On the endpoint regularity of discrete maximal operators
aa r X i v : . [ m a t h . C A ] S e p ON THE ENDPOINT REGULARITY OF DISCRETEMAXIMAL 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 maximaloperator Mf ( ~n ) = sup r ≥ N ( r ) X ~m ∈ Ω r (cid:12)(cid:12) f ( ~n + ~m ) (cid:12)(cid:12) , where (cid:8) Ω r (cid:9) r ≥ are dilations of a convex set Ω (open, bounded and withLipschitz boudary) containing the origin and N ( r ) is the number of latticepoints inside Ω r . We prove here that the operator f
7→ ∇ Mf is boundedand continuous from l ( Z d ) to l ( Z d ). We also prove the same result for thenon-centered version of this discrete maximal operator. Introduction
Background.
For a function f ∈ L loc ( R d ) the (centered) Hardy-Littlewoodmaximal operator is defined as M f ( x ) = sup r> m ( B r ) Z B r | f ( x + y ) | d y, where B r is the ball of radius r centered at the origin and m ( B r ) is the d -dimensionalLebesgue 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 >
1, and that it satisfies a weak-typeestimate M : L ( R d ) → L weak ( R d ) at the endpoint p = 1. The same holds in thenon-centered case, when we consider the supremum over balls that simply containthe point x . In both instances we may also replace the balls by dilations of a convexset with Lipschitz boundary (since these have bounded eccentricity).Over the last years several works addressed the problem of understanding thebehavior of differentiability under a maximal operator. This program began withKinnunen [8] who investigated the action of the classical Hardy-Littlewood max-imal operator in Sobolev spaces and showed that M : W ,p ( R d ) → W ,p ( R d ) isbounded for p >
1. This paradigm that an L p -bound implies a W ,p -bound waslater extended to a local version of the maximal operator [9], to a fractional ver-sion [10] and to a multilinear version [5]. The continuity of M : W ,p → W ,p for p > Date : June 30, 2012.2000
Mathematics Subject Classification.
Primary 42B25, 46E35.
Key words and phrases.
Discrete maximal operators; Hardy-Littlewood maximal operator;Sobolev spaces; bounded variation. since we do not have sublinearity for the weak derivatives of the Hardy–Littlewoodmaximal 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 1]: is the operator f
7→ ∇
M f bounded from W , ( R d ) to L ( R d )?Observe that a bound of the type k∇ M f k L ( R d ) ≤ C (cid:0) k f k L ( R d ) + k∇ f k L ( R d ) (cid:1) (1.1)would imply, via a dilation invariance argument, the bound k∇ M f k L ( R d ) ≤ C k∇ f k L ( R d ) , (1.2)and so the fundamental question would be to compare the variation of M f with thevariation of the original function f (perhaps having the additional information that f is integrable). In the work [18], Tanaka obtained the bound (1.2) in dimension d = 1 for the non-centered Hardy-Littlewood maximal operator with constant C =2. This was later improved by Aldaz and P´erez L´azaro [1] who obtained (1.2) withthe sharp C = 1 under the minimal assumption that f is of bounded variation(still, only in dimension d = 1 and for the non-centered maximal operator). Theprogress in the centered case is very recent and also only in dimension d = 1. In[11] O. Kurka showed that if f is of bounded variation on R thenVar( M f ) ≤ C Var( f )for some constant C >
1, where Var( f ) denotes the total variation of the function f . This result was later adapted to the one-dimensional discrete setting by Temur[19]. It is likely that the sharp constant in Kurkas inequality should be C = 1, butthis remains an open problem. Regularity results of similar flavour for the heatflow maximal operator and the Poisson maximal operator were obtained in [6].1.2. The discrete analogue.
We address here this problem in the discrete setting.We shall generally denote by ~n = ( n , n , ...., n d ) a vector in Z d and for a function f : Z d → R we define its l p -norm as usual: k f k l p ( Z d ) = X ~n ∈ Z d (cid:12)(cid:12) f ( ~n ) (cid:12)(cid:12) p /p , if 1 ≤ p < ∞ , and k f k l ∞ ( Z d ) = sup ~n ∈ Z d (cid:12)(cid:12) f ( ~n ) (cid:12)(cid:12) . The gradient ∇ f of a discrete function f will be the vector ∇ f ( ~n ) = (cid:18) ∂f∂x ( ~n ) , ∂f∂x ( ~n ) , ..., ∂f∂x d ( ~n ) (cid:19) , where ∂f∂x i ( ~n ) := f ( ~n + ~e i ) − f ( ~n ) , and ~e i = (0 , , ..., , ...,
0) is the canonical i -th base vector.Now let Ω ⊂ R d be a bounded open subset that is convex with Lipschitz bound-ary. Let us assume that ~ ∈ int(Ω) and normalize it so that ~e d ∈ ∂ Ω. We now
ISCRETE MAXIMAL OPERATORS 3 define the set that will play the role of the “ball of center ~x and radius r ” in ourmaximal operators. For r > r ( ~x ) = (cid:8) ~x ∈ Z d ; r − ( ~x − ~x ) ∈ Ω (cid:9) , and for r = 0 we put Ω ( ~x ) = (cid:8) ~x (cid:9) . Whenever ~x = ~ r = Ω r (cid:0) ~ (cid:1) for simplicity. For instance, to workwith regular l p -balls one should consider Ω = (cid:8) ~x ∈ R d ; | ~x | p < (cid:9) .From now on we use the letter M to denote the centered discrete maximaloperator associated to Ω given by M f ( ~n ) = sup r ≥ N ( r ) X ~m ∈ Ω r (cid:12)(cid:12) f ( ~n + ~m ) (cid:12)(cid:12) , (1.3)where N ( r ) is the number of lattice points in the set Ω r . We define the non-centereddiscrete maximal operator f M associated to Ω in a similar way, by writing f M f ( ~n ) = sup r ≥ N ( ~x , r ) X ~m ∈ Ω r ( ~x ) | f ( ~m ) | , (1.4)where the supremum is taken over all “balls” Ω r ( ~x ) such that ~n ∈ Ω r ( ~x ), and N ( ~x , r ) denotes the number of lattice points in the set Ω r ( ~x ).These convex Ω-balls have roughly the same behavior as the regular balls, fromthe geometric and arithmetic points of view. For instance, we have the followingasymptotics [12, Chapter VI §
