Regularity and continuity of the multilinear strong maximal operators
aa r X i v : . [ m a t h . C A ] J a n REGULARITY AND CONTINUITY OF THE MULTILINEAR STRONGMAXIMAL OPERATORS
FENG LIU, QINGYING XUE ∗ , AND K ˆOZ ˆO YABUTA Abstract.
Let m ≥
1, in this paper, our object of investigation is the regularity and andcontinuity properties of the following multilinear strong maximal operator M R ( ~f )( x ) = sup R ∋ xR ∈R m Y i =1 | R | Z R | f i ( y ) | dy, where x ∈ R d and R denotes the family of all rectangles in R d with sides parallel to the axes.When m = 1, denote M R by M R . Then, M R coincides with the classical strong maximalfunction initially studied by Jessen, Marcinkiewicz and Zygmund. We showed that M R isbounded and continuous from the Sobolev spaces W ,p ( R d ) × · · · × W ,p m ( R d ) to W ,p ( R d ),from the Besov spaces B p ,qs ( R d ) × · · · × B p m ,qs ( R d ) to B p,qs ( R d ), from the Triebel-Lizorkinspaces F p ,qs ( R d ) × · · · × F p m ,qs ( R d ) to F p,qs ( R d ). As a consequence, we further showed that M R is bounded and continuous from the fractional Sobolev spaces W s,p ( R d ) × · · · × W s,p m ( R d )to W s,p ( R d ) for 0 < s ≤ < p < ∞ . As an application, we obtain a weak typeinequality for the Sobolev capacity, which can be used to prove the p -quasicontinuity of M R .In addition, we proved that M R ( ~f ) is approximately differentiable a.e. if ~f = ( f , . . . , f m )with each f j ∈ L ( R d ) being approximately differentiable a.e. The discrete type of the strongmaximal operators has also been considered. We showed that this discrete type of the maximaloperators enjoys somewhat unexpected regularity properties. Introduction
Hardy-Littlewood maximal functions.
Let f ∈ L ( R d ) with d ≥ M be thewell-known Hardy-Littlewood maximal operator defined on R n as follows. M f ( x ) = sup r> | B r ( x ) | Z B r ( x ) | f ( y ) | dy, where B r ( x ) is the open ball in R d centered at x with radius r and | B r ( x ) | denotes the volumeof B r ( x ). Analogously, the uncentered maximal function f M f at a point x is defined by takingthe supremum of averages over open balls that contain the point. It was well known that themaximal functions and their purpose in differentiation on R were first introduced by Hardy andLittlewood [24], and on R d were treated by Wiener [55]. The celebrated theorem of Hardy-Littlewood-Wiener states that the operator M is of type ( p, p ) for 1 < p ≤ ∞ and weak type(1 , Key words and phrases.
Multilinear strong maximal operators, discrete multilinear strong maximal operators,Sobolev spaces, regularity, Tribel-Lizorkin spaces and Besov spaces, mixed Lebesgue spaces and Sobolev spaces,Sobolev capacity.The first author was supported partly by NSFC (No. 11701333) and SP-OYSTTT-CMSS (No. Sxy2016K01).The second author was partly supported by NSFC (Nos. 11471041, 11671039) and NSFC-DFG (No. 11761131002).The third named author was supported partly by Grant-in-Aid for Scientific Research (C) Nr. 15K04942, JapanSociety for the Promotion of Science.Corresponding author: Qingying Xue Email: [email protected]. ∗ , AND K ˆOZ ˆO YABUTA maximal functions and their variants are often used to control some other important operatorsand give some good absolute size estimates (see [7], [35] and [36]).There is a basic question in the theory of Hardy-Littlewood maximal operators: How does theHardy-Littlewood maximal operator preserve the smoothness properties of a function? Achieve-ments have been made in this direction in the past few years. Among them is the nice work ofKinnunen [29] in 1997, where the regularity properties of maximal operators on the W ,p spaceshas been studied. Recall that the Sobolev spaces W ,p ( R d ), 1 ≤ p ≤ ∞ , are defined by W ,p ( R d ) := { f : R d → R : k f k ,p = k f k L p ( R d ) + k∇ f k L p ( R d ) < ∞} , where ∇ f = ( D f, . . . , D d f ) is the weak gradient of f . Kinnunen showed that M is boundedfrom W ,p ( R d ) to W ,p ( R d ) for 1 < p ≤ ∞ . It was noticed that the W ,p -bound for f M alsoholds by a simple modification of Kinnunen’s arguments or Theorem 1 of [23]. Later on, theresult of Kinnunen has been extended to a local version in [30], to a fractional version in [31], toa multisublinear version in [12, 41] and to a one-sided version in [40]. Whether the continuity for M on W ,p ( R d ) space holds or not is another certainly nontrivial problem, since the maximaloperator is not necessarily sublinear at the derivative level. This problem was first posed byHaj lasz and Onninen [23] and was later settled affirmatively by Luiro [46].Due to the lack of reflexivity of L , it makes the understanding of the W , ( R d ) regularitymore subtler. One interesting question was raised by Haj lasz and Onninen in [23]: Is theoperator f
7→ |∇M f | bounded from W , ( R d ) to L ( R d )? A complete answer was addressedonly in dimension d = 1 in [2, 34, 39, 51] and partial progress on the general case d ≥ M is bounded on L p ( R d ) = W ,p ( R d ) and W ,p ( R d ) for p >
1. Thereforea natural question arises: what is the properties of M on the fractional Sobolev spaces W s,p ( R d )defined by the Bessel potentials when 0 < s <
1? This question was first studied by Korry [33]who observed that M : W s,p ( R d ) → W s,p ( R d ) is bounded for all 0 < s < < p < ∞ .Notice that F p, s ( R d ) = W s,p ( R d ) for any s > < p < ∞ (see [20]). It may be furtherexpected that M still enjoys the boundedness on Triebel-Lizorkin spaces F p,qs ( R d ). This wasdone by Korry [32], who indeed proved that M is bounded on the inhomogeneous Triebel-Lizorkin spaces F p,qs ( R d ) and Besov spaces B p,qs ( R d ) for all 0 < s < < p, q < ∞ .Recently, Luiro [47] established the continuity of M on F p,qs ( R d ) for all 0 < s <
Over the past few decades, many celebratedworks have been done in the study of the maximal functions associated with different kindsof basis. These bases mainly including: some differentiation bases (balls or cubes, rectangleswith some restrictions see [25], [57] and [58]), translation in-variant basis of rectangles [14], basisformed by convex sets, using rectangles with a side parallel to some direction (lacunary parabolicset of directions in [50], Cantor set of directions in [26], arbitrary set of directions [1], [27]). Inthis paper, we will focus on the translation in-variant basis of rectangles studied by C´ordobaand Fefferman [14].Let ~f = ( f , . . . , f m ) be an m -dimensional vector of locally integrable functions and R denotesthe collection of all open rectangles R ⊂ R d with sides parallel to the coordinate axes. In 2011,Grafakos, Liu, P´erez and Torres [21] introduced and studied the weighted strong and endpoint EGULARITY AND CONTINUITY OF THE STRONG MAXIMAL OPERATORS 3 estimates for the multilinear strong maximal function M R , which is defined by(1.1) M R ( ~f )( x ) = sup R ∋ xR ∈R m Y i =1 | R | Z R | f i ( y i ) | dy i , where x ∈ R d and R denotes the family of all rectangles in R d with sides parallel to the axes.Whenever m = 1, we simply denote M R by M R . Then M R coincides with the classical strongmaximal operator. As the most prototypical representative of the multi-parameter operators, M R can be looked as a geometric maximal operator which commutes with full d -parametergroup of dilations ( x , x , . . . , x d ) → ( δ x , δ x , . . . , δ d x d ). It was proved by Garc´ıa-Cuerva andRubio de Francia that M R is bounded on L p ( R d ) for all 1 < p < ∞ (see [17, p.452]). In 1935, amaximal theorem was given by Jessen, Marcinkiewicz and Zygmund in [25]. They pointed outthat unlike the classical Hardy-Littlewood maximal operator, the strong maximal function isnot of weak type (1 , L (log + L )( R d )to L ( R d ). Subsequently, an additional proof of the maximal theorem was given by C´ordobaand Fefferman in 1975, using an alternative geometric method [14]. The basis of the work ofC´ordoba and Fefferman is a selection theorem for families of rectangles in R d . Some delicateproperties of rectangles in R d were also quantified in that study.Furthermore, if m = 1 and d = 1, the operator M R = f M . It was known that f M is boundedand continuous on W ,p ( R ) for 1 < p < ∞ . It follows from [2, 39] that if f ∈ W , ( R ), then f M f is absolutely continuous on R and it holds that k ( f M f ) ′ k L ( R ) ≤ k f ′ k L ( R ) . For d ≥
1, Aldaz andP´erez L´azaro [3] considered a class of local strong maximal operator and proved that it mapsBV( U ) into L ( U ), where U is an open set of R d and BV( U ) is a subclass of L ( U ) functions.See [19, Definition 1.3] and [4, Definition 3.4] for instance.The results in [21] indicate that M R is bounded from L p ( R d ) × · · · × L p m ( R d ) to L p ( R d )for all 1 < p , . . . , p m , p ≤ ∞ and 1 /p = P mi =1 /p i . Moreover, for ~f = ( f , . . . , f m ) with each f i ∈ L p i ( R d ), the following norm inequality holds(1.2) k M R ( ~f ) k L p ( R d ) . p ,...,p m m Y i =1 k f i k L pi ( R d ) . It is well known that the geometry of rectangles in R d is more intricate than that of cubes orballs, even when both classes of sets are restricted to have sides parallel to the axes. Even for m = 1, a basic observation is that M f ( x ) . d M R f ( x ) for all x ∈ R d . However, there doesnot exist any constant C > M R f ( x ) ≤ C M f ( x ) for all x ∈ R d . This indicatesfully that the strong maximal functions are uncontrollable. For these reasons, this makes theinvestigation of the strong maximal functions very complex, but also quite interesting.Based on the facts concerning the previous results on the Hardy-Littlewood maximal opera-tors, it is therefore a natural question to ask whether the multilinear strong maximal operatorsare bounded and continuous on the products of the first order Sobolev spaces W ,p ( R d ) orthe fractional Sobolev spaces W s,p ( R d ) or on its generalizations F p,qs ( R d ) and B p,qs ( R d ). Thisis the main motivation of this work. In the first part of this work, the regularityand conti-nuity properties of the strong maximal functions will be studied. We will show that M R isbounded and continuous from the Sobolev spaces W ,p ( R d ) × · · · × W ,p m ( R d ) to W ,p ( R d ),from the Besov spaces B p ,qs ( R d ) × · · · × B p m ,qs ( R d ) to B p,qs ( R d ), from the Triebel-Lizorkin spaces F p ,qs ( R d ) × · · · × F p m ,qs ( R d ) to F p,qs ( R d ). We further showed that M R is bounded and continuousfrom the fractional Sobolev spaces W s,p ( R d ) × · · · × W s,p m ( R d ) to W s,p ( R d ) for 0 < s < < p < ∞ . As an application, we obtain a weak type inequality for the Sobolev capacity, whichcan be used to prove p -quasicontinuity of the strong maximal function of a Sobolev function. In FENG LIU, QINGYING XUE ∗ , AND K ˆOZ ˆO YABUTA addition, we also show that M R ( ~f ) is approximately differentiable a.e. if ~f = ( f , . . . , f m ) witheach f j ∈ L ( R d ) being approximately differentiable a.e.1.3. Discrete multilinear strong maximal operators.
Another aim of this paper is toinvestigate the regularity properties of the discrete multilinear strong maximal operators. Fora vector-valued function ~f = ( f , . . . , f m ) with each f j being a discrete function defined on Z d ,we define the discrete multilinear strong maximal operator M R by(1.3) M R ( ~f )( ~n ) = sup R ∋ ~nR ∈R N ( R ) m m Y i =1 X ~k ∈ R ∩ Z n | f i ( ~k ) | , where N ( R ) is the number of elements in the set R ∩ Z d . When m = 1, the operator M R reducesto the discrete strong maximal operator M R .Let us recall some pertinent definitions, notations and backgrounds. We shall generally denoteby ~n = ( n , n , . . . , n d ) a vector in Z d . For a discrete function f : Z d → R , we define the ℓ p ( Z d )-norm for 1 ≤ p < ∞ by k f k ℓ p ( Z d ) = ( P ~n ∈ Z d | f ( ~n ) | p ) /p and ℓ ∞ ( Z d )-norm by k f k ℓ ∞ ( Z d ) =sup ~n ∈ Z d | f ( ~n ) | . Next, we recall the definitions of discrete Sobolev space W ,p ( Z d ) and BV q ( Z d )function class. Definition 1.1 ( Discrete Sobolev space W ,p ( Z d ), ([6])) . For 1 ≤ l ≤ d , let ~e l be thecanonical l -th base vector defined by ~e l = (0 , . . . , , , , . . . , D l f ( ~n ) be the partialderivative of f given by D l f ( ~n ) = f ( ~n + ~e l ) − f ( ~n ) and ∇ f be the gradient of f defined by ∇ f ( ~n ) = ( D f ( ~n ) , . . . , D d f ( ~n )). Then, the discrete Sobolev spaces is defined by W ,p ( Z d ) := { f : Z d → R | k f k ,p = k f k ℓ p ( Z d ) + k∇ f k ℓ p ( Z d ) < ∞} . Note that(1.4) k∇ f k ℓ p ( Z d ) ≤ d k f k ℓ p ( Z d ) for 1 ≤ p ≤ ∞ . It follows that(1.5) k f k ℓ p ( Z d ) ≤ k f k ,p ≤ (2 d + 1) k f k ℓ p ( Z d ) for 1 ≤ p ≤ ∞ . This implies that the discrete Sobolev space W ,p ( Z d ) is just ℓ p ( Z d ) with an equivalent norm.It might make our efforts to study the W ,p ( Z d ) regularity of discrete maximal operators seemalmost vacuous since any ℓ p -bound automatically implies a W ,p -bound. However, the endpoint p = 1 is highly nontrivial because of the lack of ℓ -bound for discrete strong maximal operators.To investigate the endpoint regularity of M R , we now introduce the following function class. Definition 1.2 (BV( Z d ) function class , ([9])) . We denote by BV( Z d ) the set of all functionsof bounded variation defined on Z d , where the total variation of f : Z d → R is defined byVar( f ) = k∇ f k ℓ ( Z d ) . (1.4) together with (1.5) and a simple example f ( ~n ) = 1 yields thatBV( Z d ) ( ℓ ( Z d ) = W , ( Z d ) . Recently, the investigation of the regularity of discrete maximal operators has also attracted theattention of many authors (see [6, 9, 10, 37, 40, 43, 45, 49, 52] et al.). Recall that the discreteuncentered version of maximal function is defined by
M f ( ~n ) = sup r> ,~n ∈ B r N ( B r ) X ~k ∈ B r ∩ Z d | f ( ~k ) | , EGULARITY AND CONTINUITY OF THE STRONG MAXIMAL OPERATORS 5 where the surpremum is taken over all open balls B r in R d containing the point ~n with radius r and N ( B r ) denotes the number of lattice points in the set B r . We denote the centered versionof discrete maximal function by f M .When d = 1, the regularity properties of the discrete maximal type operators were studiedby Bober et al. [6], Temur [52] and Madrid [49], Carneiro and Madrid [10] and Liu [37]. Thefollowing sharp inequalities have been established.(1.6) Var( f M f ) ≤ Var( f )and(1.7) Var( M f ) ≤ k f k ℓ ( Z ) . For d ≥
1, Carneiro and Hughes [9] proved that M maps ℓ ( Z d ) into BV( Z d ) boundedly andcontinuously. In (1.3), if one replace the rectangles R by balls B r , then we denote M R by M .Still more recently, the results in [9] was extended by Liu and Wu [43] as follows. Theorem A ( [43] ) . Let d ≥ . Then M maps ℓ ( Z d ) × · · · × ℓ ( Z d ) into BV( Z d ) boundedlyand continuously. It is observed that M ( ~f )( ~n ) . d,m M R ( ~f )( ~n ) for all ~n ∈ Z d . Specially, M R = M when d = 1.However, when d ≥
2, there does not exist any constant
C > M R ( ~f )( ~n ) ≤ C M ( ~f )( ~n )for all ~n ∈ Z d . Based on the above analysis, it is interesting and natural to ask whetherthe discrete strong maximal operators still enjoy some sort of regularity properties. We willshow that the discrete type of the strong maximal operators does enjoy somewhat unexpectedregularities in the end of next part.1.4. Main results.
We now state our main results as follows.
