sj y j From the convexity of Ω and the fact that e d Ω wealsohaveω rj x j Ω sj y j Therefore 25 Mf n,b j A yj,s j f n,b j = N y j,s j and 26 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 f m, { A xj,r j f n,a j A yj,s j f n,b j } Consider a point p = p,p d Z d Thetermf p,p d will only contribute positively to a summand on the right-hand side of 26 if p,p d Ωrj x j In this case, since n,a j Ωrj x j, using 23 we have 27 p,p d n,a j 2 λr j 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

11 DISCRETE MAXIMAL OPERATORS Setup Since f k f f k f and the maximal operator only sees the absolute value of a function, we may assume without loss of generality that f k 0 for all k, andthatf 0 It suffices to prove the result for each partial derivative, ie, that 3 Mf k Mf x i x i 0 l Z d as k, for each i =, 2,,d As before, we shall prove it for i = d and the other cases are analogous 32 A discrete version of Luiro s lemma For a function g l Z d andapoint 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, ie, 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 analog of [3, Lemma 22] 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 mapping r A r f n for r 0 From the fact that f l Z d together with 22 we can see that A r f n 0asr Therefore, the set of values in the image {A r f n; r 0} such that A r f n 2Mf n is a finite set There exists then a second larger value which falls short of the maximum by a quantity we define as ɛ n, ie, if A r f n >Mf n ɛ n thena 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 wehave 32 Mf n =A s f n =A s f k na s f f k n Mf k nɛ Now given r k Rf k n we can use 32 to obtain A rk f n =A rk f k na rk f f k n = Mf k na 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 33 Reduction via the Brezis Lieb lemma Given ɛ>0, we can find k 0 such that f k f l ɛ, andusinglemma3forafixed n Z d,wecanchoosek k 0 so that we also have Rf k n Rf n for k k Takinganyr k Rf k n wehave 33 Mf n Mf k n = A rk f n A rk f k n ɛ, for k k and thus Mf k n Mf n ask The same can be said replacing n by n e d and thus we find that 34 x d Mf k n x d Mf n

12 256 EMANUEL CARNEIRO AND KEVIN HUGHES 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 34 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 35 lim k Mf k x d = Mf l Z d x d l Z d l Z d The reduction to 35 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 34 we have 36 Mf x d lim inf l Z d k Mf k x d l Z d 35 Upper bound Given ɛ>0 we shall prove that there exists k 0 = k 0 ɛ such that for k k 0 we have 37 Mf k x d Mf l Z d x d ɛ 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 36 would prove that the limit exists and 35 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 38,n d x d l Z d n 2R n d 2R n 2R n d >2R := S S 2 S 3 We shall bound S, S 2 and S 3 separately Mf k n,n d x d n >2R n d Z,

13 DISCRETE MAXIMAL OPERATORS 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 39 f k f l Z d ɛ, for k k Using33wehavethat for any n with n 2R Thus 30 S = n 2R n d 2R Mf k n Mf n x d x d 2ɛ, Mf k n,n d x d n 2R n d 2R Mf n,n d x d 2ɛ 4R d Mf x d 2ɛ 4R d l Z d 37 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 28 3 S 2 = Mf k n,l x d n >2R 2 l= n >2R j= { Arj f k n,a j Asj f k n,b j } 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 32,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, λ p n 2 p d a j 2 /2 d aj a j c

14 258 EMANUEL CARNEIRO AND KEVIN HUGHES Using Lemma 2 we see that the sum on the right-hand side of 32 is majorized by the sum with the sequence a j = j This gives us 2f k p 33,p d n >2R j= λ p n 2 j 2 /2 d c c 2 c d, λ p n 2 j 2 /2 d c We now evaluate this contribution in two distinct sets First, we consider the case when p,p d BR, for which we have p n R Imposing the condition that 34 λ R>c 2 we can ensure that the contribution of f k p,p d is majorized by 2f k p,p d λ n c d 35 λ n c d n R := 2 f k p,p d hr The fact that hr 0asR 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 23 If we then sum up these contributions and plug them in on the right-hand side of 3 we find 36 S 2 2 hr χ BR f k l Z d 2 C χ BR cf k l Z d 38 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 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