2, Theorem 2] for the number of lattice points N ( ~x , r ) = C Ω r d + O (cid:0) r d − (cid:1) (1.5)as r → ∞ , where C Ω = m (Ω) is the d -dimensional volume of Ω, and the constantimplicit 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 N ( r ) = (2 ⌊ r ⌋ + 1) d ).As in the continuous case, both M and f M are of strong type ( p, p ), if p > ,
1) (see for instance [17, Chapter X]). It is then natural toask 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 k f k l p + k∇ f k l p is equivalent to the norm k f k l p , and thus the question of whether M and f M are bounded in discrete Sobolev spaces is trivially true for p >
1. Onthe other hand, the regularity at the endpoint case p = 1 is a very interesting topicand the main objective of this paper is to present the folllowing result. Theorem 1 (Endpoint regularity of discrete maximal operators) . Let d ≥ andconsider M and f M as defined in (1.3) and (1.4) . (i) (Centered case) The operator f
7→ ∇
M f is bounded and continuous from l (cid:0) Z d (cid:1) to l (cid:0) Z d (cid:1) . (ii) (Non-centered case) The operator f
7→ ∇ f M f is bounded and continuousfrom l (cid:0) Z d (cid:1) to l (cid:0) Z d (cid:1) . The boundedness part in Theorem 1 provides a positive answer to the questionof Haj lasz and Onninen [7, Question 1] in the discrete setting, in all dimensions andfor this general family of centered or non-centered maximal operators with convexΩ-balls. The insight for this part was originated in a joint work of the authors
CARNEIRO AND HUGHES with J. Bober and L. B. Pierce [2] where the case d = 1 was treated, and it hastwo main ingredients: (i) a double counting argument to evaluate the maximalcontribution of each point mass of f to k∇ M f k l ; (ii) a summability argumentover the sequence of local maxima and local minima of M f . The technique is nowrefined to contemplate the n -dimensional case and this general family of operators.The continuity result is a novelty in the endpoint regularity theory. Luiro’sframework [13] for the continuity of the classical Hardy-Littlewood maximal oper-ator in the Sobolev space W ,p ( R d ), for p >
1, is not adaptable since it relies onthe L p -boundedness of this operator (which we do not have here), and we will onlybe able to use a few ingredients of it. The heart of our proof lies instead on thetwo core ideas mentioned above for the boundedness part and a useful applicationof the Brezis-Lieb lemma [4]. Remark 1:
One might ask if inequality (1.2) holds in the discrete case, whichwould be a stronger result than our Theorem 1. This has only been proved indimension d = 1 for the non-centered maximal operator (see [2]) with sharp constant C = 1 (i.e. the non-centered maximal function does not increase the variation ofa function). Note that the dilation invariance argument to deduce (1.2) from (1.1)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 f (0) =1 and f ( n ) = 0 for n = 0, we obtain M f ( n ) = 1 / (2 | n | + 1) and thus ( M f ) ′ ( n ) = O (cid:0) | n | − (cid:1) . Examples like this may raise the question on whether ∇ M f belongs toa better l p space (i.e. p <
1) when f ∈ l . It turns out that the general answeris negative, and Theorem 1 is sharp in this sense. To see this consider a function f ∈ l ( Z ) such that f / ∈ l p ( Z ) for any p <
1, for example f ( n ) = 1 / (cid:0) n log ( n + 1) (cid:1) for n ≥
1, and zero otherwise. Now choose a sequence 1 = a < a < a < a < .... of natural numbers such that(i) a ≥ a n +1 − a n > a n − a n − + 2, for any n ≥ f (1) > k f k a − a )+1 .(iv) f (1)3 > k f k a − a − .(v) f ( n ) > k f k a n − a n − )+1 , for any n ≥ f ( n )3 > k f k a n − a n − +1)+1 , for any n ≥ g : Z → R given by g ( a n ) = f ( n ) for n ≥
1, and zero other-wise. Note that k g k l = k f k l . Conditions (i)-(vi) above guarantee that, for theone-dimensional discrete centered Hardy-Littlewood maximal operator M , we have M g ( a n ) = f ( n ) and M g ( a n + 1) = f ( n )3 , for n ≥
1. Thus (
M g ) ′ ( a n ) = f ( n )3 , andthus ( M g ) ′ / ∈ l p ( Z ) for any p < Remark 3:
Another interesting variant would be to consider the spherical maximaloperator [3, 16] and its discrete analogue [15]. The non-endpoint regularity of thecontinuous operator in Sobolev spaces was proved in [7] and it would be interestingto investigate what happens in the endpoint case, both in the continuous and inthe discrete settings.
ISCRETE MAXIMAL OPERATORS 5 Proof of Theorem 1 - Boundedness
Centered case.
We start with some arithmetic and geometric properties ofthe sets Ω r . From (1.5) we can find a constant c depending only on the dimension d and the set Ω such that N ( ~x , r ) ≤ C Ω (cid:0) r + c (cid:1) d , (2.1)and N ( ~x , r ) ≥ max n C Ω (cid:0) max { r − c , } (cid:1) d , o =: C Ω (cid:0) r − c (cid:1) d + . (2.2)Over (2.2) it should be clear that if ~x ∈ Z d we can take r ≥
0, and if ~x / ∈ Z d we shall only be taking radii r so that the corresponding ball contains at least onelattice point to calculate the average. We define c > c as the constant such that C Ω ( c − c ) d = 1 . Since Ω is bounded, there exists λ > ⊂ B λ (note that λ ≥ ~e d ∈ Ω). This means that if ~p ∈ Ω r ( ~x ) then (cid:12)(cid:12) ~p − ~x (cid:12)(cid:12) ≤ λr. (2.3)These constants c , c and λ will be fixed throughout the rest of the paper.2.1.1. Set up.