Theorem 1.1 ( Properties on Sobolev spaces ) . Let < p , . . . , p m , p < ∞ and /p = P mi =1 /p i . Then M R is bounded and continuous from W ,p ( R d ) ×· · ·× W ,p m ( R d ) to W ,p ( R d ) .Moreover, if ~f = ( f , . . . , f m ) with each f i ∈ W ,p i ( R d ) , then, for ≤ l ≤ d , it holds that | D l M R ( ~f )( x ) | . m,d,p ,...,p m m X µ =1 M R ( ~f lµ )( x ) , a . e . x ∈ R d , where ~f lµ = ( f , . . . , f µ − , D l f µ , f µ +1 , . . . , f m ) . Remark 1.3.
The case p = ∞ is also valid in Theorem 1.1, which follows from the similararguments to those used in [29, Remark (iii)]. Theorem 1.2 ( Properties on Besov spaces ) . Let < p , . . . , p m , p, q < ∞ , /p = P mi =1 /p i and < s < . Then M R is bounded and continuous from B p ,qs ( R d ) × · · · × B p m ,qs ( R d ) to B p,qs ( R d ) . Theorem 1.3 ( Properties on Triebel-Lizorkin spaces ) . Let < p , . . . , p m , p, q < ∞ , /p = P mi =1 /p i and < s < . Then M R is bounded and continuous from F p ,qs ( R d ) × · · · × F p m ,qs ( R d ) to F p,qs ( R d ) . Noting that F p, s ( R d ) = W s,p ( R d ) for any s > < p < ∞ , then Theorem 1.3 impliesthe following result immediately. Corollary 1.4 ( Properties on Fractional Sobolev spaces ) . Let < p , . . . , p m , p < ∞ , /p = P mi =1 /p i and < s < . Then M R is bounded and continuous from the fractionalSobolev spaces W s,p ( R d ) × · · · × W s,p m ( R d ) to W s,p ( R d ) . FENG LIU, QINGYING XUE ∗ , AND K ˆOZ ˆO YABUTA Theorem 1.1 can be used to obtain a weak type inequality for the Sobolev capacity, which canbe further employed to prove the quasicontinuity of the strong maximal function of a Sobolevfunction. We first need to give the definition of Sobolev p -capacity. Definition 1.4 ( Sobolev p -capacity , ([28])) . For 1 < p < ∞ , the Sobolev p -capacity of theset E ⊂ R d is defined by(1.8) C p ( E ) := inf f ∈A ( E ) Z R d ( | f ( y ) | p + |∇ f ( y ) | p ) dy, where A ( E ) = { f ∈ W ,p ( R d ) : f ≥ E } . We set C p ( E ) = ∞ if A ( E ) = ∅ .It was shown in [15] that the Sobolev p -capacity is a monotone and a countably subadditiveset function. Also, it is an outer measure over R d . Definition 1.5 ( p -quasicontinuous and p -quasieverywhere , [15]) . A function f is said tobe p -quasicontinuous in R d if for every ǫ >
0, there exists a set F ⊂ R d such that C p ( F ) < ǫ and the restriction of f to R d \ F is continuous and finite. A property holds p -quasieverywhereif it holds outside a set of the Sobolev p -capacity zero. Remark 1.6.
It was known that each Sobolev function has a quasicontinuous representative,that is, for each u ∈ W ,p ( R d ), there is a p -quasicontinuous function v ∈ W ,p ( R d ) such that u = v a.e. in R d . This representative is unique in the sense that if v and w are p -quasicontinuousand v = w a.e. in R d , then w = v p -quasieverywhere in R d , see [15] for more details.In 1997, Kinnunen proved that M f is p -quasicontinuous if f ∈ W ,p ( R d ) for any 1 < p < ∞ .Motivated by Kinnunen’s work [29], we shall prove the following result: Theorem 1.5 ( p -quasicontinuity ) . Let < p , . . . , p m < ∞ , and /p = P mi =1 /p i . Supposethat ~f = ( f , . . . , f m ) with each f i ∈ W ,p i ( R d ) , then M R ( ~f ) is p -quasicontinuous. In 2010, Haj lasz and Mal´y [22] proved that M f is approximately differentiable a.e. providedthat f ∈ L ( R d ). Motivated by Haj lasz and Mal´y’s work, we shall establish the following result: Theorem 1.6.
Let ~f = ( f , . . . , f m ) with each f j ∈ L ( R d ) being approximately differentiablea.e., then M R ( ~f ) is approximately differentiable a.e. Remark 1.7.
Since every function in W , ( R d ) space is approximately differentiable a.e., thusTheorem 1.6 yields that if each f j ∈ W , ( R d ), then M R ( ~f ) is approximately differentiable a.e.However, it is unknown that whether M R ( ~f ) is weak differentiable when each f j ∈ W , ( R d ),even in the case m = 1 and d ≥ Theorem 1.7 ( Properties of discrete strong maximal functions ) . Let d ≥ and m ≥ .Then M R is bounded and continuous from ℓ ( Z d ) × · · · × ℓ ( Z d ) to BV( Z d ) . Equivalently, theoperator ~f
7→ ∇ M R ( ~f ) is bounded and continuous from ℓ ( Z d ) ×· · ·× ℓ ( Z d ) to ℓ ( Z d ) . Moreover,if f j ∈ ℓ ( Z d ) for ≤ j ≤ m . Then k∇ M R ( ~f ) k ℓ ( Z d ) . d m X l =1 k∇ f l k ℓ ( Z d ) Y j = l, ≤ j ≤ m k f j k ℓ ( Z d ) . Remark 1.8. we need to address the facts that:
EGULARITY AND CONTINUITY OF THE STRONG MAXIMAL OPERATORS 7 (i) M R is bounded and continuous from W ,p ( Z d ) × · · · × W ,p m ( Z d ) to W ,p ( Z d ) for all1 < p , . . . , p m , p ≤ ∞ and 1 /p = P mi =1 /p i . This conclusion is basically implied by thefollowing two facts. First, one can check that M R is bounded from ℓ p ( Z d ) ×· · ·× ℓ p m ( Z d )to ℓ p ( Z d ). Secondly, it holds easily that | M R ( ~f ) − M R ( ~g ) | ≤ P mµ =1 M R ( ~F µ ) , where ~f = ( f , . . . , f m ), ~g = ( g , . . . , g m ) and ~F µ = ( f , . . . , f µ − , f µ − g µ , g µ +1 , . . . , g m ). Thistogether with (1.5) implies the continuity for M R from W ,p ( Z d ) × · · · × W ,p m ( Z d ) to W ,p ( Z d );(ii) When d ≥
2, the operator f
7→ ∇ M R f is bounded and continuous from ℓ ( Z d ) to ℓ p ( Z d ) for 1 < p ≤ ∞ . However, the operator f
7→ ∇ M R f is not bounded from ℓ ( Z d )to ℓ ( Z d ). This conclusions are basically implied by two facts. First, one can easilycheck that the operator f
7→ ∇ M R f is bounded and continuous from ℓ ( Z d ) to ℓ p ( Z d ).Secondly, let f ( ~n ) = χ { ~ } ( ~n ). Note that k f k ℓ ( Z d ) = 1 and M R f ( ~n ) = Q di =1 ( | n i | + 1) − for each ~n = ( n , . . . , n d ) ∈ Z d . It follows that k∇ M R f k ℓ ( Z d ) = + ∞ . Thus, the operator f
7→ ∇ M R f is not bounded from ℓ ( Z d ) to ℓ ( Z d );(iii) When d ≥
2, from Remark (ii) we know that the discrete strong maximal operator M R isnot bounded from ℓ ( Z d ) to BV( Z d ). However, it was known that the discrete maximaloperator M is bounded from ℓ ( Z d ) to BV( Z d ). Thus, the regularity property of discretestrong maximal operator M R is worse than that of M when d ≥ Corollary 1.8.
Let d ≥ . Then the map f
7→ ∇ M R f is bounded from ℓ ( Z d ) to ℓ q ( Z d ) if andonly if q > . This paper will be organized as follows. Section 2 will be devoted to present the proof ofTheorem 1.1. Section 3 will be devoted to give the proofs of Theorems 1.2 and 1.3. The proofsof Theorems 1.5 and 1.6 will be given in Sections 4 and 5, respectively. In Section 6, we shallprove Theorem 1.7. Finally, we introduce some properties of u x, ~f in Section 7. We would liketo remark that the main ideas employed in the proofs of Theorems 1.1 and 1.7 are greatlymotivated by [29, 46], but our methods and techniques are more delicate and complex thanthose in [29, 46]. It should be pointed out that the main ideas in the proofs of Theorems 1.2and 1.3 are motivated by [44]. Our arguments in the proof of the bounded part in Theorem1.7 are motivated by [10], but our methods and techniques are somewhat different and directthan those in [10]. In addition, the Brezis-Lieb lemma [8] is not necessary in the proof of thecontinuity part of Theorem 1.7.Throughout this paper, if there exists a constant c > ϑ such that A ≤ cB ,we then write A . ϑ B or B & ϑ A ; and if A . ϑ B . ϑ A , we then write A ∼ ϑ B .2. Properties on Sobolev spaces
Prelimary lemmas.
We first present several preliminary lemmas, which play importantroles in the proof of Theorem 1.1. Some basic ideas will be taken from [46], where the proof forthe continuity in W ,p ( R d ) of the Hardy-Littlewood maximal operator has been given. We onlyconsider the case d = 2 and other cases are analogous and more complex. FENG LIU, QINGYING XUE ∗ , AND K ˆOZ ˆO YABUTA For A ⊂ R and x ∈ R , define d ( x, A ) := inf a ∈ A | x − a | and A ( λ ) := { x ∈ R ; d ( x, A ) ≤ λ } for λ ≥ . We denote by k f k p,A the L p -norm of f χ A for all measurable sets A ⊂ R . Let 1 /p = P mj =1 /p j and 1 < p , p , . . . , p m , p < ∞ . Let ~f = ( f , . . . , f m ) with each f j ∈ L p j ( R ). For convenience,we set R + = (0 , ∞ ) and R + = [0 , ∞ ). We also set( R + ) = { ( r , r , ,
0) : ( r , r ) ∈ R , r + r > } , ( R + ) = { (0 , , r , r ) : ( r , r ) ∈ R , r + r > } , ( R + ) , = { ( r , r , r , r ) : ( r , r , r , r ) ∈ R , r + r > , r + r > } . Define the function u ( x ,x ) , ~f : R → R by u ( x ,x ) , ~f ( r , , r , , ,
0) := 1( r , + r , ) m m Y j =1 Z x + r , x − r , | f j ( y , x ) | dy for ( r , , r , , , ∈ ( R + ) ; u ( x ,x ) , ~f (0 , , r , , r , ) := 1( r , + r , ) m m Y j =1 Z x + r , x − r , | f j ( x , y ) | dy for (0 , , r , , r , ) ∈ ( R + ) ; u ( x ,x ) , ~f ( r , , r , , r , , r , ) := Y i =1 r i, + r i, ) m m Y j =1 Z x + r , x − r , Z x + r , x − r , | f j ( y , y ) | dy dy , for ( r , , r , , r , , r , ) ∈ ( R + ) , . In particular, we denote u ( x ,x ) , ~f (0 , , ,
0) = Q mj =1 | f j ( x , x ) | . We can write M R ( ~f )( x ) = sup r , ,r , ,r , ,r , > u ( x ,x ) , ~f ( r , , r , , r , , r , ) . For a fixed point x = ( x , x ) ∈ R , we define the sets B i ( ~f )( x , x ) ( i = 1 , ,
3) by B ( ~f )( x , x ) := n ( r , , r , , r , , r , ) ∈ R : M R ( ~f )( x , x ) =lim sup ( r , ,k ,r , ,k ,r , ,k ,r , ,k ) → ( r , ,r , ,r , ,r , ) u ( x ,x ) , ~f ( r , ,k , r , ,k , r , ,k , r , ,k )for some r , ,k , r , ,k , r , ,k , r , ,k > o . B ( ~f )( x , x ) := n ( r , , r , , r , , r , ) ∈ R × { (0 , } : M R ( ~f )( x , x ) =lim sup ( r , ,k ,r , ,k ) → ( r , ,r , ) u ( x ,x ) , ~f ( r , ,k , r , ,k , ,
0) for some r , ,k , r , ,k > o . B ( ~f )( x , x ) := n ( r , r ) ∈ { (0 , } × R : M R ( ~f )( x , x ) =lim sup ( r , ,k ,r , ,k ) → ( r , ,r , ) u ( x ,x ) , ~f (0 , , r , ,k , r , ,k ) for some r , ,k , r , ,k > o . The function u ( x ,x ) , ~f enjoys the following properties: EGULARITY AND CONTINUITY OF THE STRONG MAXIMAL OPERATORS 9
Lemma 2.1.
Let ~f = ( f , . . . , f m ) with each f j ∈ L p j ( R ) for < p j < ∞ , ( j = 1 , , . . . , m ) .Then the following statements hold: (i) u ( x ,x ) , ~f are continuous on ( R + ) := { ( r , , r , , r , , r , ) ∈ R : r , + r , , r , + r , > } for all ( x , x ) ∈ R , and continuous on R for a.e. ( x , x ) ∈ R ; lim ( r , ,r , ,r , ,r , ∈ R r , r , ,r , r , →∞ u ( x ,x ) , ~f ( r , , r , , r , , r , ) = 0 , for a.e. ( x , x ) ∈ R ; B ( ~f )( x , x ) are nonempty and closed for every ( x , x ) ∈ R ; (ii) u ( x ,x ) , ~f are continuous on { ( r , , r , ) ∈ R : r , + r , > } × { (0 , } for all x ∈ R and a.e. x ∈ R , and continuous at (0 , , , for a.e. ( x , x ) ∈ R ; lim ( r , ,r , ∈ R r , r , →∞ u ( x ,x ) , ~f ( r , , r , , ,
0) = 0 , for all x ∈ R and a.e. x ∈ R ; B ( ~f )( x , x ) are nonempty and closed for a.e. ( x , x ) ∈ R ; (iii) u ( x ,x ) , ~f are continuous on { (0 , } × { ( r , , r , ) ∈ R : r , + r , > } for all x ∈ R and a.e. x ∈ R and continuous at (0 , , , for a.e. ( x , x ) ∈ R ; lim ( r , ,r , ∈ R r , r , →∞ u ( x ,x ) , ~f (0 , , r , , r , ) = 0 , for all x ∈ R and a.e. x ∈ R ; B ( ~f )( x , x ) are nonempty and closed for a.e. ( x , x ) ∈ R .Proof. (i) The first statement follows from the integrability of f j . The proof of the continuityon R for a.e. ( x , x ) ∈ R is very delicate. So, we shall prove it in the last section. We cansee easily that for any ( x , x ) ∈ R , it holds thatlim ( r , ,r , ,r , ,r , ∈ R r , r , ,r , r , →∞ u ( x ,x ) , ~f ( r , , r , , r , , r , ) = 0 , since u ( x ,x ) ,f ( r , , r , , r , , r , ) ≤ | ( r , + r , )( r , + r , ) | − /p Q mj =1 k f j k L pj ( R ) . But, when0 < r , + r , + r , + r , → ∞ , we should treat more carefully, and we shall prove it in thelast section. The last statement can be checked easily.(ii) The first statement follows from the integrability of f j . The continuity at (0 , , ,
0) willbe checked in the last section. Since u ( x ,x ) ,f ( ~r ) ≤ | r , + r , | − /p Q mj =1 k f j ( · , x ) k L pj ( R ) for any( r , , r , , , ∈ R + × { (0 , } and all x ∈ R and a.e. x ∈ R , we getlim ( r , ,r , ∈ R r , r , →∞ u ( x ,x ) , ~f ( r , , r , , ,
0) = 0 . The last statement can be checked easily.(iii) (iii) is the same as in (ii). (cid:3)
Lemma 2.2.
The following relationships between M R ( ~f ) and u ( x ,x ) , ~f are valid. (iv) M R ( ~f )( x , x ) = m Q j =1 | f j ( x , x ) | for a.e. ( x , x ) ∈ R such that ~ ∈ S i =1 B i ( f )( x , x ) ; (v) M R ( ~f )( x , x ) = u ( x ,x ) , ~f ( ~r ) for a.e. ( x , x ) ∈ R such that ~r ∈ B ( ~f )( x , x ) ∩ ( R + ) ;(vi) M R ( ~f )( x , x ) = u ( x ,x ) , ~f ( ~r ) for a.e. ( x , x ) ∈ R such that ~r ∈ B ( ~f )( x , x ) ∩ ( R + ) ; ∗ , AND K ˆOZ ˆO YABUTA (vii) M R ( ~f )( x , x ) = u ( x ,x ) , ~f ( ~r ) for a.e. ( x , x ) ∈ R such that ~r ∈ B ( ~f )( x , x ) ; (viii) M R ( ~f )( x , x ) = u ( x ,x ) , ~f ( ~r ) for a.e. ( x , x ) ∈ R such that ~r ∈ B ( ~f )( x , x ) ; (ix) M R ( ~f )( x , x ) = u ( x ,x ) , ~f ( ~r ) if ~r ∈ B ( ~f )( x ) ∩ ( R + ) , for all x ∈ R . For convenience, for any ~r = ( r , r , . . . , r d ) ∈ R d + and x = ( x , x , . . . , x d ) ∈ R d , we let R ~r ( x ) = { y = ( y , y , . . . , y d ) ∈ R d ; | y i − x i | < r i , i = 1 , , . . . , d } . Lemma 2.3.