15 DISCRETE MAXIMAL OPERATORS 259 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 = wehave 37 S 3 { 2 Arj f k n,a j Asj f k n },b j n 2R j= The contribution of a generic point f k p,p d to the right-hand side of 37 following the calculation 2 has an upper bound of 38 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, 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 =, thesumon the right-hand side of 38 is maximized when a j =2R j for j We would then have the upper bound 39 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 c d c 2 c d, λ p n 2 p d 2R j 2 /2 d c 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, andifwechooser

16 260 EMANUEL CARNEIRO AND KEVIN HUGHES satisfying 34 the contribution of f k p,p d will be less than or equal to 2f k p { 4R d 320,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 Cf k p,p d as in 23 Plugging these upper bounds in 37 we find { } 32 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 d By symmetry the same bound holds for S 3 39 Conclusion Putting together 38, 30, 36 and 32 we obtain Mf k x d 322 Mf l Z d x d 2ɛ 4R d l Z d { } 4R d 4 λ R c d 6hR 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 34, { } 4R d λ R c d 6hR ɛ 3 f l Z d, and Then we choose ɛ such that 324 ɛ χ BR cf l Z d ɛ 36 C ɛ 64R d, and this generates k as described in 39 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 325 χ BR f k l Z d f k l Z d f l Z d and 326 χ BR cf k l Z d χ BR cf l Z d χ BR cf k f l Z d ɛ 8 C

17 DISCRETE MAXIMAL OPERATORS 26 Plugging into 322 gives us Mf k x d Mf l Z d x d ɛ l Z d 3 ɛ 3 ɛ 3, for all k k 0, and the proof is now complete 32 Non-centered case We will indicate here the basic changes that have to be made in comparison with the centered case argument Forafunctiong l Z d andapoint 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, ie, Rg n = x, r Rd R ; Mg n =A x,r g n = N x, r m Ω r x g m The proof of the following result is essentially the same as in Lemma 3 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 in the appropriate places Acknowledgments The first author acknowledges support from CNPq-Brazil grants 47352/20 8 and /20 2, and FAPERJ grant E 26/0300/202 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 [] JM 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 , electronic [2] J Bober, E Carneiro, K Hughes and LB Pierce, On a discrete version of Tanaka s theorem for maximal functions, Proc Amer Math Soc , [3] J Bourgain, Averages in the plane over convex curves and maximal operators, J Anal Math [4] H Brézis and E Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc Amer Math Soc , [5] E Carneiro and D Moreira, On the regularity of maximal operators, Proc Amer Math Soc , [6] E Carneiro and B F Svaiter, On the variation of maximal operators of convolution type, J Funct Anal , [7] P Haj lasz and J Onninen, On boundedness of maximal functions in Sobolev spaces, Ann Acad Sci Fenn Math , [8] J Kinnunen, The Hardy Littlewood maximal function of a Sobolev function, Israel J Math , 7 24 [9] J Kinnunen and P Lindqvist, The derivative of the maximal function, J Reine Angew Math , 6 67 [0] J Kinnunen and E Saksman, Regularity of the fractional maximal function, Bull London Math Soc , [] O Kurka, On the variation of the Hardy-Littlewood maximal function, preprint at

18 262 EMANUEL CARNEIRO AND KEVIN HUGHES [2] S Lang, Algebraic number theory, Vol 0 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2nd ed, 994, ISBN [3] H Luiro, Continuity of the maximal operator in Sobolev spaces, Proc Amer Math Soc , electronic [4], On the regularity of the Hardy Littlewood maximal operator on subdomains of R n, Proc Edinb Math Soc , [5] A Magyar, EM Stein, and S Wainger, Discrete analogues in harmonic analysis: spherical averages, Ann Math , [6] EM Stein, Maximal functions I Spherical means, Proc Natl Acad Sci USA , [7], Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Vol 43 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ 993, ISBN With the assistance of Timothy S Murphy, Monographs in Harmonic Analysis, III [8] H Tanaka, A remark on the derivative of the one-dimensional Hardy Littlewood maximal function, Bull Aust Math Soc , [9] F Temur, On regularity of the discrete Hardy-Littlewood maximal function, preprint at IMPA Estrada Dona Castorina, 0, Rio de Janeiro, RJ, Brazil address: carneiro@impabr Mathematics Princeton University Fine Hall, Washington Road, Princeton, NJ , USA address: kjhughes@mathprincetonedu