We want to show that (cid:13)(cid:13) ∇ M f k l ( Z d ) ≤ C k f k l ( Z d ) (2.4)for a suitable C that might depend on d and Ω in principle. We assume withoutloss of generality that f ≥
0. It suffices to prove that (cid:13)(cid:13)(cid:13)(cid:13) ∂∂x i M f (cid:13)(cid:13)(cid:13)(cid:13) l ( Z d ) ≤ e C k f k l ( Z d ) , for any i = 1 , , ..., d . We will work with i = d (the other cases are analo-gous). Let us write each ~n = ( n , n , ..., n d ) ∈ Z d as ~n = ( n ′ , n d ), where n ′ =( n , n , ..., n d − ) ∈ Z d − . For each n ′ ∈ Z d − we will consider the sum over the lineperpendicular to Z d − passing through n ′ , i.e. ∞ X l = −∞ (cid:12)(cid:12)(cid:12)(cid:12) ∂∂x d M f (cid:0) n ′ , l (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) = ∞ X l = −∞ (cid:12)(cid:12) M f (cid:0) n ′ , l + 1 (cid:1) − M f (cid:0) n ′ , l (cid:1)(cid:12)(cid:12) . For a discrete function g : Z → R we say that a point a is a local maximum of g if g ( a − ≤ g ( a ) and g ( a + 1) < g ( a ). Analogously, we say that a point b is a localminimum of g if g ( b − ≥ g ( b ) and g ( b + 1) > g ( b ). We let { a i } i ∈ Z and { b i } i ∈ Z bethe sequences of local maxima and local minima of M f (cid:0) n ′ , · (cid:1) ordered as follows: ... < b − < a − < b < a < b < a < .... Observe that this sequence (that depends on n ′ ) might be finite (either on one sideor both). In this case, since M f ∈ l weak ( Z d ), it would terminate in a local maximumand 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 isinfinite on both sides. In this case we have ∞ X l = −∞ (cid:12)(cid:12)(cid:12)(cid:12) ∂∂x d M f (cid:0) n ′ , l (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) = 2 ∞ X j = −∞ (cid:8) M f (cid:0) n ′ , a j (cid:1) − M f (cid:0) n ′ , b j (cid:1)(cid:9) . (2.5) CARNEIRO AND HUGHES
The double counting argument.
Let r j be the minimum radius such that thesupremum in (1.3) is attained for the point (cid:0) n ′ , a j (cid:1) , i.e. M f (cid:0) n ′ , a j (cid:1) = A r j f (cid:0) n ′ , a j (cid:1) := 1 N ( r j ) X ~m ∈ Ω rj f (cid:0)(cid:0) n ′ , a j (cid:1) + ~m (cid:1) . (2.6)If we consider the radius s j = r j + ( a j − b j ) centered at the point (cid:0) n ′ , b j (cid:1) we obtain M f (cid:0) n ′ , b j (cid:1) ≥ A s j f (cid:0) n ′ , b j (cid:1) = 1 N ( r j + ( a j − b j )) X ~m ∈ Ω sj f (cid:0)(cid:0) n ′ , b j (cid:1) + ~m (cid:1) . (2.7)The observation that motivates this particular choice of the radius s j is thatΩ r j (cid:0)(cid:0) n ′ , a j (cid:1)(cid:1) ⊂ Ω s j (cid:0)(cid:0) n ′ , b j (cid:1)(cid:1) , which follows from the convexity of Ω and the factthat ~e d ∈ ∂ Ω.From (2.5), (2.6) and (2.7) we obtain (cid:13)(cid:13)(cid:13)(cid:13) ∂∂x d M f (cid:13)(cid:13)(cid:13)(cid:13) l ( Z d ) = X n ′ ∈ Z d − ∞ X l = −∞ (cid:12)(cid:12)(cid:12)(cid:12) ∂∂x d M f (cid:0) n ′ , l (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X n ′ ∈ Z d − ∞ X j = −∞ (cid:8) A r j f (cid:0) n ′ , a j (cid:1) − A s j f (cid:0) n ′ , b j (cid:1)(cid:9) , (2.8)where a j = a j ( n ′ ) and b j = b j ( n ′ ). We now consider a general point ~p = ( p , p ,...., p d ) ∈ Z d , also represented as ~p = (cid:0) p ′ , p d (cid:1) with p ′ ∈ Z d − . We want to evaluatethe maximum contribution that f (cid:0) p ′ , p d (cid:1) might have to the right-hand side of (2.8).For given n ′ and j , this contribution will only be positive if the point (cid:0) p ′ , p d (cid:1) belongsto both sets Ω r j (cid:0)(cid:0) n ′ , a j (cid:1)(cid:1) and Ω s j (cid:0)(cid:0) n ′ , b j (cid:1)(cid:1) (in case the point (cid:0) p ′ , p d (cid:1) belongs onlyto Ω s j (cid:0)(cid:0) n ′ , b j (cid:1)(cid:1) or does not belong to any of these Ω-balls, the contribution isnegative or zero and we disregard it).Since (cid:0) p ′ , p d (cid:1) ∈ Ω r j (cid:0)(cid:0) n ′ , a j (cid:1)(cid:1) , from (2.3) we have (cid:12)(cid:12)(cid:0) p ′ , p d (cid:1) − (cid:0) n ′ , a j (cid:1)(cid:12)(cid:12) ≤ λr j . (2.9) ISCRETE MAXIMAL OPERATORS 7