Let Λ > , ~ Λ = (Λ , Λ) and ~ , , < p, p i < ∞ and /p = P mi =1 /p i , and ~f = ( f , . . . , f m ) with each f i ∈ L p i ( R ) . Let ~f j = ( f ,j , . . . , f m,j ) such that f i,j → f i in L p i ( R ) when j → ∞ for all i = 1 , , . . . , m . Then for all λ > and i = 1 , , , we have (2.1) lim j →∞ |{ x ∈ R ~ Λ ( ~ B i ( ~f j )( x ) * B i ( ~f )( x ) ( λ ) }| = 0 . Proof.
Without loss of generality we may assume all f i,j ≥ f i ≥
0. We shall prove (2.1)for the case i = 1 and the other cases are analogous. Let λ > >
0. We first concludethat the set { x ∈ R ; B ( ~f j )( x ) * B ( ~f )( x ) ( λ ) } is measurable for all j ≥
1. To see this, let E bethe set of all points which are not Lebesgue points of any of the functions f i,j and f i . Obviously, | E | = 0. We denote by Q + the set of positive rationals. Fix j ≥
1, we can write { x ∈ R \ E : B ( ~f j )( x ) * B ( ~f )( x ) λ } = ∞ [ i =1 ∞ \ k =1 n x ∈ R : ∃ ~r ∈ R s . t . d ( ~r, B ( ~f )( x )) > λ + 1 i and M R ( ~f j )( x ) < u x, ~f j ( ~r ) + 1 k o = ∞ [ i =1 ∞ \ k =1 [ ~t ∈ Q (cid:16)n x ∈ R : d ( ~t, B ( ~f )( x )) > λ + 1 i o \n x ∈ R : M R ( ~f j )( x ) < u x, ~f j ( ~t ) + 1 k o(cid:17) . On the other hand, for any fixed ~t ∈ Q , we have { x : d ( ~t, B ( ~f )( x )) > λ } = ∞ [ l =1 \ ~s ∈ Q ∩{ ~s : | ~s − ~t |≤ λ } n x ∈ R : M R ( ~f )( x ) > u x, ~f ( ~s ) + 1 l o . Therefore, we get the measurability of { x ∈ R ; B ( ~f j )( x ) * B ( ~f )( x ) ( λ ) } for any j ≥ x ∈ R ~ Λ ( ~ γ ( x ) ∈ N \ { } such that(2.2) u x, ~f ( ~r ) < M R ( ~f )( x ) − γ ( x ) , when d ( ~r, B ( ~f )( x )) > λ. Actually, if (2.2) does not hold, then for a.e. x ∈ R ~ Λ ( ~ { ~r k } ∞ k =1 such that lim k →∞ u x, ~f ( ~r k ) = M R ( ~f )( x ) and d ( ~r k , B ( ~f )( x )) > λ. Hence, we may choose a subsequence { ~s k } ∞ k =1 of { ~r k } ∞ k =1 such that ~s k → ~r as k → ∞ . It followsthat ~r ∈ B ( ~f )( x ) and d ( ~r, B ( ~f )( x )) ≥ λ , which is a contradiction. Therefore, (2.2) holds. Let A ,j := { x ∈ R : | M R ( ~f j )( x ) − M R ( ~f )( x ) | ≥ (4 γ ) − } ,A ,j := { x ∈ R : | u x, ~f j ( ~r ) − u x, ~f ( ~r ) | ≥ (2 γ ) − if d ( ~r, B ( ~f )( x )) > λ } , EGULARITY AND CONTINUITY OF THE STRONG MAXIMAL OPERATORS 11 A ,j := { x ∈ R : u x, ~f j ( ~r ) < M R ( ~f j )( x ) − (4 γ ) − , if d ( ~r, B ( ~f )( x )) > λ } . Given ǫ ∈ (0 , γ = γ (Λ , λ, ǫ ) ∈ N \ { } and a measurableset E with | E | < ǫ such that(2.3) R ~ Λ ( ~ ⊂ (cid:8) x ∈ R : u x, ~f ( ~r ) < M R ( ~f )( x ) − γ − , if d ( ~r, B ( ~f )( x )) > λ (cid:9) ∪ E ⊂ A ,j ∪ A ,j ∪ A ,j ∪ E . Let ¯ A be the set of all points x such that x is a Lebesgue point of all f j . Note that | R \ ¯ A | = 0.One can easily check that A ,j ∩ ¯ A ⊂ { x ∈ R : B ( ~f j )( x ) ⊂ B ( ~f )( x ) ( λ ) } . This together with(2.3) yields that(2.4) { x ∈ R ~ Λ ( ~ B ( ~f j )( x ) * B ( ~f )( x ) ( λ ) } ⊂ A ,j ∪ A ,j ∪ E ∪ ( R \ ¯ A ) . Since f i,j → f i in L p i ( R ) when j → ∞ , then there exists N i = N i ( ǫ, γ ) ∈ N such that(2.5) k f i,j − f i k L pi ( R ) < ǫγ and k f i,j k L pi ( R ) ≤ k f i k L pi ( R ) + 1 ∀ j ≥ N i . Moreover, it holds that(2.6) | M R ( ~f j )( x ) − M R ( ~f )( x ) |≤ sup R ∋ xR ∈R | R | m (cid:12)(cid:12)(cid:12) m Y i =1 Z R f i,j ( y ) dy − m Y i =1 Z R f i ( y ) dy (cid:12)(cid:12)(cid:12) ≤ m X l =1 sup R ∋ xR ∈R | R | m l − Y µ =1 Z R f µ ( y ) dy m Y ν = l +1 Z R f ν,j ( y ) dy Z R | f l,j ( y ) − f l ( y ) | dy ≤ m X l =1 M R ( ~F lj )( x ) , where ~F lj = ( f , . . . , f l − , f l,j − f l , f l +1 ,j , . . . , f m,j ). Let N = max ≤ j ≤ m N j .Then, for any j ≥ N ,we get from (2.5) and (2.6) that(2.7) | A ,j | ≤ (4 γ ) p k M R ( ~f j ) − M R ( ~f ) k pL p ( R ) ≤ (4 γm ) p m X l =1 l − Y µ =1 k f µ k pL pµ ( R ) m Y ν = l +1 k f ν,j k pL pν ( R d ) k f l,j − f l k pL pl ( R ) . m,p ,...,p m ,p ǫ. Since | u x, ~f j ( ~r ) − u x, ~f ( ~r ) | ≤ m X l =1 M R ( ~F lj )( x ) . Similarly, | A ,j | . m,p ,...,p m ,p ǫ for any j ≥ N . This together with (2.4) and (2.7) yields (2.1). (cid:3) For any fixed h > f i ∈ L p i ( R ) with 1 < p i < ∞ , define( f i ) lh ( x ) = ( f i ) lτ ( h ) ( x ) − f i ( x ) h and ( f i ) lτ ( h ) ( x ) = f i ( x + h~e l ) , l = 1 , . It is well known that for l = 1 , < p i < ∞ , ( f i ) lτ ( h ) → f i in L p i ( R ) when h →
0, and if f i ∈ W ,p i ( R ) we have ( f i ) lh → D l f i in L p i ( R ) when h → A, B be two subsets ∗ , AND K ˆOZ ˆO YABUTA of R , we define the Hausdorff distance of A and B by π ( A, B ) := inf { δ > A ⊂ B ( δ ) and B ⊂ A ( δ ) } . Applying Lemma 2.3, we can get the following corollary.
Corollary 2.4.
Let f i ∈ L p i ( R ) with < p i < ∞ . Then for all Λ > , λ > , i = 1 , , and l = 1 , , we have lim h → |{ x ∈ R ~ Λ ( ~ π ( B i ( ~f )( x ) , B i ( ~f )( x + h~e l )) > λ }| = 0 . Proof.
Fix i ∈ { , , } . It suffices to show that(2.8) lim h → |{ x ∈ R ~ Λ ( ~
0) : B i ( ~f )( x ) * B i ( ~f )( x + h~e l ) ( λ ) or B i ( ~f )( x + h~e l ) * B i ( ~f )( x ) ( λ ) }| = 0 . One can easily check that B i ( ~f )( x + h~e l ) = B i ( ~f lτ ( h ) )( x ) and B i ( ~f )( x ) = B i ( ~f lτ ( − h ) )( x + h~e l ) . Here ~f lτ ( h ) = ( f ( x + h~e l ) , . . . , f m ( x + h~e l )). It follows that(2.9) { x ∈ R ~ Λ ( ~
0) : B i ( ~f )( x ) * B i ( ~f )( x + h~e l ) ( λ ) } = { x ∈ R ~ Λ ( ~
0) : B i ( ~f lτ ( − h ) )( x + h~e l ) * B i ( ~f )( x + h~e l ) ( λ ) }⊂ { x ∈ R ~ Λ+1 ( ~
0) : B i ( ~f lτ ( − h ) )( x ) * B i ( ~f )( x ) ( λ ) } − h~e l where ~ Λ + 1 = (Λ + 1 , Λ + 1) and | h | ≤
1. Moreover,(2.10) { x ∈ R ~ Λ ( ~
0) : B i ( ~f )( x + he l ) * B i ( ~f )( x ) ( λ ) } = { x ∈ R ~ Λ ( ~
0) : B i ( ~f lτ ( h ) )( x ) ( λ ) * B i ( ~f )( x ) ( λ ) } . We note that ( f i ) lτ ( h ) → f i in L p i ( R ) when h →
0. By Lemma 2.3, it yields that(2.11) lim h → |{ x ∈ R ~ Λ+1 ( ~
0) : B i ( ~f lτ ( − h ) )( x ) * B i ( ~f )( x ) ( λ ) }| = 0and(2.12) lim h → |{ x ∈ R ~ Λ ( ~
0) : B i ( ~f lτ ( h ) )( x ) ( λ ) * B i ( ~f )( x ) ( λ ) }| = 0 . Now, it is easy to see that (2.8) follows from (2.9)-(2.12). (cid:3)
We now state some formulas for the derivatives of the multilinear strong maximal functions,which provide a foundation for our analysis in the continuity part of Theorem 1.1.
Lemma 2.5.
Let f i ∈ W ,p i ( R ) with < p i < ∞ . Then for any l = 1 , and almost every ( x , x ) ∈ R , we have (i) For all ~r ∈ B ( ~f )( x , x ) with r , + r , > , r , + r , > , it holds that (2.13) D l M R ~f ( x , x )= m P µ =1 r , + r , ) m ( r , + r , ) m m Y j = µ, ≤ j ≤ m Z x + r , x − r , Z x + r , x − r , | f j ( y , y ) | dy dy × Z x + r , x − r , Z x + r , x − r , D l | f µ ( y , y ) | dy dy (ii) For all ~r ∈ B ( ~f )( x , x ) ∪ B ( ~f )( x , x ) with r , + r , > , r , = r , = 0 , we have (2.14) D l M R ~f ( x , x )= m X µ =1 r , + r , ) m m Y j = µ, ≤ j ≤ m Z x + r , x − r , | f j ( y , x ) | dy Z x + r , x − r , D l | f µ ( y , x ) | dy (iii) For all ~r ∈ B ( ~f )( x , x ) ∪ B ( ~f )( x , x ) with r , = r , = 0 , r , + r , > , it holds (2.15) D l M R ~f ( x , x )= m X µ =1 r , + r , ) m m Y j = µ, ≤ j ≤ m Z x + r , x − r , | f j ( x , y ) | dy Z x + r , x − r , D l | f µ ( x , y ) | dy (iv) If ~ ∈ B i ( ~f )( x , x ) for i = 1 , , , then, (2.16) D l M R ~f ( x , x ) = D l | f | ( x , x ) . Proof.