Using (2.1), (2.2) and (2.9), we can estimate the maximum contribution of f (cid:0) p ′ , p d (cid:1) ,for given n ′ and j , on the associated summand on right-hand side of (2.8) as f (cid:0) p ′ , p d (cid:1) (cid:18) N ( r j ) − N ( r j + a j − b j ) (cid:19) ≤ f (cid:0) p ′ , p d (cid:1) (cid:18) N ( r j ) − N ( r j + a j − a j − ) (cid:19) ≤ f (cid:0) p ′ , p d (cid:1) (cid:18) C Ω ( r j − c ) d + − C Ω ( r j + a j − a j − + c ) d (cid:19) ≤ f (cid:0) p ′ , p d (cid:1) C Ω (cid:16) λ − (cid:0) | p ′ − n ′ | + ( p d − a j ) (cid:1) / − c (cid:17) d + − min ( C Ω ( c + a j − a j − + c ) d , C Ω (cid:16) λ − (cid:0) | p ′ − n ′ | + ( p d − a j ) (cid:1) / + a j − a j − + c (cid:17) d , (2.10)In the last inequality of (2.10) we have used (2.9) and the fact that the function g ( x ) = (cid:18) C Ω ( x − c ) d − C Ω ( x + a j − a j − + c ) d (cid:19) is decreasing as x → ∞ , for x ≥ c . If we sum (2.10) over all j and then over all n ′ ∈ Z d − we find an upper bound for the contribution of f (cid:0) p ′ , p d (cid:1) to the right-handside of (2.8). This is given by2 f (cid:0) p ′ , p d (cid:1) X n ′ ∈ Z d ∞ X j = −∞ C Ω (cid:16) λ − (cid:0) | p ′ − n ′ | + ( p d − a j ) (cid:1) / − c (cid:17) d + − min ( C Ω ( c + a j − a j − + c ) d , C Ω (cid:16) λ − (cid:0) | p ′ − n ′ | + ( p d − a j ) (cid:1) / + a j − a j − + c (cid:17) d . (2.11)2.1.3. The summability argument.
We now prove that the double sum in (2.11) isbounded independently of the the point (cid:0) p ′ , p d (cid:1) and the increasing sequence { a j } .For this we may assume p ′ = 0 (since the sum is over all n ′ ∈ Z d − we can justchange variables here to m ′ = n ′ + p ′ ). We also assume p d = 0, since we mayconsider the increasing sequence a ′ j = a j + p d . The problem becomes then to CARNEIRO AND HUGHES bound S ( { a j } ) = X n ′ ∈ Z d − ∞ X j = −∞ C Ω (cid:16) λ − (cid:0) | n ′ | + a j (cid:1) / − c (cid:17) d + − min C Ω ( c + a j − a j − + c ) d , C Ω (cid:16) λ − (cid:0) | n ′ | + a j (cid:1) / + a j − a j − + c (cid:17) d (2.12)independently of the increasing sequence { a j } of integers. The key tool is the lemmabelow. Lemma 2 (Summability lemma) . For any increasing sequence { a j } j ∈ Z of integersconsider the sum S ( { a j } ) given by (2.12) . The sum S ( { a j } ) is maximized for thesequence a j = j , and in this case the sum is finite.Proof. Suppose we have two terms in the sequence, say a and a that are notconsecutive. Let us prove that if we introduce a term ˜ a in the sequence, with a < ˜ a < a , the overall sum does not decrease. For this it is sufficient to see that C Ω (cid:16) λ − (cid:0) | n ′ | + a (cid:1) / − c (cid:17) d + − min C Ω ( c + a − a + c ) d , C Ω (cid:16) λ − (cid:0) | n ′ | + a (cid:1) / + a − a + c (cid:17) d ≤ C Ω (cid:16) λ − (cid:0) | n ′ | + a (cid:1) / − c (cid:17) d + − min C Ω ( c + a − ˜ a + c ) d , C Ω (cid:16) λ − (cid:0) | n ′ | + a (cid:1) / + a − ˜ a + c (cid:17) d + C Ω (cid:16) λ − (cid:0) | n ′ | + ˜ a (cid:1) / − c (cid:17) d + − min C Ω ( c +˜ a − a + c ) d , C Ω (cid:16) λ − (cid:0) | n ′ | + ˜ a (cid:1) / +˜ a − a + c (cid:17) d , ISCRETE MAXIMAL OPERATORS 9 and this is true if and only if min C Ω ( c + a − ˜ a + c ) d , C Ω (cid:16) λ − (cid:0) | n ′ | + a (cid:1) / + a − ˜ a + c (cid:17) d − min C Ω ( c + a − a + c ) d , C Ω (cid:16) λ − (cid:0) | n ′ | + a (cid:1) / + a − a + c (cid:17) d ≤ C Ω (cid:16) λ − (cid:0) | n ′ | + ˜ a (cid:1) / − c (cid:17) d + − min C Ω ( c +˜ a − a + c ) d , C Ω (cid:16) λ − (cid:0) | n ′ | + ˜ a (cid:1) / +˜ a − a + c (cid:17) d . The last inequality can be verified from the fact that g ( x ) = 1 C Ω x d − C Ω ( x + ˜ a − a ) d is decreasing as x → ∞ , for x ≥
0, and the fact that λ − (cid:0) | n ′ | + a (cid:1) / + (cid:0) a − ˜ a (cid:1) ≥ λ − (cid:0) | n ′ | + ˜ a (cid:1) / . The latter follows by calling a = ˜ a + t (note that t ≥ t to check the sign (here we make use ofthe fact that λ ≥
1, since we might have | ˜ a | > | a | ).Therefore the required sum (2.12) is bounded by above by the sum consideringthe particular sequence a j = j . This gives us S = X n ′ ∈ Z d − ∞ X j = −∞ C Ω (cid:16) λ − (cid:0) | n ′ | + j (cid:1) / − c (cid:17) d + − min C Ω ( c + 1+ c ) d , C Ω (cid:16) λ − (cid:0) | n ′ | + j (cid:1) / + 1+ c (cid:17) d = X ~n ∈ Z d C Ω (cid:0) λ − | ~n | − c (cid:1) d + − min ( C Ω ( c +1+ c ) d , C Ω (cid:0) λ − | ~n | +1+ c (cid:1) d ) ≤ X λ − | ~n |≤ c X λ − | ~n | >c C Ω (cid:0) λ − | ~n | − c (cid:1) d − C Ω (cid:0) λ − | ~n | + 1 + c (cid:1) d ! = e C ( d, Ω) < ∞ . (2.13) (cid:3) Conclusion.