We may assume without loss of generality that all f i ≥
0, since | f i | ∈ W ,p i ( R ) if f i ∈ W ,p i ( R ). Fix Λ > l ∈ { , } . Invoking Corollary 2.4, for any i ∈ { , , } , we can choosea sequence { s i,k } ∞ k =1 , s i,k > s i,k → k →∞ π ( B i ( ~f )( x ) , B i ( ~f )( x + s i,k ~e l )) = 0for a.e. x ∈ R ~ Λ ( ~ M R ( ~f ) ∈ W ,p ( R )and k ( M R ( ~f )) ls i,k − D l M R ( ~f ) k L p ( R ) → k → ∞ . We also see that k ( f µ ) ls i,k − D l f µ k L p ( R ) → k → ∞ , k ( f µ ) lτ ( s i,k ) − f µ k L p ( R ) → k → ∞ , kM R (( f µ ) ls i,k − D l f µ ) k L p ( R ) → k → ∞ , kM R (( f µ ) lτ ( s i,k ) − f µ ) k L p ( R ) → k → ∞ , k f M j (( f µ ) ls i,k − D l f µ ) k L p ( R ) → k → ∞ ( j = 1 , , k f M j (( f µ ) lτ ( s i,k ) − f µ ) k L p ( R ) → k → ∞ ( j = 1 , , where f M j is the one dimensional uncentered Hardy-Littlewood maximal operator with respectto the variable x j ( j = 1 , { h i,k } ∞ k =1 of { s i,k } ∞ k =1 anda measurable set A i, ⊂ R ~ Λ ( ~
0) such that | R ~ Λ ( ~ \ A i, | = 0 and(i) ( f µ ) lh i,k ( x ) → D l f µ ( x ), ( f µ ) lτ ( h i,k ) ( x ) → f µ ( x ), M R (( f µ ) lh i,k − D l f µ ) → M R (( f µ ) lτ ( h i,k ) − f µ ) → f M j (( f µ ) lh i,k − D l f µ )( x ) → j = 1 , f M j (( f µ ) lτ ( h i,k ) − f µ ) → j = 1 ,
2) and( M R ( ~f )) lh i,k ( x ) → D l M R ( ~f )( x ) when k → ∞ for any x ∈ A i, ;(ii) lim k →∞ π ( B i ( ~f ))( x ) , B i ( ~f )( x + h i,k ~e l )) = 0 for any x ∈ A i, .Let A i, := ∞ T k =1 { x ∈ R : M R ( ~f )( x + h i,k ~e l ) ≥ u x + h i,k ~e l , ~f (0 , , , } ,A i, := { x ∈ R : M R ( ~f )( x ) = u x, ~f (0 , , ,
0) if (0 , , , ∈ B i ( ~f ( x ) } ,A i, := ∞ T k =1 { x ∈ R : M R ( ~f )( x + h i,k ~e l ) = u x + h i,k ~e l , ~f (0 , , , , , , ∈ B i ( ~f )( x + h i,k ~e l ) } , ∗ , AND K ˆOZ ˆO YABUTA A i, := ∞ T k =1 { x ∈ R : M R ( ~f )( x + h i,k ~e l ) = u x + h i,k ~e l , ~f ( r , , r , , , r , , r , , , ∈ B ( ~f )( x + h i,k ~e l ) and r , + r , > } ,A i, : = { x ∈ R : M R ( ~f )( x ) = u x, ~f ( r , , r , , ,
0) if ( r , , r , , , ∈ B ( ~f )( x )and r , + r , > } ,A i, := ∞ T k =1 { x ∈ R : M R ( ~f )( x + h i,k ~e l ) = u x + h i,k ~e l , ~f (0 , , r , , r , )if (0 , , r , , r , ) ∈ B ( ~f )( x + h i,k ~e l ) and r , + r , > } ,A i, : = { x ∈ R : M R ( ~f )( x ) = u x, ~f (0 , , r , , r , ) if (0 , , r , , r , ) ∈ B ( ~f )( x )and r , + r , > } ,A i, := ∞ T k =1 { x ∈ R : M R ( ~f )( x + h i,k ~e l ) = u x + h i,k ~e l , ~f ( ~r ) if ~r ∈ B ( ~f )( x + h i,k ~e l ) } ; A i, := { x ∈ R : M R ( ~f )( x ) = u x, ~f ( ~r ) if ~r ∈ B ( ~f )( x ) } ; A i, := ∞ T k =1 { x ∈ R : M R ( ~f )( x + h i,k ~e l ) = u x + h i,k ~e l , ~f ( ~r ) if ~r ∈ B ( ~f )( x + h i,k ~e l ) } ; A i, := { x ∈ R : M R ( ~f )( x ) = u x, ~f ( ~r ) if ~r ∈ B ( f )( x ) } . Let A i = T k =1 A i,k and A = A ∪ A ∪ A . Note that | R ~ Λ ( ~ \ A | = 0. Let x = ( x , x ) ∈ A be a Lebesgue point of all f µ and D l f µ and ~r = ( r , , r , , r , , r , ) ∈ B i ( ~f )( x ). There exists ~r ik =( r , ,i,k , r , ,i,k , r , ,i,k , r , ,i,k ) ∈ B i ( ~f )( x + h i,k ~e l ) such that lim k →∞ ( r , ,i,k , r , ,i,k , r , ,i,k , r , ,i,k ) =( r , , r , , r , , r , ). Furthermore, we assume that x is a Lebesgue point of D l f µ ( · , x ) for i = 2, x is a Lebesgue point of D l f µ ( x , · ) for i = 3, and k D l f µ ( x , · ) k L pµ ( R ) , k D l f µ ( · , x ) k L pµ ( R ) < ∞ . Case A ( r , + r , > r , + r , > ~r ∈ B ( ~f )( x ) and this happenswhen x ∈ A . Without loss of generality we may assume that all r , , ,k > r , , ,k > r , , ,k > r , , ,k >
0. Denote [ x − r , , ,k , x + r , , ,k ] × [ x − r , , ,k , x + r , , ,k ] by R k and dy dy = d~y . Then, noting ~r k ∈ B ( ~f )( x + h ,k ~e l ) and using Lemma 2.2 (ix), we have(2.17) D l M R ( ~f )( x ) = lim k →∞ h ,k ( M R ( ~f )( x + h ,k ~e l ) − M R ( ~f )( x )) ≤ lim k →∞ h ,k ( u x + h ,k ~e l , ~f ( ~r k ) − u x, ~f ( ~r k ))= lim k →∞ m X µ =1 r , , ,k + r , , ,k ) m ( r , , ,k + r , , ,k ) m Z Z R k ( f µ ) lh ,k ( y , y ) dy dy × µ − Y ν =1 Z Z R k ( f ν ) lτ ( h ,k ) ( y , y ) dy dy m Y w = µ +1 Z Z R k f w ( y , y ) dy dy = m X µ =1 r , + r , ) m ( r , + r , ) m Z x + r , x − r , Z x + r , x − r , D l f µ ( y , y ) d~y × µ − Y ν =1 Z x + r , x − r , Z x + r , x − r , f ν ( y , y ) d~y m Y w = µ +1 Z x + r , x − r , Z x + r , x − r , f w ( y , y ) d~y. Here, we used the fact that lim k →∞ ~r k = ~r and ( f µ ) lτ ( h ,k ) χ R k → f µ χ [ x − r , ,x + r , ] × [ x − r , ,x + r , ] and ( f µ ) lh ,k χ R k → D l f µ χ [ x − r , ,x + r , ] × [ x − r , ,x + r , ] in L ( R ) as k → ∞ . Then EGULARITY AND CONTINUITY OF THE STRONG MAXIMAL OPERATORS 15 (2.18) D l M R ~f ( x ) ≤ m X µ =1 r , + r , ) m ( r , + r , ) m Z x + r , x − r , Z x + r , x − r , D l f µ ( y , y ) dy dy × Y ν = µ, ≤ ν ≤ m Z x + r , x − r , Z x + r , x − r , f ν (( y , y )) dy dy . On the other hand, using Lemma 2.2 (ix), we have(2.19) D l M R ( ~f )( x ) = lim k →∞ h ,k ( M R ~f ( x + h ,k ~e l ) − M R ( ~f )( x )) ≥ lim k →∞ h ,k ( u x + h ,k ~e l , ~f ( ~r ) − u x, ~f ( ~r ))= m X µ =1 r , + r , ) m ( r , + r , ) m Z x + r , x − r , Z x + r , x − r , D l f µ ( y , y ) dy dy × Y ≤ ν = µ ≤ m Z x + r , x − r , Z x + r , x − r , f ν (( y , y )) dy dy . Combining (2.19) with (2.18) yields (2.13) for a.e. x ∈ R ~ Λ ( ~ ∩ A . Case B ( r , + r , > r , = r , = 0). We consider the following two cases.(i) ( r , , r , , , ∈ B ( ~f )( x ). This happens in the case x ∈ A . Without loss of generalitywe may assume that all r , , ,k , r , , ,k >
0. We notice that r , , ,k = r , , ,k = 0 for all k ≥ ~r k ∈ B ( ~f )( x + h ,k ~e l ) and using Lemma 2.2 (vii), we have(2.20) D l M R ( ~f )( x ) = lim k →∞ h ,k ( M R ( ~f )( x + h ,k ~e l ) − M R ( ~f )( x )) ≤ lim k →∞ h ,k ( u x + h ,k ~e l , ~f ( r , , ,k , r , , ,k , , − u x, ~f ( r , , ,k , r , , ,k , , ≤ lim k →∞ h ,k m X µ =1 r , , ,k + r , , ,k ) m Z x + r , , ,k x − r , , ,k ( f µ ) lh ,k ( y , x ) dy × µ − Y ν =1 Z x + r , , ,k x − r , , ,k ( f ν ) lτ ( h ,k ) ( y , x ) dy m Y w = µ +1 Z x + r , , ,k x − r , , ,k f w ( y , x ) dy ≤ m X µ =1 r , + r , ) m Z x + r , x − r , D l f µ ( y , x ) dy Y ν = µ, ≤ ν ≤ m Z x + r , x − r , f ν ( y , x ) dy . Here we used the fact that lim k →∞ r , , ,k = r , , lim k →∞ r , , ,k = r , and( f µ ) lh ,k ( · , x ) χ [ x − r , , ,k ,x + r , , ,k ] → D l f µ ( · , x ) χ [ x − r , ,x + r , ] in L ( R ) as k → ∞ . Further more, using Lemma 2.2 (vii), we have(2.21) D l M R ( ~f )( x ) ≥ lim k →∞ h ,k ( u x + h ,k e l , ~f ( r , , r , , , − u x, ~f ( r , , r , , , ≥ lim k →∞ m X µ =1 r , , ,k + r , , ,k ) m Z x + r , , ,k x − r , , ,k ( f µ ) lh ,k ( y , x ) dy × µ − Y ν =1 Z x + r , , ,k x − r , , ,k ( f ν ) lτ ( h ,k ) (( y , x )) dy m Y w = µ +1 Z x + r , , ,k x − r , , ,k f w (( y , x )) dy
16 FENG LIU, QINGYING XUE ∗ , AND K ˆOZ ˆO YABUTA ≥ m X µ =1 r , + r , ) m Z x + r , x − r , D l f µ ( y , x ) dy Y ν = µ, ≤ ν ≤ m Z x + r , x − r , f ν ( y , x ) dy . (2.21) together with (2.20) yields (2.14) for a.e. x ∈ R ~ Λ ( ~ ∩ A .(ii) ( r , , r , , , ∈ B ( ~f )( x ). This happens in the case x ∈ A . Assume that r , , ,k , r , , ,k >
0. As in the case A, noting x ∈ A ⊂ A , , we have(2.22) D l M R ( ~f )( x ) ≤ lim k →∞ m X µ =1 r , , ,k + r , , ,k ) m Z x + r , , ,k x − r , , ,k ( f µ ) lh ,k ( y , x ) dy × µ − Y ν =1 Z x + r , , ,k x − r , , ,k ( f ν ) lτ ( h ,k ) ( y , x ) dy m Y w = µ +1 Z x + r , , ,k x − r , , ,k f w ( y , x ) dy . We claim that the limits of the right side will tend to m X µ =1 r , + r , ) m Z x + r , x − r , D l f µ ( y , x ) dy Y ν = µ, ≤ ν ≤ m Z x + r , x − r , f ν ( y , x ) dy . To see this, we only consider the limit of the following parts, since the same reasoning appliesto the other terms. 1 r , , ,k + r , , ,k Z x + r , , ,k x − r , , ,k ( f µ ) lh ,k ( y , x ) dy . Now, we know from the property (i) for x ∈ A that(2.23) lim k →∞ (cid:12)(cid:12)(cid:12)(cid:12) r , , ,k + r , , ,k Z x + r , , ,k x − r , , ,k ( f µ ) lh ,k ( y , x ) − D l f µ ( y , x )) dy (cid:12)(cid:12)(cid:12)(cid:12) ≤ lim k →∞ f M (( f µ ) lh ,k − D l f µ )( x , x )+ lim k →∞ (cid:12)(cid:12)(cid:12)(cid:12) r , , ,k + r , , ,k Z x + r , , ,k x − r , , ,k D l f µ ( y , x ) − D l f µ ( y , x )) dy (cid:12)(cid:12)(cid:12)(cid:12) = 0 . We see moreover that(2.24) lim k →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) r , , ,k + r , , ,k − r , + r , (cid:17) Z x + r , , ,k x − r , , ,k D l f µ ( y , x ) dy (cid:12)(cid:12)(cid:12)(cid:12) ≤ lim k →∞ ( r , , ,k + r , , ,k ) (cid:12)(cid:12)(cid:12) r , , ,k + r , , ,k − r , + r , (cid:12)(cid:12)(cid:12) f M ( D l f µ )( x , x ) = 0 . Noting that k D ℓ f µ )( · , x ) k L p ( R )) < ∞ , we get(2.25) lim k →∞ (cid:12)(cid:12)(cid:12)(cid:12) r , + r , Z x + r , , ,k x − r , , ,k D l f µ ( y , x ) dy − r , + r , Z x + r , x − r , D l f µ ( y , x ) dy (cid:12)(cid:12)(cid:12)(cid:12) ≤ lim k →∞ ( | r , , ,k − r , | + | r , , ,k − r , | ) /p ′ r , + r , × (cid:18)Z x − min { r , ,r , , ,k } x − max { r , ,r , , ,k } + Z x +max { r , ,r , , ,k } x +min { r , ,r , , ,k } | D l f µ ( y , x )) | p dy (cid:19) /p ≤ C lim k →∞ ( | r , , ,k − r , | + | r , , ,k − r , | ) /p ′ r , + r , k D l f µ ( · , x ) k L p ( R )) = 0 . From (2.22) to (2.25) , it follows that
EGULARITY AND CONTINUITY OF THE STRONG MAXIMAL OPERATORS 17 (2.26)lim k →∞ r , , ,k + r , , ,k Z x + r , , ,k x − r , , ,k ( f µ ) lh ,k ( y , x ) dy = 1 r , + r , Z x + r , x − r , D l f µ ( y , x ) dy , and hence we verified the claim.On the other hand, noting x ∈ A ⊂ A , , by the same reasoning as in the case A, we get(2.27) D l M R ( ~f )( x ) ≥ m X µ =1 r , + r , ) m Z x + r , x − r , , D l f µ ( y , x ) dy µ − Y ν =1 Z x + r , x − r , f ν ( y , x ) dy × m Y w = µ +1 Z x + r , x − r , f w ( y , x ) dy The above claim and (2.27) yield (2.14) for a.e. x ∈ R ~ Λ ( ~ ∩ A . Case C ( r , = r , = 0 and r , + r , > x ∈ R ~ Λ ( ~ ∩ ( A ∪ A ). Case D ( ~r = (0 , , , , , , ∈ B ( ~f )( x ). Then x ∈ A . The lower bound of D l M R ( ~f )( x )follows from(2.28) D l M R ( ~f )( x )= lim k →∞ h ,k ( M R ( ~f )( x + h ,k ~e l ) − M R ( ~f )( x )) ≥ lim k →∞ h ,k (cid:16) m Y µ =1 f µ ( x + h ,k ~e l ) − m Y µ =1 f µ ( x ) (cid:17) ≥ m X µ =1 lim k →∞ h ,k ( f µ ( x + h ,k ~e l ) − f µ ( x )) (cid:16) µ − Y ν =1 f ν ( x ) (cid:17)(cid:16) m Y j = µ +1 f j ( x + h ,k ~e l ) (cid:17) ≥ m X µ =1 D l f µ ( x ) (cid:16) Y i = µ, ≤ i ≤ m f i ( x ) (cid:17) . To get the upper bound of D l M R ( ~f )( x ), note that lim k →∞ r , , ,k = 0, lim k →∞ r , , ,k = 0 and r , , ,k = r , , ,k = 0 for all k ≥
1. If r , ,k + r , ,k = 0 for infinitely many k , then by Lemma 2.2(iv). one obtains that(2.29) D l M R ( ~f )( x ) = lim k →∞ h ,k ( M R ( ~f )( x + h ,k ~e l ) − M R ( ~f )( x )) ≤ lim k →∞ h ,k (cid:16) m Y µ =1 f µ ( x + h ,k ~e l ) − m Y µ =1 f µ ( x ) (cid:17) ≤ m X µ =1 D l f µ ( x ) (cid:16) Y ν = µ, ≤ ν ≤ m f ν ( x ) (cid:17) . If there exists k ∈ N such that r , , ,k + r , , ,k > k ≥ k . Then (2.20) gives that D l M R ( ~f )( x ) ≤ m X µ =1 lim k →∞ r , , ,k + r , , ,k Z x + r , , ,k x − r , , ,k ( f µ ) lh ,k ( y , x ) dy (cid:16) µ − Y ν =1 r , , ,k + r , , ,k ∗ , AND K ˆOZ ˆO YABUTA (2.30) × Z x + r , , ,k x − r , , ,k f ν ( y , x ) dy (cid:17)(cid:16) m Y j = ν +1 r , , ,k + r , , ,k Z x + r , , ,k x − r , , ,k ( f j ) lτ ( h ,k ) ( y , x ) dy (cid:17) . Since x is a Lebesgue point for D l f µ ( · , x ), we have(2.31) lim k →∞ (cid:12)(cid:12)(cid:12) r , , ,k + r , , ,k Z x + r , , ,k x − r , , ,k ( f µ ) lh ,k ( y , x ) dy − D l f µ ( x , x ) (cid:12)(cid:12)(cid:12) ≤ lim k →∞ r , , ,k + r , , ,k Z x + r , , ,k x − r , , ,k | ( f µ ) lh ,k ( y , x ) − D l f µ ( y , x )) | dy ≤ lim k →∞ f M (( f µ ) lh ,k − D l f µ )( x )+ lim k →∞ r , , ,k + r , , ,k Z x + r , , ,k x − r , , ,k | D l f µ ( y , x ) − D l f µ ( y , x )) | dy = 0 . Similarly, it holds that lim k →∞ r , , ,k + r , , ,k R x + r , , ,k x − r , , ,k ( f µ ) lτ ( h ,k ) ( y , x ) dy = f µ ( x , x ).We get from (2.30) and (2.31) that(2.32) D l M R ( ~f )( x ) ≤ m P µ =1 D l f µ ( x , x ) (cid:16) Q ≤ ν = µ ≤ m f ν ( x , x ) (cid:17) (2.32) together with (2.28)-(2.29) yields (2.16) in the case ~ ∈ B ( ~f )( x ) for a.e. x ∈ R ~ Λ ( ~ , , , ∈ B ( ~f )( x ). We can get (2.16) for almost x ∈ R ~ Λ ( ~
0) similarly.(iii) Assume that (0 , , , ∈ B ( ~f )( x ). In the case x ∈ A . Note that(2.33) D l M R ( ~f )( x ) = lim k →∞ h ,k ( M R ( ~f )( x + h ,k ~e l ) − M R ( ~f )( x )) ≥ m X µ =1 D l f µ ( x ) (cid:16) Y ν = µ, ≤ ν ≤ m f ν ( x ) (cid:17) . Below we estimate the upper bound of D l M R ( ~f )( x ). We consider the following four cases:(a) If ( r , ,k , r , ,k , r , ,k , r , ,k ) = (0 , , ,
0) for infinitely many k , then D l M R ( ~f )( x ) = lim k →∞ h ,k ( M R ( ~f )( x + h ,k ~e l ) − M R ( ~f )( x )) ≤ m X µ =1 lim k →∞ h ,k ( f µ ( x + h ,k ~e l ) − f µ ( x )) (cid:16) µ − Y ν =1 f ν ( x ) (cid:17)(cid:16) m Y j = µ +1 f j ( x + h ,k ~e l ) (cid:17) ≤ m X µ =1 D l f µ ( x , x ) (cid:16) Y i = µ, ≤ i ≤ m f i ( x , x ) (cid:17) . This leads to the desired results.(b) Denote [ x − r , , ,k , x + r , , ,k ] × [ x − r , , ,k , x + r , , ,k ] by R k . If there exists k ∈ N such that r , , ,k + r , , ,k > r , , ,k + r , , ,k > k ≥ k . Then (2.17) gives that(2.34) D l M R ( ~f )( x ) ≤ m X µ =1 lim k →∞ r , , ,k + r , , ,k )( r , , ,k + r , , ,k ) Z Z R k ( f µ ) lh ,k ( y , y ) dy dy × (cid:16) µ − Y ν =1 r , , ,k + r , , ,k )( r , , ,k + r , , ,k ) Z Z R k f ν ( y , y ) dy dy (cid:17) EGULARITY AND CONTINUITY OF THE STRONG MAXIMAL OPERATORS 19 × (cid:16) m Y j = µ +1 r , , ,k + r , , ,k )( r , , ,k + r , , ,k ) Z Z R k ( f j ) lτ ( h ,k ) ( y , y ) dy dy . Since ( x , x ) is a Lebesgue point for D l f µ , then(2.35) lim k →∞ (cid:12)(cid:12)(cid:12) r , , ,k + r , , ,k )( r , , ,k + r , , ,k ) Z Z R k ( f µ ) lh ,k ( y , y ) dy dy − D l f µ ( x , x ) (cid:12)(cid:12)(cid:12) ≤ lim k →∞ M R (( f µ ) lh ,k − D l f µ )( x , x )+ lim k →∞ (cid:12)(cid:12)(cid:12) r , , ,k + r , , ,k )( r , , ,k + r , , ,k ) Z Z R k D l f µ ( y , y ) − D l f µ ( x , x ) dy dy (cid:12)(cid:12)(cid:12) = 0 . Similarly, we have(2.36) lim k →∞ (cid:12)(cid:12)(cid:12) r , , ,k + r , , ,k )( r , , ,k + r , , ,k ) Z Z R k ( f µ ) lτ ( h ,k ) ( y , y ) dy dy − f µ ( x , x ) (cid:12)(cid:12)(cid:12) = 0 . (2.34) together with (2.35)-(2.36) yields the desired estimate.(c) If there exists k ∈ N such that r , , ,k + r , , ,k > k ≥ k and r , , ,k = r , , ,k = 0for infinitely many k . Then we may have D l M R ( ~f )( x ) ≤ m X µ =1 D l f µ ( x , x ) (cid:16) Y ν = µ, ≤ ν ≤ m f ν ( x , x ) (cid:17) . This shows the desired upper bounds.(d) If there exists k ∈ N such that r , , ,k + r , , ,k > k ≥ k and r , , ,k = r , , ,k = 0for infinitely many k , we can get the upper bounds by the arguments similar to those used inthe case (c).(2.33) together with (a)-(d) yields (2.16) for almost every x ∈ R ~ Λ ( ~ (cid:3) Proof of Theorem 1.1.