We have proved that the contribution of a generic point f ( p , p ,..., p d ) to the right-hand side of (2.8) is at most a constant 2 e C = 2 e C ( d, Ω) andtherefore, when we sum over all points, we get (cid:13)(cid:13)(cid:13) ∂∂x d M f (cid:13)(cid:13)(cid:13) l ( Z d ) ≤ e C k f k l ( Z d ) . Since the same holds for any direction we obtain the desired inequality (2.4).2.2.
Non-centered case.
We will indicate here the basic modifications that haveto be made in comparison with the proof for the centered case. The set up isthe same up to the beginning of the double counting argument. For a given point (cid:0) n ′ , a j (cid:1) we can pick a point ~x j and a radius r j such that (cid:0) n ′ , a j (cid:1) ∈ Ω r j ( ~x j ) and theaverage over the set Ω r j ( ~x j ) realizes the supremum in the maximal function, i.e., f M f (cid:0) n ′ , a j (cid:1) = A ( ~x j ,r j ) f (cid:0) n ′ , a j (cid:1) := 1 N ( ~x j , r j ) X ~m ∈ Ω rj ( ~x j ) f ( ~m ) . (2.14)This is guaranteed since any maximizing sequence (cid:0) ~x kj , r kj (cid:1) of the right-hand sideof (2.14) must be stationary. In fact, we should have the sequence (cid:0) ~x kj , r kj (cid:1) trappedin a bounded subset (cid:12)(cid:12) ~x kj (cid:12)(cid:12) ≤ R and r kj ≤ R , for some R > f ∈ l ( Z d )), andthen we would have only a finite number of subsets of Z d to choose from for thesum in (2.14).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 r j ( ~y j ) ⊂ Ω s j ( ~y j ). From the convexity of Ω andthe fact that ~e d ∈ ∂ Ω we also have Ω r j ( ~x j ) ⊂ Ω s j ( ~y j ). Therefore f M f (cid:0) n ′ , b j (cid:1) ≥ A ( ~y j ,s j ) f (cid:0) n ′ , b j (cid:1) = 1 N ( ~y j , s j ) X ~m ∈ Ω sj ( ~y j ) f ( ~m ) , (2.15)and (cid:13)(cid:13)(cid:13)(cid:13) ∂∂x d f M f (cid:13)(cid:13)(cid:13)(cid:13) l ( Z d ) = X n ′ ∈ Z d − ∞ X l = −∞ (cid:12)(cid:12)(cid:12)(cid:12) ∂∂x d f M f (cid:0) n ′ , l (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X n ′ ∈ Z d − ∞ X j = −∞ (cid:8) A ( ~x j ,r j ) f (cid:0) n ′ , a j (cid:1) − A ( ~y j ,s j ) f (cid:0) n ′ , b j (cid:1)(cid:9) . (2.16)Consider a point ~p = (cid:0) p ′ , p d (cid:1) ∈ Z d . The term f (cid:0) p ′ , p d (cid:1) will only contributepositively to a summand on the right-hand side of (2.16) if (cid:0) p ′ , p d (cid:1) ∈ Ω r j ( ~x j ). Inthis case, since (cid:0) n ′ , a j (cid:1) ∈ Ω r j ( ~x j ), using (2.3) we have (cid:12)(cid:12)(cid:0) p ′ , p d (cid:1) − ( n ′ , a j ) (cid:12)(cid:12) ≤ λ r j . (2.17)The rest of the proof is the same.3. Proof of Theorem 1 - Continuity
Centered case.
We want to show that if f k → f in l ( Z d ) then ∇ M f k →∇ M f in l ( Z d ). ISCRETE MAXIMAL OPERATORS 11
Set up.
Since (cid:12)(cid:12) | f k | − | f | (cid:12)(cid:12) ≤ (cid:12)(cid:12) f k − f (cid:12)(cid:12) and the maximal operator only sees theabsolute value we may assume without loss of generality that f k ≥ k , andthat f ≥
0. It suffices to prove the result for each partial derivative, i.e. that (cid:13)(cid:13)(cid:13)(cid:13) ∂∂x i M f k − ∂∂x i M f (cid:13)(cid:13)(cid:13)(cid:13) l ( Z d ) → k → ∞ , for each i = 1 , , ..., d . We shall prove it for i = d and the other casesare analogous.3.1.2. A discrete version of Luiro’s lemma.
For a function g ∈ l ( Z d ) and a point ~n ∈ Z d let us define R g ( ~n ) as the set of all radii that realize the supremum in themaximal function at the point ~n , i.e. R g ( ~n ) = r ∈ [0 , ∞ ); M g ( ~n ) = A r | g | ( ~n ) = 1 N ( r ) X ~m ∈ Ω r (cid:12)(cid:12) g (cid:0) ~n + ~m (cid:1)(cid:12)(cid:12) . The next lemma gives us information about the convergence of these sets of radii.It can be seen as the discrete analogue of [13, Lemma 2.2].
Lemma 3.
Let f k → f in l ( Z d ) . Given R > there exists k = k ( R ) such that,for k ≥ k , we have R f k ( ~n ) ⊂ R f ( ~n ) for each ~n ∈ B R .Proof. Fix ~n ∈ B R and consider the application r A r f ( ~n ) for r ≥
0. From thefact that f ∈ l ( Z d ) together with (2.2) we can see that A r f ( ~n ) → r → ∞ .Therefore the set of values in the image { A r f ( ~n ); r ≥ } such that A r f ( ~n ) ≥ M f ( ~n ) is a finite set. There exists then a “second larger” value which falls shortof the maximum by a quantity we define as ǫ ( ~n ), i.e. if A r f ( ~n ) > M f ( ~n ) − ǫ ( ~n )then A r f ( ~n ) = M f ( ~n ) and r ∈ R f ( ~n ). Define ǫ = 13 min (cid:8) ǫ ( ~n ); ~n ∈ B R (cid:9) . Since f k → f in l ( Z d ), we have f k → f in l ∞ ( Z d ). Pick k such that for k ≥ k we have k f k − f k l ∞ ≤ ǫ . For any ~n ∈ B R if we take s ∈ R f ( ~n ) we have M f ( ~n ) = A s f ( ~n ) = A s f k ( ~n ) + A s ( f − f k )( ~n ) ≤ M f k ( ~n ) + ǫ. (3.2)Now given r k ∈ R f k ( ~n ) we can use (3.2) to obtain A r k f ( ~n ) = A r k f k ( ~n ) + A r k ( f − f k )( ~n )= M f k ( ~n ) + A r k ( f − f k )( ~n ) ≥ M f k ( ~n ) − ǫ ≥ M f ( ~n ) − ǫ , and from the definition of ǫ and ǫ ( ~n ) we conclude that r k ∈ R f ( ~n ). (cid:3) Reduction via the Brezis–Lieb lemma.
Given ǫ >
0, we can find k such that k f k − f k l ∞ ≤ ǫ , and using Lemma 3 for a fixed ~n ∈ Z d , we can choose k ≥ k sothat we also have R f k ( ~n ) ⊂ R f ( ~n ) for k ≥ k . Taking any r k ∈ R f k ( ~n ) we have (cid:12)(cid:12) M f ( ~n ) − M f k ( ~n ) (cid:12)(cid:12) = (cid:12)(cid:12) A r k f ( ~n ) − A r k f k ( ~n ) (cid:12)(cid:12) ≤ ǫ , (3.3)for k ≥ k and thus M f k ( ~n ) → M f ( ~n ) as k → ∞ . The same can be said replacing ~n by ~n + ~e d and thus we find that ∂∂x d M f k ( ~n ) → ∂∂x d M f ( ~n ) (3.4) pointwise as k → ∞ . Since (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂∂x d M f k ( ~n ) (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) ∂∂x d M f k ( ~n ) − ∂∂x d M f ( ~n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) ∂∂x d M f ( ~n ) (cid:12)(cid:12)(cid:12)(cid:12) and the latter is in l ( Z d ) from the boundedness part of the theorem, an applicationof the dominated convergence theorem with (3.4) gives uslim k →∞ ((cid:13)(cid:13)(cid:13)(cid:13) ∂∂x d M f k (cid:13)(cid:13)(cid:13)(cid:13) l ( Z d ) − (cid:13)(cid:13)(cid:13)(cid:13) ∂∂x d M f k − ∂∂x d M f (cid:13)(cid:13)(cid:13)(cid:13) l ( Z d ) ) = (cid:13)(cid:13)(cid:13)(cid:13) ∂∂x d M f (cid:13)(cid:13)(cid:13)(cid:13) l ( Z d ) . Therefore, to prove (3.1) it suffices to show thatlim k →∞ (cid:13)(cid:13)(cid:13)(cid:13) ∂∂x d M f k (cid:13)(cid:13)(cid:13)(cid:13) l ( Z d ) = (cid:13)(cid:13)(cid:13)(cid:13) ∂∂x d M f (cid:13)(cid:13)(cid:13)(cid:13) l ( Z d ) . (3.5)The reduction to (3.5) is the content of the Brezis-Lieb lemma [4] in the case p = 1.We henceforth focus our efforts in proving (3.5).3.1.4. Lower bound.
From Fatou’s lemma and (3.4) we have (cid:13)(cid:13)(cid:13)(cid:13) ∂∂x d M f (cid:13)(cid:13)(cid:13)(cid:13) l ( Z d ) ≤ lim inf k →∞ (cid:13)(cid:13)(cid:13)(cid:13) ∂∂x d M f k (cid:13)(cid:13)(cid:13)(cid:13) l ( Z d ) . (3.6)3.1.5. Upper bound.
Given ǫ > k = k ( ǫ ) suchthat for k ≥ k we have (cid:13)(cid:13)(cid:13)(cid:13) ∂∂x d M f k (cid:13)(cid:13)(cid:13)(cid:13) l ( Z d ) ≤ (cid:13)(cid:13)(cid:13)(cid:13) ∂∂x d M f (cid:13)(cid:13)(cid:13)(cid:13) l ( Z d ) + ǫ. (3.7)This would imply thatlim sup k →∞ (cid:13)(cid:13)(cid:13)(cid:13) ∂∂x d M f k (cid:13)(cid:13)(cid:13)(cid:13) l ( Z d ) ≤ (cid:13)(cid:13)(cid:13)(cid:13) ∂∂x d M f (cid:13)(cid:13)(cid:13)(cid:13) 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 chosenlater) and consider the cube (cid:8) ~x ∈ R d ; | ~x | ∞ ≤ R (cid:9) . Let us continue writing ~n ∈ Z d as ~n = (cid:0) n ′ , n d (cid:1) with n ′ ∈ Z d − . We write the required sum in the following way (cid:13)(cid:13)(cid:13)(cid:13) ∂∂x d M f k (cid:13)(cid:13)(cid:13)(cid:13) l ( Z d ) = X | n ′|∞≤ R | n d | ∞ ≤ R (cid:12)(cid:12)(cid:12)(cid:12) ∂∂x d M f k (cid:0) n ′ , n d (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) + X | n ′|∞ > R n d ∈ Z (cid:12)(cid:12)(cid:12)(cid:12) ∂∂x d M f k (cid:0) n ′ , n d (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) + X | n ′|∞≤ R | n d | ∞ > R (cid:12)(cid:12)(cid:12)(cid:12) ∂∂x d M f k (cid:0) n ′ , n d (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) := S + S + S . (3.8)We shall bound S , S and S separately. ISCRETE MAXIMAL OPERATORS 13