We divide the proof into three steps:
Step 1: The boundedness part . Let 1 < p , . . . , p m , p < ∞ and 1 /p = P mi =1 /p i . Let ~f =( f , . . . , f m ) with each f i ∈ W ,p i ( R d ). For a function u and y ∈ R d we define u h ( x ) = u ( x + h ).According to [18, Section 7.11] we know that u ∈ W ,p ( R d ) for 1 < p < ∞ if and only if u ∈ L p ( R d ) and lim sup h → k u h − u k L p ( R d ) / | h | < ∞ . Therefore, we have(2.37) lim sup h → k ( f i ) h − f i k L p ( R d ) | h | < ∞ , ∀ ≤ i ≤ m. On the other hand, for any fixed h ∈ R d and x ∈ R d , we have(2.38) | ( M R ( ~f )) h ( x ) − M R ( ~f )( x ) | ≤ sup R ∋ xR ∈R | R | m (cid:12)(cid:12)(cid:12) m Y i =1 Z R | f i ( y + h ) | dy − m Y i =1 Z R | f i ( y ) | dy (cid:12)(cid:12)(cid:12) ≤ m X i =1 sup R ∋ xR ∈R | R | m Z R | f i ( y + h ) − f i ( y ) | dy × (cid:16) i − Y µ =1 Z R | f µ ( y ) | dy (cid:17)(cid:16) m Y ν = µ +1 Z R | f i ( y + h ) | dy (cid:17) ≤ m X i =1 M R ( ~f ih )( x ) , ∗ , AND K ˆOZ ˆO YABUTA where ~f ih = ( f , . . . , f i − , ( f i ) h − f i , ( f i +1 ) h , . . . , ( f m ) h ). (2.38) together with (1.2) yields that(2.39) k ( M R ( ~f )) h − M R ( ~f ) k L p ( R d ) ≤ m X i =1 k M R ( ~f ih ) k L p ( R d ) . m,d,p ,...,p m m X i =1 k ( f i ) h − f i k L pi ( R d ) Y µ = i, ≤ µ ≤ m k f µ k L pµ ( R d ) . We get from (2.39) and (2.37) that lim sup h → k ( M R ( ~f )) h − M R ( ~f ) k Lp ( R d ) | h | < ∞ . This together withthe fact that M R ( ~f ) ∈ L p ( R d ) yields that M R ( ~f ) ∈ W ,p ( R d ). Step 2: Pointwise estimate for M R ( ~f ) . Let s k ( k = 1 , , . . . ) be an enumeration of positiverational numbers. We can write M R ( ~f )( x ) = sup ~r ∈ ( { s k } ∞ k =1 ) d | R ~r ( x ) | m m Y i =1 Z R ~r ( x ) | f i ( y ) | dy, where ~r = ( r − , . . . , r − d ; r +1 , . . . , r + d ) and R ~r ( x ) = ( x − r − , x + r +1 ) ×· · ·× ( x d − r − d , x d + r + d ). Fixing k ≥
1, we let E k = { ~r = ( r − , . . . , r − d ; r +1 , . . . , r + d ) ∈ R d + ; r − i , r + i ∈ { s , . . . , s k } , i = 1 , , . . . , d } .For k ∈ { , , . . . } , we define the operator T k by T k ( ~f )( x ) = max ~r ∈ E k | R ~r ( x ) | m m Y i =1 Z R ~r ( x ) | f i ( y ) | dy. For any h ∈ R d , we can write | T k ( ~f )( x + h ) − T k ( ~f )( x ) | ≤ m X i =1 max ~r ∈ E k | R ~r ( x ) | m Z R ~r ( x ) | f i ( y + h ) − f i ( y ) | dy × (cid:16) i − Y µ =1 Z R ~r ( x ) | f µ ( y ) | dy (cid:17)(cid:16) m Y ν = µ +1 Z R ~r ( x ) | f i ( y + h ) | dy (cid:17) . This yields that(2.40) | D l ( T k ( ~f ))( x ) | ≤ m X i =1 T k ( ~f li )( x ) ≤ m X i =1 M R ( ~f li )( x ) , for a.e. x ∈ R d . Here ~f li = ( f , . . . , f i − , D l f i , f i +1 , . . . , f m ). For all k ≥
1, by (2.40) and (1.2), it holds that k T k ( ~f ) k ,p ≤ k T k ( ~f ) k L p ( R d ) + d X l =1 k D l T k ( ~f ) k L p ( R d ) ≤ k M R ( ~f ) k L p ( R d ) + d X l =1 m X i =1 k M R ( ~f li ) k L p ( R d ) . m,p ,...,p m m Q i =1 k f i k ,p i . This yields that { T k ( ~f ) } k is a bounded sequence in W ,p ( R d ) which converges to M R ( ~f )pointwisely. The weak compactness of Sobolev spaces implies that { D l ( T k ( ~f )) } k convergesto D l ( M R ( ~f )) weakly in L p ( R d ). This together with (2.40) implies that | D l M R ( ~f )( x ) | ≤ m X i =1 M R ( ~f li )( x ) , for a.e. x ∈ R d . EGULARITY AND CONTINUITY OF THE STRONG MAXIMAL OPERATORS 21
Combining this with (1.2) yields that k∇ M R ( ~f ) k L p ( R d ) ≤ d X l =1 k D l M R ( ~f ) k L p ( R d ) ≤ d X l =1 m X i =1 k M R ( ~f li ) k L p ( R d ) . m,d,p ,...,p m m X i =1 d X l =1 k D l f i k L pi ( R d ) Y j = i, ≤ j ≤ m k f j k L pj ( R d ) . Therefore, it holds that(2.41) k M R ( ~f ) k ,p = k M R ( ~f ) k L p ( R d ) + k∇ M R ( ~f ) k L p ( R d ) ≤ C m,d,p ,...,p m m Y i =1 k f i k ,p i . Step 3: The continuity part.
For convenience, we only prove the case d = 2 and the case d > ~f = ( f , . . . , f m )with each f i ∈ W ,p i ( R ) for 1 < p i < ∞ . Let ~f j = ( f ,j , . . . , f m,j ) such that f i,j → f i in W ,p i ( R ) when j → ∞ . Let 1 < p < ∞ and 1 /p = P mi =1 /p i . It follows from (2.6) that k M R ( ~f j ) − M R ( ~f ) k L p ( R ) → j → ∞ . Thus, it suffices to show that, for any l = 1 , , . . . , d ,it holds that(2.42) k D l M R ( ~f j ) − D l M R ( ~f ) k L p ( R ) → j → ∞ . Without loss of generality we may assume that all f i,j ≥ f i ≥ ǫ > l = 1 ,
2, letting ~f il = ( f , . . . , f i − , D l f i , f i +1 , . . . , f m ), there exists Λ > P mi =1 k M R ( ~f il ) k p,B < ǫ with B = R \ R ~ Λ ( ~ ~ Λ = (Λ , Λ). By the absolutecontinuity, there exists η > P mi =1 k M R ( ~f il ) k p,A < ǫ whenever A is a measurablesubset of R ~ Λ ( ~
0) such that | A | < η . As we already observed, for a.e. x ∈ R , we notice that:(i) u x, ~f il is continuous on R and lim ( r , ,r , ,r , ,r , ∈ R r , r , r , r , →∞ u x, ~f il ( r , , r , , r , , r , ) = 0;(ii) u x, ~f il ( r , , r , , ,
0) is continuous on R and lim ( r , ,r , ∈ R r , r , →∞ u x, ~f il ( r , , r , , ,
0) = 0;(iii) u x, ~f il (0 , , r , , r , ) is continuous on R and lim ( r , ,r , ∈ R r , r , →∞ u x, ~f il (0 , , r , , r , ) = 0 . Then, it follows that for a.e. x ∈ R , the function P mi =1 u x, ~f il ( · , · , · , · ) is uniformly con-tinuous on R ; the function P mi =1 u x, ~f il ( · , · , ,
0) is uniformly continuous on R ; the function P mi =1 u x, ~f il (0 , , · , · ) is uniformly continuous on R . Hence, we can find δ x > | ~r − ~r | < δ x , then (cid:12)(cid:12) m P i =1 u x, ~f il ( ~r ) − m P i =1 u x, ~f il ( ~r ) (cid:12)(cid:12) < | R ~ Λ ( ~ | − /p ǫ ;(v) If | r , , − r , , | + | r , , − r , , | < δ x , then (cid:12)(cid:12) m X i =1 u x, ~f il ( r , , , r , , , , − m X i =1 u x, ~f il ( r , , , r , , , , (cid:12)(cid:12) < | R ~ Λ ( ~ | − /p ǫ ;(vi) If | r , , − r , , | + | r , , − r , , | < δ x , then (cid:12)(cid:12) m X i =1 u x, ~f il (0 , , r , , , r , , ) − m X i =1 u x, ~f il (0 , , r , , , r , , ) (cid:12)(cid:12) < | R ~ Λ ( ~ | − /p ǫ. ∗ , AND K ˆOZ ˆO YABUTA Now we can write R ~ Λ ( ~
0) = (cid:16) ∞ [ i =1 n x ∈ R ~ Λ ( ~
0) : δ x > i o(cid:17) [ N , where |N | = 0. It follows that there exists δ > |{ x ∈ R ~ Λ ( ~
0) : | P mi =1 u x, ~f il ( ~r ) − P mi =1 u x, ~f il ( ~r ) | ≥ | R ~ Λ ( ~ | − /p ǫ for some ~r , ~r with | ~r − ~r | < δ }| =: | B , | < η ;(2.44) |{ x ∈ R ~ Λ ( ~
0) : | P mi =1 u x, ~f il ( r , , , r , , , , − P mi =1 u x, ~f il ( r , , , r , , , , | ≥ | R ~ Λ ( ~ | − /p ǫ for some r , , , r , , , r , , , r , , with | r , , − r , , | + | r , , − r , , | < δ }| =: | B , | < η ;(2.45) |{ x ∈ R ~ Λ ( ~
0) : | P mi =1 u x, ~f il (0 , , r , , , r , , ) − P mi =1 u x, ~f il (0 , , r , , , r , , ) | ≥ | R ~ Λ ( ~ | − /p ǫ for some r , , , r , , , r , , , r , , with | r , , − r , , | + | r , , − r , , | < δ }| =: | B , | < η . Applying Lemma 2.3, there exists j ∈ N such that for i = 1 , , |{ x ∈ R ~ Λ ( ~ B i ( ~f j )( x ) * B i ( ~f )( x ) ( δ ) }| =: | B i,j | < η j ≥ j . Let ~f i,jl = ( f ,j , . . . , f i − ,j , D l f i,j , f i +1 ,j , . . . , f m,j ) . Fix i = 1 , ,
3. Invoking Lemma 2.5, for a.e. x ∈ R , j ≥ j , and for any ~r ∈ B i ( ~f j )( x ) and ~r ∈ B i ( ~f )( x ) with i = 1 , ,
3, we have(2.47) (cid:12)(cid:12)(cid:12) D l M R ( ~f j )( x ) − D l M R ( ~f )( x ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) m X i =1 u x, ~f i,jl ( ~r ) − m X i =1 u x, ~f il ( ~r ) (cid:12)(cid:12)(cid:12) ≤ m X i =1 | u x, ~f i,jl ( ~r ) − u x, ~f il ( ~r ) | + (cid:12)(cid:12)(cid:12) m X i =1 u x, ~f il ( ~r ) − m X i =1 u x, ~f il ( ~r ) (cid:12)(cid:12)(cid:12) . If x / ∈ B ∪ B ,i ∪ B i,j , we choose ~r ∈ B i ( ~f j )( x ) and ~r ∈ B i ( ~f )( x ) such that | ~r − ~r | < δ and(2.48) (cid:12)(cid:12)(cid:12) m X i =1 u x, ~f il ( ~r ) − m X i =1 u x, ~f il ( ~r ) (cid:12)(cid:12)(cid:12) < | R ~ Λ ( ~ | − /p ǫ. On the other hand, for any ~r ∈ B i ( ~f j )( x ) and ~r ∈ B i ( ~f )( x ), one may obtain that(2.49) (cid:12)(cid:12)(cid:12) m X i =1 u x, ~f il ( ~r ) − m X i =1 u x, ~f il ( ~r ) (cid:12)(cid:12)(cid:12) ≤ m X i =1 M R ( ~f il )( x ) , for a.e. x ∈ R . To get the estimate of P mi =1 | u x, ~f i,jl ( ~r ) − u x, ~f il ( ~r ) | , we consider the following cases: EGULARITY AND CONTINUITY OF THE STRONG MAXIMAL OPERATORS 23
Case 1.
For simplicity, we denote RR R = R x + r , x − r , R x + r , x − r , . If ~r = ( r , , r , , r , , r , ) ∈ R with r , + r , > r , + r , >
0. Then | u x, ~f i,jl ( ~r ) − u x, ~f il ( ~r ) | = Y w =1 ( r w, + r w, ) − m (cid:12)(cid:12)(cid:12)(cid:16) i − Y µ =1 Z Z R f µ,j ( y , y ) dy dy (cid:17)(cid:16) Z Z R D l f i,j ( y , y ) dy dy (cid:17) × (cid:16) m Y ν = i +1 Z Z R f ν,j ( y , y ) dy dy (cid:17) − (cid:16) i − Y µ =1 Z Z R f µ ( y , y ) dy dy (cid:17)(cid:16) Z Z R D l f i ( y , y ) dy dy (cid:17)(cid:16) m Y ν = i +1 Z Z R f ν ( y , y ) dy dy (cid:17)(cid:12)(cid:12)(cid:12) ≤ i − X µ =1 2 Y w =1 ( r w, + r w, ) − m (cid:16) µ − Y ℓ =1 Z Z R f ℓ ( y , y ) dy y (cid:17) Z Z R | f µ,j − f µ | ( y , y ) dy dy × (cid:16) i − Y κ = µ +1 Z Z R f κ,j ( y , y ) dy dy (cid:17) Z Z R D l f i,j ( y , y ) dy dy × (cid:16) m Y τ = i +1 Z Z R f τ,j ( y , y ) dy dy (cid:17) + m X ν = i +1 2 Y w =1 ( r w, + r w, ) − m (cid:16) i − Y ℓ =1 Z Z R f ℓ ( y , y ) dy dy (cid:17) × Z Z R D l f i ( y , y ) dy dy (cid:16) ν − Y κ = i +1 Z Z R f κ ( y , y ) dy dy (cid:17) Z Z R | f ν,j − f ν | ( y , y ) dy dy × (cid:16) m Y τ = ν +1 Z Z R f τ,j ( y , y ) dy dy (cid:17) + Y w =1 ( r w, + r w, ) − m (cid:16) i − Y κ =1 Z Z R f κ ( y , y ) dy dy (cid:17) × Z Z R | D l f i,j − D l f i | ( y , y ) dy dy (cid:16) m Y τ = i +1 Z Z R f τ,j ( y , y ) dy dy (cid:17) ≤ i − X µ =1 M R ( ~F lµ,j )( x ) + m X ν = i +1 M R ( ~G lν,j )( x ) + M R ( ~H li,j )( x ) =: G li,j ( x ) , where ~F lµ,j = ( f , . . . , f µ − , f µ,j − f µ , f µ +1 ,j , . . . , f i − ,j , D l f i,j , f i +1 ,j , . . . , f m,j ) , and ~G lν,j , ~H li,j are defined by ~G lν,j = ( f , . . . , f i − , D l f i , f i +1 , . . . , f ν − , f ν,j − f ν , f ν +1 ,j , . . . , f m,j ) and ~H li,j =( f , . . . , f i − , D l f i,j − D l f i , f i +1 ,j , . . . , f m,j ). Case 2. If ~r = (0 , , , | u x, ~f i,jl ( ~r ) − u x, ~f il ( ~r ) | ≤ i − X µ =1 (cid:16) µ − Y ℓ =1 f ℓ ( x ) (cid:17) ( f µ,j − f µ )( x ) (cid:16) i − Y κ = µ +1 f κ,j ( x ) (cid:17) D l f i,j ( x ) (cid:16) m Y τ = i +1 f τ,j ( x ) (cid:17) + m X ν = i +1 (cid:16) i − Y ℓ =1 f ℓ ( x ) (cid:17) D l f i ( x ) (cid:16) ν − Y κ = i +1 f κ ( x ) (cid:17) ( f ν,j − f ν )( x ) (cid:16) m Y τ = ν +1 f τ,j ( x ) (cid:17) + (cid:16) i − Y κ =1 f κ ( x ) (cid:17) ( D l f i,j − D l f i )( x ) (cid:16) m Y τ = i +1 f τ,j ( x ) (cid:17) . ∗ , AND K ˆOZ ˆO YABUTA Case 3. If ~r = (0 , , r , , r , ) ∈ R for r , + r , >
0, then | u x, ~f i,jl ( ~r ) − u x, ~f il ( ~r ) | = 1( r , + r , ) m (cid:12)(cid:12)(cid:12)(cid:16) i − Y µ =1 Z x + r , x − r , f µ,j ( x , y ) dy (cid:17) × (cid:16) Z x + r , x − r , D l f i,j ( x , y ) dy (cid:17)(cid:16) m Y ν = i +1 Z x + r , x − r , f ν,j ( x , y ) dy (cid:17) − (cid:16) i − Y µ =1 Z x + r , x − r , f µ ( x , y ) dy (cid:17) × (cid:16) Z x + r , x − r , D l f i ( x , y ) dy (cid:17)(cid:16) m Y ν = i +1 Z x + r , x − r , f ν ( x , y ) dy (cid:17)(cid:12)(cid:12)(cid:12) ≤ i − X µ =1 r , + r , ) m (cid:16) µ − Y ℓ =1 Z x + r , x − r , f ℓ ( x , y ) dy (cid:17) Z x + r , x − r , | f µ,j − f µ ) | ( x , y ) dy × (cid:16) i − Y κ = µ +1 Z x + r , x − r , f κ,j ( x , y ) dy (cid:17) Z x + r , x − r , D l f i,j ( x , y ) dy × (cid:16) m Y τ = i +1 Z x + r , x − r , f τ,j ( x , y ) dy (cid:17) + m X ν = i +1 r , + r , ) m (cid:16) i − Y ℓ =1 Z x + r , x − r , f ℓ ( x , y ) dy (cid:17) Z x + r , x − r , D l f i ( x , y ) dy × (cid:16) ν − Y κ = i +1 Z x + r , x − r , f κ ( x , y ) dy (cid:17) Z x + r , x − r , | f ν,j − f ν | ( x , y ) dy × (cid:16) m Y τ = ν +1 Z x + r , x − r , f τ,j ( x , y ) dy (cid:17) + 1( r , + r , ) m (cid:16) i − Y κ =1 Z x + r , x − r , f κ ( x , y ) dy (cid:17) Z x + r , x − r , | D l f i,j − D l f i | ( x , y ) dy × (cid:16) m Y τ = i +1 Z x + r , x − r , f τ,j ( x , y ) dy (cid:17) . Case 4. If ~r = ( r , , r , , , ∈ R with r , + r , >
0. Then, similarly as in Case 3, wecan obtain | u x, ~f i,jl ( ~r ) − u x, ~f il ( ~r ) |≤ i − X µ =1 r , + r , ) m (cid:16) µ − Y ℓ =1 Z x + r , x − r , f ℓ ( y , x ) dy (cid:17) Z x + r , x − r , | f µ,j − f µ ) | ( y , x ) dy × (cid:16) i − Y κ = µ +1 Z x + r , x − r , f κ,j ( y , x ) dy (cid:17) Z x + r , x − r , D l f i,j ( y , x ) dy × (cid:16) m Y τ = i +1 Z x + r , x − r , f τ,j ( y , x ) dy (cid:17) EGULARITY AND CONTINUITY OF THE STRONG MAXIMAL OPERATORS 25 + m X ν = i +1 r , + r , ) m (cid:16) i − Y ℓ =1 Z x + r , x − r , f ℓ ( y , x ) dy (cid:17) Z x + r , x − r , D l f i ( y , x ) dy × (cid:16) ν − Y κ = i +1 Z x + r , x − r , f κ ( y , x ) dy (cid:17) Z x + r , x − r , | f ν,j − f ν | ( y , x ) dy × (cid:16) m Y τ = ν +1 Z x + r , x − r , f τ,j ( y , x ) dy (cid:17) + 1( r , + r , ) m (cid:16) i − Y κ =1 Z x + r , x − r , f κ ( y , x ) dy (cid:17) Z x + r , x − r , | D l f i,j − D l f i | ( y , x ) dy × (cid:16) m Y τ = i +1 Z x + r , x − r , f τ,j ( y , x ) dy (cid:17) . Together with the above cases, we obtain(2.50) m X i =1 | u x, ~f i,jl ( ~r ) − u x, ~f il ( ~r ) | ≤ m X i =1 G i,jl ( x ) =: G jl ( x ) , for any ~r ∈ [0 , ∞ ) . Note that lim j →∞ kG i,jl k L p ( R d ) = 0 . It follows that there exists j ∈ N such that(2.51) kG jl k L p ( R ) < ǫ, ∀ j ≥ j . Observe from (2.43)-(2.46) that | B ,i ∪ B i,j | < η for all j ≥ j and i = 1 , ,
3. These factstogether with (2.47)-(2.51) imply that k D l M R ( ~f j ) − D l M R ( ~f ) k L p ( R ) ≤ kG jl k L p ( R ) + (cid:13)(cid:13)(cid:13) m X i =1 M R ( ~f il ) (cid:13)(cid:13)(cid:13) p,B + (cid:13)(cid:13)(cid:13) m X i =1 M R ( ~f il ) (cid:13)(cid:13)(cid:13) p,B , ∪ B ,j + (cid:13)(cid:13)(cid:13) m X i =1 M R ( ~f il ) (cid:13)(cid:13)(cid:13) p,B , ∪ B ,j + (cid:13)(cid:13)(cid:13) m X i =1 M R ( ~f il ) (cid:13)(cid:13)(cid:13) p,B , ∪ B ,j + (cid:13)(cid:13) | R ~ Λ ( ~ | − /p ǫ (cid:13)(cid:13) p, ( B ∪ ( ∪ i =1 B ,i ∪ B i,j )) c ≤ ǫ for all j ≥ max { j , j } , which leads to (2.42). (cid:3) Properties on Besov and Triebel-Lizorkin spaces
This section will be devoted to give the proofs of Theorems 1.2 and 1.3. In what follows, welet ∆ ζ f denote the difference of f , i.e. ∆ ζ f ( x ) = f ( x + ζ ) − f ( x ) for all x, ζ ∈ R d . We also let R d = { ζ ∈ R d ; 1 / < | ζ | ≤ } .To prove Theorems 1.2 and 1.3, we need the following characterizations of homogeneousTriebel-Lizorkin spaces ˙ F p,qs ( R d ) and homogeneous Besov spaces ˙ B p,qs ( R d ). Lemma 3.1. ( [56] ). (i) Let < s < , < p < ∞ , < q ≤ ∞ and ≤ r < min( p, q ) . Then k f k ˙ F p,qs ( R d ) ∼ (cid:13)(cid:13)(cid:13)(cid:16) X k ∈ Z ksq (cid:16) Z R d | ∆ − k ζ f | r dζ (cid:17) q/r (cid:17) /q (cid:13)(cid:13)(cid:13) L p ( R d ) ; ∗ , AND K ˆOZ ˆO YABUTA (ii) Let < s < , ≤ p < ∞ , ≤ q ≤ ∞ and ≤ r ≤ p . Then (3.1) k f k ˙ B p,qs ( R d ) ∼ (cid:16) X k ∈ Z ksq (cid:13)(cid:13)(cid:13)(cid:16) Z R d | ∆ − k ζ f | r dζ (cid:17) /r (cid:13)(cid:13)(cid:13) qL p ( R d ) (cid:17) /q . Proof of Theorem 1.2.
Note that k f k B p,qs ( R d ) ∼ k f k ˙ B p,qs ( R d ) + k f k L p ( R d ) for s > < p, q < ∞ . For a measurable function g : R d × Z × R d → R , we define k g k p,q := (cid:16) X k ∈ Z ksq (cid:16) Z R d Z R d | g ( x, k, ζ ) | p dxdζ (cid:17) q/p (cid:17) /q . Using (3.1) with r = p and Fubini’s theorem, we have(3.2) k f k ˙ B p,qs ( R d ) ∼ k ∆ − k ζ f k p,q . Let 0 < s < < p , . . . , p m , p, q < ∞ with 1 /p = P mi =1 /p i . Let ~f = ( f , . . . , f m ) witheach f j ∈ B p j ,qs ( R d ). Fix ζ ∈ R d , it is clear that M R ( ~f )( x + ζ ) = sup R ∋ x + ζR ∈R | R | m m Y i =1 Z R | f i ( y ) | dy = sup R ∋ xR ∈R | R | m m Y i =1 Z R | f i ( y + ζ ) | dy. One can easily check that(3.3) | ∆ − k ζ ( M R ( ~f ))( x ) | ≤ m X l =1 M R ( ~f k,ζl )( x ) , where ~f k,ζl = ( f , . . . , f l − , ∆ k ζ f l , f k,ζl +1 , . . . , f k,ζm ) and f k,ζj ( x ) = f j ( x +2 − k ζ ) for all l +1 ≤ j ≤ m .Then we get from (3.2)-(3.3) and Minkowski’s inequality that(3.4) k M R ( ~f ) k ˙ B p,qs ( R d ) . (cid:16) X k ∈ Z ksq (cid:16) Z R d Z R d | ∆ − k ζ M R ( ~f )( x ) | p dxdζ (cid:17) q/p (cid:17) /q . m X l =1 (cid:16) X k ∈ Z ksq (cid:13)(cid:13)(cid:13) k M R ( ~f k,ζl ) k L p ( R d ) (cid:13)(cid:13)(cid:13) qL p ( R d ) (cid:17) /q . m X l =1 (cid:16) X k ∈ Z ksq (cid:13)(cid:13)(cid:13) l − Y i =1 k f i k L pi ( R d ) k ∆ − k ζ f l k L pl ( R d ) m Y j = l +1 k f k,ζj k L pj ( R d ) (cid:13)(cid:13)(cid:13) qL p ( R d ) (cid:17) /q . m X l =1 Y i = l, ≤ i ≤ m k f i k L pi ( R d ) (cid:16) X k ∈ Z ksq (cid:13)(cid:13)(cid:13) k ∆ − k ζ f l k L pl ( R d ) (cid:13)(cid:13)(cid:13) qL pl ( R d ) (cid:17) /q . m X l =1 Y i = l, ≤ i ≤ m k f i k L pi ( R d ) k f l k ˙ B pl,qs ( R d ) . (3.4) together with (1.2) implies that k M R ( ~f ) k B p,qs ( R d ) ≤ C m Y i =1 k f i k B pi,qs ( R d ) . This completes the proof of the boundedness part.
EGULARITY AND CONTINUITY OF THE STRONG MAXIMAL OPERATORS 27
We now prove the continuity part. Let ~f j = ( f ,j , . . . , f m,j ) and f i,j → f i in B p i ,qs ( R d ) as j → ∞ . It is known that f i,j → f i in ˙ B p i ,qs ( R d ) and in L p i ( R d ) as j → ∞ . One can check that(3.5) | M R ( ~f j ) − M R ( ~f ) | ≤ m X l =1 M R ( ~f l ) . Here ~f l = ( f , . . . , f l − , f l,j − f l , f l +1 ,j , . . . , f m,j ). It follows from (3.5) that M R ( ~f j ) → M R ( ~f ) in L p ( R d ) as j → ∞ . Therefore, it suffices to show that M R ( ~f j ) → M R ( ~f ) in ˙ B p,qs ( R d ) as j → ∞ .We will prove this claim by contradiction.Without loss of generality, we may assume that there exists c > k M R ( ~f j ) − M R ( ~f ) k ˙ B p,qs ( R d ) > c, for every j. It is obvious that k ∆ − k ζ ( M R ( ~f j ) − M R ( ~f )) k L p ( R d ) → j → ∞ for every ( k, ζ ) ∈ Z × R d .By (3.3), for every ( x, k, ζ ) ∈ R d × Z × R d , we have(3.6) | ∆ − k ζ ( M R ( ~f j ) − M R ( ~f ))( x ) |≤ | ∆ − k ζ ( M R ( ~f j ))( x ) | + | ∆ − k ζ ( M R ( ~f ))( x ) |≤ m X l =1 | M R ( ~f k,ζl,j )( x ) − M R ( ~f k,ζl )( x ) | + 2 m X l =1 M R ( ~f k,ζl )( x ) . Here ~f k,ζl is given as in (3.3) and ~f k,ζl,j = ( f ,j , . . . , f l − ,j , ∆ − k ζ f l,j , f k,ζl +1 ,j , . . . , f k,ζm,j ) with f k,ζi,j ( x ) = f i,j ( x + 2 − k ζ ) for all l + 1 ≤ i ≤ m . From the third inequality to the last one in (3.4), we obtain(3.7) (cid:13)(cid:13)(cid:13) m X l =1 M R ( ~f k,ζl ) (cid:13)(cid:13)(cid:13) p,q . (cid:16) X k ∈ Z ksq (cid:16) Z R d Z R d (cid:12)(cid:12)(cid:12) m X l =1 M R ( ~f k,ζl )( x ) (cid:12)(cid:12)(cid:12) p dxdζ (cid:17) q/p (cid:17) /q . m X l =1 (cid:16) X k ∈ Z ksq (cid:13)(cid:13)(cid:13) k M R ( ~f k,ζl ) k L p ( R d ) (cid:13)(cid:13)(cid:13) qL p ( R d ) (cid:17) /q . m X l =1 Y i = l, ≤ i ≤ m k f i k L pi ( R d ) k f l k ˙ B pl,qs ( R d ) . One can also verify that(3.8) | M R ( ~f k,ζl,j ) − M R ( ~f k,ζl ) | ≤ l − X µ =1 M R ( ~I k,ζµ,j ) + m X ν = l +1 M R ( ~J k,ζν,j ) + M R ( ~K k,ζi,j ) , where ~I k,ζµ,j = ( f , . . . , f µ − , f µ,j − f µ , f µ +1 ,j , . . . , f l − ,j , ∆ − k ζ f l,j , f k,ζl +1 ,j , . . . , f k,ζm,j ) ,~J k,ζν,j = ( f , . . . , f l − , ∆ k ζ f l , f k,ζl +1 , . . . , f k,ζν − , f k,ζν,j − f k,ζν , f k,ζν +1 ,j , . . . , f k,ζm,j ) ,~K k,ζi,j = ( f , . . . , f l − , ∆ − k ζ ( f l,j − f l ) , f k,ζl +1 ,j , . . . , f k,ζm,j ) . By (3.7) and (3.8), one can deduce that (cid:13)(cid:13)(cid:13) m X l =1 | M R ( ~f k,ζl,j ) − M R ( ~f k,ζl ) | (cid:13)(cid:13)(cid:13) p,q → j → ∞ . Thus, we can extract a subsequence such that P ∞ j =1 k P ml =1 | M R ( ~f k,ζl,j ) − M R ( ~f k,ζl ) |k p,q < ∞ . ∗ , AND K ˆOZ ˆO YABUTA Let H ( x, k, ζ ) = ∞ X j =1 (cid:12)(cid:12)(cid:12) m X l =1 M R ( ~f k,ζl,j )( x ) − M R ( ~f k,ζl )( x ) (cid:12)(cid:12)(cid:12) + 2 m X l =1 M R ( ~f k,ζl )( x ) . It is easily to check that k H k p,q < ∞ . By (3.6), we get(3.9) | ∆ − k ζ ( M R ( ~f j ) − M R ( ~f ))( x ) | ≤ H ( x, k, ζ ) for a . e . ( x, k, ζ ) ∈ R d × Z × R d . Since k H k p,q < ∞ , we have R R d | H ( x, k, ζ ) p dx < ∞ for a.e. ( k, ζ ) ∈ Z × R d . By (3.9) and thedominated convergence theorem, for a.e. ( k, ζ ) ∈ Z × R d , it holds that(3.10) lim j →∞ Z R d | ∆ − k ζ ( M R ( ~f j ) − M R ( ~f ))( x ) | p dx = 0 . Using (3.9) and the fact k H k p,q < ∞ again, we have(3.11) Z R d | ∆ − k ζ ( M R ( ~f j ) − M R ( ~f ))( x ) | p dx ≤ Z R d H ( x, k, ζ ) p dx, for a.e ( k, ζ ) ∈ Z × R d and(3.12) Z R d Z R d H ( x, k, ζ ) p dxdζ < ∞ for every k ∈ Z . It follows from (3.10)-(3.12) and the dominated convergence theorem that(3.13) lim j →∞ (cid:16) Z R d Z R d | ∆ − k ζ ( M R ( ~f j ) − M R ( ~f ))( x ) | p dxdζ (cid:17) /p = 0For every k ∈ Z , by (3.9) and the fact k H k p,q < ∞ again, we have(3.14) (cid:16) Z R d Z R d | ∆ − k ζ ( M R ( ~f j ) − M R ( ~f ))( x ) | p dxdζ (cid:17) /p ≤ (cid:16) Z R d Z R d H ( x, k, ζ ) p dxdζ (cid:17) /p and(3.15) (cid:16) X k ∈ Z ksq (cid:16) Z R d Z R d H ( x, k, ζ ) p dxdζ (cid:17) q/p (cid:17) /q < ∞ . Using (3.14)-(3.15) and the dominated convergence theorem again, one may obtain k ∆ − k ζ ( M R ( ~f j ) − M R ( ~f )) k p,q = (cid:16) X k ∈ Z ksq (cid:16) Z R d Z R d | ∆ − k ζ ( M R ( ~f j ) − M R ( ~f ))( x ) | p dxdζ (cid:17) q/p (cid:17) /q → j → ∞ . By (3.2), this yields that k M R ( ~f j ) − M R ( ~f ) k ˙ B p,qs ( R d ) → j → ∞ , which gives a contradiction.The proof of Theorem 1.2 is finished. (cid:3) Proof of Theorem 1.3.