Bound for S . Let us pick ǫ > k = k ( ǫ , R ) such that R f k ( ~n ) ⊂ R f ( ~n ) for each ~n with | ~n | ∞ ≤ R + 1 and k f k − f k l ∞ ( Z d ) ≤ ǫ , (3.9)for k ≥ k . Using (3.3) we have that (cid:12)(cid:12)(cid:12)(cid:12) ∂∂x d M f k ( ~n ) − ∂∂x d M f ( ~n ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ǫ , for any ~n with | ~n | ∞ ≤ R . Thus S = X | n ′|∞≤ R | n d | ∞ ≤ R (cid:12)(cid:12)(cid:12)(cid:12) ∂∂x d M f k (cid:0) n ′ , n d (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X | n ′|∞≤ R | n d | ∞ ≤ R (cid:12)(cid:12)(cid:12)(cid:12) ∂∂x d M f (cid:0) n ′ , n d (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) + 2 ǫ (4 R + 1) d ≤ (cid:13)(cid:13)(cid:13)(cid:13) ∂∂x d M f (cid:13)(cid:13)(cid:13)(cid:13) l ( Z d ) + 2 ǫ (4 R + 1) d . (3.10)3.1.7. Bound for S . Here we start with the same idea (and notation for the localmaxima and local minima over vertical lines) as in (2.8) S = X | n ′ | ∞ > R ∞ X l = −∞ (cid:12)(cid:12)(cid:12) ∂∂x d M f k (cid:0) n ′ , l (cid:1)(cid:12)(cid:12)(cid:12) ≤ X | n ′ | ∞ > R ∞ X j = −∞ (cid:8) A r j f k (cid:0) n ′ , a j (cid:1) − A s j f k (cid:0) n ′ , b j (cid:1)(cid:9) . (3.11)We find an upper bound for the contribution of a generic point f k (cid:0) p ′ , p d (cid:1) to theright-hand side of (3.11) as previously done in (2.11). This is given by2 f k (cid:0) p ′ , p d (cid:1) X | n ′ | ∞ > R ∞ X j = −∞ C Ω (cid:16) λ − (cid:0) | p ′ − n ′ | + ( p d − a j ) (cid:1) / − c (cid:17) d + − min ( C Ω ( c + a j − a j − + c ) d , C Ω (cid:16) λ − (cid:0) | p ′ − n ′ | + ( p d − a j ) (cid:1) / + a j − a j − + c (cid:17) d . (3.12) Using Lemma 2 we see that the sum on the right-hand side of (3.12) is majorizedby the sum with the sequence a j = j . This gives us2 f k (cid:0) p ′ , p d (cid:1) X | n ′ | ∞ > R ∞ X j = −∞ C Ω (cid:16) λ − (cid:0) | p ′ − n ′ | + j (cid:1) / − c (cid:17) d + − min ( C Ω ( c + 1 + c ) d , C Ω (cid:16) λ − (cid:0) | p ′ − n ′ | + j (cid:1) / + 1 + c (cid:17) d . (3.13)We now evaluate this contribution in two distinct sets. Firstly, we consider the casewhen (cid:0) p ′ , p d (cid:1) ∈ B R , for which we have | p ′ − n ′ | ≥ R . Imposing the condition that λ − R > c (3.14)we can ensure that the contribution of f k (cid:0) p ′ , p d (cid:1) is majorized by2 f k (cid:0) p ′ , p d (cid:1) X | ~n |≥ R C Ω ( λ − | ~n | − c ) d − C Ω ( λ − | ~n | + 1 + c ) d ! := 2 f k (cid:0) p ′ , p d (cid:1) h ( R ) . (3.15)The fact that h ( R ) → R → ∞ is a crucial point in this proof and shall be usedwhen we choose R at the end. Secondly, when (cid:0) p ′ , p d (cid:1) / ∈ B R the contribution willsimply be bounded by 2 e Cf k (cid:0) p ′ , p d (cid:1) as we found in (2.13). If we then sum up thesecontributions and plug them in on the right-hand side of (3.11) we find S ≤ h ( R ) k χ B R f k k l ( Z d ) + 2 e C k χ B Rc f k k l ( Z d ) . (3.16)3.1.8. Bound for S . We start by noting that S = X | n ′ | ∞ ≤ R ∞ X l =2 R +1 (cid:12)(cid:12)(cid:12)(cid:12) ∂∂x d M f k (cid:0) n ′ , l (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) + X | n ′ | ∞ ≤ R − R − X l = −∞ (cid:12)(cid:12)(cid:12)(cid:12) ∂∂x d M f k (cid:0) n ′ , l (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) := S +3 + S − . Let us provide an upper bound for S +3 . The upper bound for S − is analogous. Weconsider the sequence of local maxima { a j } and local minima { b j } for M f k (cid:0) n ′ , l (cid:1) when l ≥ R + 1. In this situation we do have a first local maximum a (whichmight be the endpoint 2 R + 1) and we order this sequence as follows:2 R + 1 ≤ a < b < a < b < a ... If the sequence terminates, it will be in a local maximum since
M f k ∈ l weak ( Z d ),and we can just truncate the sum in the argument below. Keeping the notation asbefore (and including for convenience a = b = −∞ ) we have S +3 ≤ X | n ′ | ∞ ≤ R ∞ X j =1 (cid:8) A r j f k (cid:0) n ′ , a j (cid:1) − A s j f k (cid:0) n ′ , b j (cid:1)(cid:9) . (3.17) ISCRETE MAXIMAL OPERATORS 15