Given an operator T acting on functions in R , we denote by T j , j = 1 , , . . . , d , the operator defined on functions in R d by letting T act on the j -th variablewhile keeping the remaining variables fixed, namely T j f ( x ) = T ( f ( x , x , . . . , x j − , · , x j +1 , . . . , x d ))( x j ) for x ∈ R d . We also define the operator T by T f ( x ) = T ◦ T ◦ . . . ◦ T d f ( x ) . We need the following lemma.
Lemma 3.2. If T is bounded on L p ( R , ℓ q ( L r ( R d ))) for some < p, q, r < ∞ , then the operator T is bounded on L p ( R d , ℓ q ( L r ( R d ))) . EGULARITY AND CONTINUITY OF THE STRONG MAXIMAL OPERATORS 29
Proof.
For all j = 1 , . . . , d , we shall prove the following inequality(3.16) (cid:13)(cid:13)(cid:13)(cid:16) X i ∈ Z k T j f i,ζ k qL r ( R d ) (cid:17) /q (cid:13)(cid:13)(cid:13) L p ( R d ) ≤ k T k (cid:13)(cid:13)(cid:13)(cid:16) X i ∈ Z k f i,ζ k qL r ( R d ) (cid:17) /q (cid:13)(cid:13)(cid:13) L p ( R d ) . Here k T k represents the operator norm of T on L p ( R , ℓ q ( L r ( R d ))). We only prove (3.16) for j = 1 and the other cases are analogous. We may write (cid:13)(cid:13)(cid:13)(cid:16) X i ∈ Z k T f i,ζ k qL r ( R d ) (cid:17) /q (cid:13)(cid:13)(cid:13) pL p ( R d ) = Z R d (cid:16) X i ∈ Z k T f i,ζ k qL r ( R d ) (cid:17) p/q dx = Z R d − (cid:16) Z R (cid:16) X i ∈ Z (cid:16) Z R d | T ( f i,ζ ( · , x , . . . , x d ))( x ) | r dζ (cid:17) q/r (cid:17) p/q dx (cid:17) dx . . . dx d ≤ k T k p Z R d − (cid:16) Z R (cid:16) X i ∈ Z k f i,ζ ( x , x , . . . , x d ) k qL r ( R d ) (cid:17) p/q dx (cid:17) dx . . . dx d = k T k p (cid:13)(cid:13)(cid:13)(cid:16) X i ∈ Z k f i,ζ k qL r ( R d ) (cid:17) /q (cid:13)(cid:13)(cid:13) pL p ( R d ) , which leads to (3.16) for j = 1. (3.16) together with the definition of T yields that (cid:13)(cid:13)(cid:13)(cid:16) X i ∈ Z kT f i,ζ k qL r ( R d ) (cid:17) /q (cid:13)(cid:13)(cid:13) L p ( R d ) ≤ k T k d (cid:13)(cid:13)(cid:13)(cid:16) X i ∈ Z k f i,ζ k qL r ( R d ) (cid:17) /q (cid:13)(cid:13)(cid:13) L p ( R d ) . This proves Lemma 3.2. (cid:3)
The following vector-valued inequalities of the one dimensional uncentered Hardy-Littlewoodmaximal function will be very useful in the proof of Theorem 1.3.
Lemma 3.3. ( [56] ). For any < p, q, r < ∞ , it holds that (cid:13)(cid:13)(cid:13)(cid:16) X j ∈ Z kM f j,ζ k qL r ( R d ) (cid:17) /q (cid:13)(cid:13)(cid:13) L p ( R ) . p,q,r (cid:13)(cid:13)(cid:13)(cid:16) X j ∈ Z k f j,ζ k qL r ( R d ) (cid:17) /q (cid:13)(cid:13)(cid:13) L p ( R ) . Applying Lemmas 3.2 and 3.3, we can get the following
Lemma 3.4.
For any < p, q, r < ∞ , it holds that (cid:13)(cid:13)(cid:13)(cid:16) X j ∈ Z kM R f j,ζ k qL r ( R d ) (cid:17) /q (cid:13)(cid:13)(cid:13) L p ( R d ) . p,q,r (cid:13)(cid:13)(cid:13)(cid:16) X j ∈ Z k f j,ζ k qL r ( R d ) (cid:17) /q (cid:13)(cid:13)(cid:13) L p ( R d ) . Proof.
For j = 1 , . . . , d , we define the operator M j by M j f ( x , x , . . . , x d ) = sup a Proof of Theorem 1.3. Let 0 < s < < p , . . . , p m , p, q < ∞ with 1 /p = P mi =1 /p i . Let ~f = ( f , . . . , f m ) with each f j ∈ F p j ,qs ( R d ). One can easily check that (3.3) also holds. We getfrom (3.3) that(3.19) | ∆ − k ζ ( M R ( ~f ))( x ) |≤ m X l =1 M R (∆ − k ζ f l )( x ) l − Y µ =1 M R f µ ( x ) m Y ν = l +1 M R ( f k,ζν )( x )= m X l =1 M R (∆ − k ζ f l )( x ) l − Y µ =1 M R f µ ( x ) m Y ν = l +1 M R (∆ − k ζ f ν + f ν )( x ) ≤ X ∅6 = τ ⊂ τ m Y µ ∈ τ M R (∆ − k ζ f µ )( x ) Y ν ∈ τ ′ M R f ν ( x ) , where τ m = { , , . . . , m } and τ ′ = τ m \ τ for τ ⊂ τ m .Thus, Lemma 3.1 (i), (3.19) and the Minkowski inequality yield that(3.20) k M R ( ~f ) k ˙ F p,qs ( R d ) . (cid:13)(cid:13)(cid:13)(cid:16) X k ∈ Z ksq (cid:16) Z R d | ∆ − k ζ ( M R ( ~f )) | dζ (cid:17) q (cid:17) /q (cid:13)(cid:13)(cid:13) L p ( R d ) . X ∅6 = τ ⊂ τ m (cid:13)(cid:13)(cid:13)(cid:16) X k ∈ Z ksq (cid:16) Z R d Y µ ∈ τ M R (∆ − k ζ f µ ) Y ν ∈ τ ′ M R f ν dζ (cid:17) q (cid:17) /q (cid:13)(cid:13)(cid:13) L p ( R d ) . We shall prove the following estimate.(3.21) (cid:13)(cid:13)(cid:13)(cid:16) X k ∈ Z ksq (cid:16) Z R d Y µ ∈ τ M R (∆ − k ζ f µ ) Y ν ∈ τ ′ M R f ν dζ (cid:17) q (cid:17) /q (cid:13)(cid:13)(cid:13) L p ( R d ) = (cid:13)(cid:13)(cid:13)(cid:16) X k ∈ Z ksq (cid:16) Z R d Y µ ∈ τ M R (∆ − k ζ f µ ) dζ (cid:17) q (cid:17) /q Y ν ∈ τ ′ M R f ν (cid:13)(cid:13)(cid:13) L p ( R d ) . Y µ ∈ τ k f µ k F pµ,qs ( R d ) Y ν ∈ τ ′ k f ν k F pν,qs ( R d ) Let 1 /p τ = P µ ∈ τ /p µ . Then, using H¨older’s inequality and the L p bounds for M R we have (cid:13)(cid:13)(cid:13)(cid:16) X k ∈ Z ksq (cid:16) Z R d Y µ ∈ τ M R (∆ − k ζ f µ ) dζ (cid:17) q (cid:17) /q Y ν ∈ τ ′ M R f ν (cid:13)(cid:13)(cid:13) L p ( R d ) ≤ (cid:13)(cid:13)(cid:13)(cid:16) X k ∈ Z ksq (cid:16) Z R d Y µ ∈ τ M R (∆ − k ζ f µ ) dζ (cid:17) q (cid:17) /q (cid:13)(cid:13)(cid:13) L pτ ( R d ) (cid:13)(cid:13)(cid:13) Y ν ∈ τ ′ M R f ν (cid:13)(cid:13)(cid:13) L pτ ′ ( R d ) ≤ (cid:13)(cid:13)(cid:13)(cid:16) X k ∈ Z ksq Y µ ∈ τ (cid:13)(cid:13)(cid:13) M R (∆ − k ζ f µ ) (cid:13)(cid:13)(cid:13) qL pµ/pτ ( R d ) (cid:17) /q (cid:13)(cid:13)(cid:13) L pτ ( R d ) Y ν ∈ τ ′ (cid:13)(cid:13)(cid:13) M R f ν (cid:13)(cid:13)(cid:13) L pν ( R d ) ≤ (cid:13)(cid:13)(cid:13) Y µ ∈ τ (cid:16) X k ∈ Z ksq (cid:13)(cid:13)(cid:13) M R (∆ − k ζ f µ ) (cid:13)(cid:13)(cid:13) p µ q/p τ L pµ/pτ ( R d ) (cid:17) p τ /p µ q (cid:13)(cid:13)(cid:13) L pτ ( R d ) Y ν ∈ τ ′ (cid:13)(cid:13)(cid:13) f ν (cid:13)(cid:13)(cid:13) L pν ( R d ) ≤ Y i ∈ τ (cid:13)(cid:13)(cid:13)(cid:16) X k ∈ Z ksq (cid:13)(cid:13)(cid:13) M R (∆ − k ζ f µ ) (cid:13)(cid:13)(cid:13) p µ q/p τ L pµ/pτ ( R d ) (cid:17) p τ /p µ q (cid:13)(cid:13)(cid:13) L pµ ( R d ) Y ν ∈ τ ′ (cid:13)(cid:13)(cid:13) f ν (cid:13)(cid:13)(cid:13) L pν ( R d )EGULARITY AND CONTINUITY OF THE STRONG MAXIMAL OPERATORS 31 = Y µ ∈ τ (cid:13)(cid:13)(cid:13)(cid:16) X k ∈ Z k ( p τ s/p µ )( p µ q/p τ ) (cid:13)(cid:13)(cid:13) M R (∆ − k ζ f µ ) (cid:13)(cid:13)(cid:13) p µ q/p τ L pµ/pτ ( R d ) (cid:17) p τ /p µ q (cid:13)(cid:13)(cid:13) L pµ ( R d ) Y ν ∈ τ ′ k f ν k L pν ( R d ) . Y µ ∈ τ (cid:13)(cid:13)(cid:13)(cid:16) X k ∈ Z k ( p τ s/p µ )( p µ q/p τ ) k ∆ − k ζ f µ k p µ q/p τ L pµ/pτ ( R d ) (cid:17) p τ /p µ q (cid:13)(cid:13)(cid:13) L pµ ( R d ) Y ν ∈ τ ′ k f ν k L pν ( R d ) . Y µ ∈ τ k f µ k ˙ F pµ,pµq/pτpτ s/pµ ( R d ) Y ν ∈ τ ′ k f i k L pν ( R d ) ≤ Y µ ∈ τ k f µ k F pµ,pµq/pτpτ s/pµ ( R d ) Y ν ∈ τ ′ k f ν k L pν ( R d ) ≤ Y µ ∈ τ k f µ k F pµ,qs ( R d ) Y ν ∈ τ ′ k f ν k F pν,qs ( R d ) . In the last estimate, we have used p µ > p τ and the inclusion property of Triebel-Lizorkin spaces.In the 6th estimate, we used Lemma 3.4. Thus, (3.21) holds. It follows from (3.20)-(3.21) that(3.22) k M R ( ~f ) k F p,qs ( R d ) ≤ C m Y i =1 k f i k F pi,qs ( R d ) . This completes the proof of the boundedness part.Below we prove the continuity part. Let f i,j → f i in F p i ,qs ( R d ) as j → ∞ . It is known that that f i,j → f i in ˙ F p i ,qs ( R d ) and in L p i ( R d ) as j → ∞ . By (3.5), it follows that M R ( ~f j ) → M R ( ~f ) in L p ( R d ) as j → ∞ . Therefore, it suffices to show that M R ( ~f j ) → M R ( ~f ) in ˙ F p,qs ( R d ) as j → ∞ .Again, we will prove this claim by contradiction. Without loss of generality we may assumethat, for every j , there exists c > k M R ( ~f j ) − M R ( ~f ) k ˙ F p,qs ( R d ) > c. For a measurable function g : R d × Z × R d → R , we define k g k E sp,q := (cid:16) Z R d (cid:16) X k ∈ Z ksq (cid:16) Z R d | g ( x, k, ζ ) | dζ (cid:17) q (cid:17) p/q dx (cid:17) /p . By Lemma 3.1, we see that if 1 ≤ r < min( p, q ), then k f k ˙ F p,qs ( R d ) ∼ k ∆ − k ζ f k E sp,q . By (3.6) and(3.8), we get(3.23) | ∆ − k ζ ( M R ( ~f j ) − M R ( ~f )) |≤ m X l =1 (cid:16) l − X µ =1 M R ( ~I k,ζµ,j ) + m X ν = l +1 M R ( ~J k,ζν,j ) + M R ( ~K k,ζi,j ) (cid:17) + 2 m X l =1 M R ( ~f k,ζl ) , where ~f k,ζl is given as in (3.19) and ~I k,ζµ,j , ~J k,ζν,j and ~K k,ζi,j are given as in (3.8).Notice that M R ( ~I k,ζµ,j ) ≤ µ − Y i =1 M R f i M R ( f µ,j − f µ ) l − Y ℓ = µ +1 M R f ℓ,j M R (∆ − k ζ f l,j ) m Y w = l +1 M R f k,ζw,j = µ − Y i =1 M R f i M R ( f µ,j − f µ ) l − Y ℓ = µ +1 M R f ℓ,j M R (∆ − k ζ f l,j ) × m Y w = l +1 M R (∆ − k ζ f w,j + f w,j ) ∗ , AND K ˆOZ ˆO YABUTA ≤ µ − Y i =1 M R f i M R ( f µ,j − f µ ) l − Y ℓ = µ +1 M R f ℓ,j M R (∆ − k ζ f l,j ) × m Y w = l +1 ( M R (∆ − k ζ f w,j ) + M R f w,j ) . This together with the arguments similar to those used in deriving (3.21) yields that(3.24) k M R ( ~I k,ζµ,j ) k E sp,q . k f µ,j − f µ k F pµ,qs ( R d ) µ − Y i =1 k f i k F pi,qs ( R d ) m Y w = µ +1 k f w,j k F pw,qs ( R d ) . Similarly, we can conclude that(3.25) k M R ( ~J k,ζν,j ) k E sp,q . k f ν,j − f ν k F pν,qs ( R d ) ν − Y i =1 k f i k F pi,qs ( R d ) m Y w = ν +1 k f w,j k F pw,qs ( R d ) ;(3.26) k M R ( ~K k,ζi,j ) k E sp,q . k f l,j − f l k F pl,qs ( R d ) l − Y ℓ =1 k f ℓ k F pℓ,qs ( R d ) m Y w = l +1 k f w,j k F pw,qs ( R d ) ;(3.27) k M R ( ~f k,ζl ) k E sp,q . m Y ℓ =1 k f ℓ k F pℓ,qs ( R d ) . It follows from (3.24)-(3.27) that (cid:13)(cid:13)(cid:13) l − X µ =1 M R ( ~I k,ζµ,j ) + m X ν = l +1 M R ( ~J k,ζν,j ) + M R ( ~K k,ζi,j ) (cid:13)(cid:13)(cid:13) E sp,q → j → ∞ . Therefore, one can extract a subsequence, we still denote it by j , such that(3.28) ∞ X j =1 (cid:13)(cid:13)(cid:13) l − X µ =1 M R ( ~I k,ζµ,j ) + m X ν = l +1 M R ( ~J k,ζν,j ) + M R ( ~K k,ζi,j ) (cid:13)(cid:13)(cid:13) E sp,q < ∞ . Let G ( x, k, ζ ) = m X l =1 ∞ X j =1 l − X µ =1 M R ( ~I k,ζµ,j )( x ) + m X ν = l +1 M R ( ~J k,ζν,j )( x )+ M R ( ~K k,ζi,j )( x ) + 2 m X l =1 M R ( ~f k,ζl )( x ) . We get from (3.27) and (3.28) that k G k E sp,q < ∞ . Furthermore by (3.23), one obtains