The contribution of a generic point f k (cid:0) p ′ , p d (cid:1) to the right-hand side of (3.17) (fol-lowing the calculation (2.11)), has an upper bound of2 f k (cid:0) p ′ , p d (cid:1) X | n ′ | ∞ ≤ R ∞ X j =1 C Ω (cid:16) λ − (cid:0) | p ′ − n ′ | + ( p d − a j ) (cid:1) / − c (cid:17) d + − min ( C Ω ( c + a j − a j − + c ) d , C Ω (cid:16) λ − (cid:0) | p ′ − n ′ | + ( p d − a j ) (cid:1) / + a j − a j − + c (cid:17) d . (3.18)Following the ideas of Lemma 2, keeping the constraint that a = −∞ , the sum onthe right-hand side of (3.18) is maximized when a j = 2 R + j for j ≥
1. We wouldthen have the upper bound2 f k (cid:0) p ′ , p d (cid:1) X | n ′ | ∞ ≤ R C Ω (cid:16) λ − (cid:0) | p ′ − n ′ | + ( p d − R − (cid:1) / − c (cid:17) d + + 2 f k (cid:0) p ′ , p d (cid:1) X | n ′ | ∞ ≤ R ∞ X j =2 C Ω (cid:16) λ − (cid:0) | p ′ − n ′ | + ( p d − R − j ) (cid:1) / − c (cid:17) d + − min ( C Ω ( c + 1 + c ) d , C Ω (cid:16) λ − (cid:0) | p ′ − n ′ | + ( p d − R − j ) (cid:1) / + 1 + c (cid:17) d . (3.19)Again, we evaluate this contribution separately for (cid:0) p ′ , p d (cid:1) in the sets B R and B Rc .In the first case, if (cid:0) p ′ , p d (cid:1) ∈ B R we have | p d − R − | ≥ R , and if we choose R satisfying (3.14) the contribution of f k (cid:0) p ′ , p d (cid:1) will be less than or equal to2 f k (cid:0) p ′ , p d (cid:1) ( (4 R + 1) d − C Ω ( λ − R − c ) d + X | ~n |≥ R C Ω ( λ − | ~n | − c ) d − C Ω ( λ − | ~n | + 1 + c ) d ! = 2 f k (cid:0) p ′ , p d (cid:1) ( (4 R + 1) d − C Ω ( λ − R − c ) d + h ( R ) ) . (3.20) In the second case, if (cid:0) p ′ , p d (cid:1) ∈ B Rc , we just bound the contribution of f k (cid:0) p ′ , p d (cid:1) by 2 e C f k (cid:0) p ′ , p d (cid:1) as in (2.13). Plugging these upper bounds in (3.17) we find S +3 ≤ ( (4 R + 1) d − C Ω ( λ − R − c ) d + h ( R ) ) k χ B R f k k l ( Z d ) + 2 e C k χ B Rc f k k l ( Z d ) . (3.21)By symmetry the same bound holds for S − .3.1.9. Conclusion.
Putting together (3.8), (3.10), (3.16) and (3.21) we obtain (cid:13)(cid:13)(cid:13)(cid:13) ∂∂x d M f k (cid:13)(cid:13)(cid:13)(cid:13) l ( Z d ) ≤ (cid:13)(cid:13)(cid:13)(cid:13) ∂∂x d M f (cid:13)(cid:13)(cid:13)(cid:13) l ( Z d ) + 2 ǫ (4 R + 1) d + ( R + 1) d − C Ω ( λ − R − c ) d + 6 h ( R ) ) k χ B R f k k l ( Z d ) + 6 e C k χ B Rc f k k l ( Z d ) . (3.22)We choose (in this order) R large enough so that it satisfies (3.14), ( R + 1) d − C Ω ( λ − R − c ) d + 6 h ( R ) ) ≤ ǫ (cid:0) k f k l ( Z d ) + 1 (cid:1) , (3.23)and k χ B Rc f k l ( Z d ) ≤ ǫ e C .
Then we choose ǫ such that ǫ ≤ ǫ R + 1) d , (3.24)and this generates a k as described in (3.9). We now choose k ≥ k such that forall k ≥ k we have k f k − f k l ( Z d ) ≤ min (cid:26) ǫ e C , (cid:27) , which then implies that k χ B R f k k l ( Z d ) ≤ k f k k l ( Z d ) ≤ (cid:0) k f k l ( Z d ) + 1 (cid:1) (3.25)and k χ B Rc f k k l ( Z d ) ≤ k χ B Rc f k l ( Z d ) + k χ B Rc ( f k − f ) k l ( Z d ) ≤ ǫ e C . (3.26)Plugging (3.23), (3.24), (3.25) and (3.26) into (3.22) gives us (cid:13)(cid:13)(cid:13)(cid:13) ∂∂x d M f k (cid:13)(cid:13)(cid:13)(cid:13) l ( Z d ) ≤ (cid:13)(cid:13)(cid:13)(cid:13) ∂∂x d M f (cid:13)(cid:13)(cid:13)(cid:13) l ( Z d ) + ǫ ǫ ǫ , for all k ≥ k , and the proof is now complete.3.2. Non-centered case.
We will indicate here the basic changes that have to bemade 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 e R g ( ~n ) asthe set of all pairs ( ~x, r ) ∈ R d × R + such that ~n ∈ Ω r ( ~x ) and the supremum in thenon-centered maximal function at ~n is attained for Ω r ( ~x ), i.e. e R g ( ~n ) = ( ~x, r ) ∈ R d × R + ; f M g ( ~n ) = A ( ~x,r ) | g | ( ~n ) = 1 N ( ~x, r ) X ~m ∈ Ω r ( ~x ) (cid:12)(cid:12) g (cid:0) ~m (cid:1)(cid:12)(cid:12) . ISCRETE MAXIMAL OPERATORS 17
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 > there exists k = k ( R ) such that,for k ≥ k , we have e R f k ( ~n ) ⊂ e R f ( ~n ) for each ~n ∈ B R . The rest of the proof is also similar, using (2.15), (2.16) and (2.17) in the appro-priate places. 4.
Acknowledgements
The first author acknowledges support from CNPq-Brazil grants 473152 / − / −
2. The second author acknowldges support from NSF grantDMS − References
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