that(3.29) | ∆ − k ζ ( M R ( ~f j ) − M R ( ~f ))( x ) | ≤ G ( x, k, ζ ) for every ( x, k, ζ ) ∈ R d × Z × R d . (3.29) together with the dominated convergence theorem leads to(3.30) Z R d | ∆ − k ζ ( M R ( ~f j ) − M R ( ~f ))( x ) | dζ → j → ∞ for every ( x, k, ζ ) ∈ R d × Z × R d . Since it holds that k G k E sp,q < ∞ , we immediately deduce that(3.31) (cid:16) X k ∈ Z ksq (cid:16) Z R d G ( x, k, ζ ) dζ (cid:17) q (cid:17) /q < ∞ , for a.e. x ∈ R d . EGULARITY AND CONTINUITY OF THE STRONG MAXIMAL OPERATORS 33 Using (3.29), we obtain(3.32) Z R d | ∆ − k ζ ( M R ( ~f j ) − M R ( ~f ))( x ) | dζ ≤ Z R d G ( x, k, ζ ) dζ, for a.e. x ∈ R d and k ∈ Z . (3.30)-(3.32) and the dominated convergence theorem give(3.33) (cid:16) X k ∈ Z ksq (cid:16) Z R d | ∆ − k ζ ( M R ( ~f j ) − M R ( ~f ))( x ) | dζ (cid:17) q (cid:17) /q → j → ∞ , for a.e. x ∈ R d By (3.29) again, for a.e. x ∈ R d , it is true that(3.34) (cid:16) X k ∈ Z ksq (cid:16) Z R d | ∆ − k ζ ( M R ( ~f j ) − M R ( ~f ))( x ) | dζ (cid:17) q (cid:17) /q ≤ (cid:16) X k ∈ Z ksq (cid:16) Z R d G ( x, k, ζ ) dζ (cid:17) q (cid:17) /q It follows from (3.33)-(3.34), k G k E sp,q < ∞ and the dominated convergence theorem thatlim j →∞ k ∆ − k ζ ( M R ( ~f j ) − M R ( ~f )) k E sp,q = 0 , which yields k M R ( ~f j ) − M R ( ~f ) k ˙ F p,qs ( R d ) → j → ∞ and leads to a contradiction. (cid:3) Property of p -quasicontinuity Proof. We will divide the proof of Theorem 1.5 into three steps. Step 1: A weak type inequality for the Sobolev capacity. Let us begin with a capacity inequalitythat can be used in studying the pointwise behaviour of Sobolev functions by the standardmethods (see [15]). Let ~f = ( f , . . . , f m ) with each f i ∈ W ,p i ( R d ) for 1 < p i < ∞ . Let1 < p < ∞ and 1 /p = P mi =1 /p i . For λ > 0, we set O λ = { x ∈ R d ; M R ( ~f )( x ) > λ } . Note that O λ is an open set. We get from Theorem 1.1 that(4.1) C p ( O λ ) /p ≤ λ (cid:16) Z R d ( | M R ( ~f )( x ) | p + |∇ M R ( ~f )( x ) | p ) dx (cid:17) /p ≤ λ k M R ( ~f ) k ,p . m,d,p ,...,p m m Y i =1 k f i k ,p i λ . Step 2: The continuity of M R ( ~f ) . To prove the p -quasicontinuity of M R ( ~f ), we first provethat M R ( ~f ) ∈ C ( R d ) if ~f = ( f , . . . , f m ) with each f i ∈ C ∞ ( R d ). We can write M R ( ~f )( x ) = sup ~r ∈ R d + m Y i =1 | E ~r ( x ) | Z E ~r ( x ) | f i ( y ) | dy, where ~r = ( r − , . . . , r − d ; r +1 , . . . , r + d ) and E ~r ( x ) = ( x − r − , x + r +1 ) × · · · × ( x − r − d , x + r + d ). Forfixed x, h ∈ R d , we have | M R ( ~f )( x + h ) − M R ( ~f )( x ) |≤ m X i =1 sup ~r ∈ R d + | E ~r ( x ) | m Z E ~r ( x ) | f i ( y + h ) − f i ( y ) | dy × (cid:16) i − Y µ =1 Z E ~r ( x ) | f µ ( y ) | dy (cid:17)(cid:16) m Y ν = i +1 Z E ~r ( x + h ) | f ν ( y ) | dy (cid:17) . ∗ , AND K ˆOZ ˆO YABUTA For fixed ~r ∈ R d + and i = 1 , . . . , m , by H¨older’s inequality, we obtain1 | E ~r ( x ) | m Z E ~r ( x ) | f i ( y + h ) − f i ( y ) | dy (cid:16) i − Y µ =1 Z E ~r ( x ) | f µ ( y ) | dy (cid:17)(cid:16) m Y ν = i +1 Z E ~r ( x + h ) | f ν ( y ) | dy (cid:17) ≤ | E ~r ( x ) | − /p m Y i =1 k f i k L pi ( R d ) . It follows that given ǫ > 0, there exists a constant 0 < δ ǫ < + ∞ such that1 | E ~r ( x ) | m Z E ~r ( x ) | f i ( y + h ) − f i ( y ) | dy (cid:16) i − Y µ =1 Z E ~r ( x ) | f µ ( y ) | dy (cid:17)(cid:16) m Y ν = i +1 Z E ~r ( x + h ) | f ν ( y ) | dy (cid:17) < ǫ, when | E ~r ( x ) | > δ ǫ . On the other hand, for any x, h ∈ R d and ~r ∈ R d + with | E ~r ( x ) | ≤ δ ǫ , by themean value theorem for differentials, we have1 | E ~r ( x ) | Z E ~r ( x ) | f i ( y + h ) − f i ( y ) | dy ≤ C ( f i ) | h | and there exists M i > | f i ( x ) | ≤ M i for all x ∈ R d and i = 1 , . . . , m . Then we have1 | E ~r ( x ) | m Z E ~r ( x ) | f i ( y + h ) − f i ( y ) | dy (cid:16) i − Y µ =1 Z E ~r ( x ) | f µ ( y ) | dy (cid:17)(cid:16) m Y ν = i +1 Z E ~r ( x + h ) | f ν ( y ) | dy (cid:17) ≤ C ( f i ) Y µ = i, ≤ µ ≤ m M µ | h | . Therefore, for the above ǫ > x ∈ R d , there exists γ = γ ( ǫ ) > 0, if | h | < γ , then | M R ( ~f )( x + h ) − M R ( ~f )( x ) | ≤ C ( ~f ) ǫ. Thus, it holds that M R ( ~f ) ∈ C ( R d ). Step 3: The p -quasicontinuity of M R ( ~f ) . Suppose that f i ∈ W ,p i ( R d ), we can choose asequence of functions { f i,k } k ≥ ⊂ C ∞ ( R d ) such that f i,k → f i in W ,p i ( R d ). This yields thatthere exists a large K ∈ N such that(4.2) k f i,k − f i k ,p i ≤ − k , ∀ k ≥ K and 1 ≤ i ≤ m. Fix k ≥ K . Let ~f k = ( f ,k , . . . , f m,k ) and E k = { x ∈ R d : | M R ( ~f k )( x ) − M R ( ~f )( x ) | > − k } . By (2.6), we have(4.3) | M R ( ~f k )( x ) − M R ( ~f )( x ) | ≤ m X l =1 M R ( ~F lk )( x ) , where ~F lk ( x ) = ( f , . . . , f l − , f l,k − f l , f l +1 ,k , . . . , f m,k ). Then, by (4.1)-(4.3), we have(4.4) ( C p ( E k )) /p . m,d,p ,...,p m k m X l =1 l − Y µ =1 k f µ k ,p µ k f l,k − f l k ,p l m Y ν = l +1 k f ν,k k ,p ν . m,d,p ,...,p m − k . Let G k = S ∞ i = k E i with k ≥ K . Then by subadditivity and (4.4), it holds that C p ( G k ) ≤ ∞ X i = k C p ( E i ) . m,d,p ,...,p m ∞ X i = k − ip . m,d,p ,...,p m (1 − k ) p p − , ∀ k ≥ K , EGULARITY AND CONTINUITY OF THE STRONG MAXIMAL OPERATORS 35 which leads to lim k →∞ C p ( G k ) = 0. On the other hand, for x ∈ R d \ G k ,(4.5) | M R ( ~f k )( x ) − M R ( ~f )( x ) | ≤ − k ∀ k ≥ K . This implies that { M R ( ~f k ) } converges to M R ( ~f ) uniformly in R d \ G k . By Step 2, we see that M R ( ~f k ) ∈ C ( R d ). It follows that M R ( ~f ) is continuous in R d \ G k . We notice that M R ( ~f K )( x ) < ∞ for all x ∈ R d . This together with (4.5) implies that M R ( ~f ) is finite in R d \ G k . Hence, M R ( ~f ) is q -quasicontinuous. (cid:3) Approximate differentiability of M R This section is devoted to proving Theorem 1.6. Let us recall some definitions and presentsome useful lemmas.Let f be a real-valued function defined on a set E ⊂ R d . We say that f is approximatelydifferentiable at x ∈ E if there is a vector L = ( L , L , . . . , L d ) ∈ R d such that for any ǫ > A ǫ = n x ∈ R d : | f ( x ) − f ( x ) − L ( x − x ) || x − x | < ǫ o has x as a density point. If this is the case, then x is a density point of E and L is uniquelydetermined. The vector L is called the approximate differential of f at x and is denotedby ∇ f ( x ). Note that every function f ∈ W , ( R d ) is approximately differentiable a.e. Itwas pointed out in [22] that M f is approximately differentiable a.e. under the assumptionthat f ∈ W , ( R d ). However, it is unknown that whether f ∈ W , ( R d ) implies the weakdifferentiability of M f when d ≥ 2. The relationship between approximate differentiability andweak differentiability is still not clear.To prove Theorem 1.6, we need the following lemma, which provides several characterizationsof a.e. approximate differentiability of a function. Lemma 5.1. ([54]) Let f : E → R be measurable, E ⊂ R d . Then the following conditions areequivalent: (i) f is approximately differentiable a.e. (ii) For any ǫ > , there is a closed set F ⊂ E and a locally Lipschitz function g : R d → R such that f = g on x ∈ F and | E \ F | < ǫ . (iii) For any ǫ > , there is a closed set F ⊂ E and a function g ∈ C ( R d ) such that f = g on x ∈ F and | E \ F | < ǫ . Lemma 5.2. Let ~f = ( f , . . . , f m ) with each f j ∈ L ( R d ) . Let ~ε = ( ε , . . . , ε d ) with each ε i > .The truncated multilinear strong maximal operator M ~ε R is defined by M ~ε R ( ~f )( x ) = sup ( r − ,...,r − d ; r +1 ,...,r + d ) ∈ R d + r + i + r − i ≥ εi, i =1 , ,...,d m Y i =1 | E ~r ( x ) | Z E ~r ( x ) | f i ( y ) | dy, where x = ( x , . . . , x d ) , ~r = ( r − , . . . , r − d ; r +1 , . . . , r + d ) and E ~r ( x ) = ( x − r − , x + r +1 ) × · · · × ( x d − r − d , x d + r + d ) . Then M ~ε R ( ~f ) is Lipschitz continuous for every ~ε ∈ R d + .Proof. Fix ~ε = ( ε , . . . , ε d ) ∈ R d + . We set ε = min ≤ i ≤ d ε i . Fix ~r = ( r − , . . . , r − d ; r +1 , . . . , r + d ) ∈ R d + with r + i + r − i ≥ ε i for 1 ≤ i ≤ d . It is obvious that r + i + r − i ≥ ε for all 1 ≤ i ≤ d .Notice that for any r ≥ a ≥ b ≥ δ ≥ 1, it is true that(5.1) (cid:16) rr + b (cid:17) δ ≥ (cid:16) aa + b (cid:17) δ ≥ − δ ba . ∗ , AND K ˆOZ ˆO YABUTA Let x = ( x , . . . , x d ) ∈ R d and y = ( y , . . . , y d ) ∈ R d . We note that(5.2) E ~r ( x ) ⊂ E ~r ′ ( y ) , where ~r ′ = ( r − + | y − x | , . . . , r − d + | y d − x d | ; r +1 + | y − x | , . . . , r + d + | y d − x d | ). (5.2) gives that(5.3) (cid:16) | E ~r ( x ) || E ~r ′ ( y ) | (cid:17) m = (cid:16) d Y j =1 r − j + r + j r − j + r + j + | y j − x j | (cid:17) m ≥ (cid:16) ε ε + | x − y | (cid:17) md ≥ − mdε | x − y | . We get from (5.2) and (5.3) that(5.4) M ~ε R ( ~f )( y ) ≥ m Y i =1 | E ~r ′ ( y ) | Z E ~r ( y ) | f i ( z ) | dz ≥ m Y i =1 | E ~r ( x ) || E ~r ′ ( y ) | | E ~r ( x ) | Z E ~r ( x ) | f i ( z ) | dz ≥ (cid:16) − mdε | x − y | (cid:17) m Y i =1 | E ~r ( x ) | Z E ~r ( x ) | f i ( z ) | dz. Taking the supremum over ~r = ( r − , . . . , r − d ; r +1 , . . . , r + d ) ∈ R d + with r + i + r − i ≥ ε i for 1 ≤ i ≤ d ,we get from (5.4) that M ~ε R ( ~f )( y ) ≥ (cid:16) − mdε | x − y | (cid:17) M ~ε R ( ~f )( x ) . It follows that(5.5) M ~ε R ( ~f )( x ) − M ~ε R ( ~f )( y ) ≤ mdε | x − y | M ~ε R ( ~f )( x ) . Similarly, we can get(5.6) M ~ε R ( ~f )( y ) − M ~ε R ( ~f )( x ) ≤ mdε | x − y | M ~ε R ( ~f )( y ) . Thus, (5.5) and (5.6) imply that | M ~ε R ( ~f )( x ) − M ~ε R ( ~f )( y ) | ≤ mdε | x − y | ( M ~ε R ( ~f )( x ) + M ~ε R ( ~f )( y )) ≤ mdε md +10 m Y j =1 k f j k L ( R d ) | x − y | . This proves Lemma 5.2. (cid:3) Proof of Theorem 1.6. Let Z j be the set of all Lebesgue points of f j and u x, ~f ( r ) defined asin Section 2. We set E = R d \ ( T mj =1 Z j ). Let x ∈ T mj =1 Z j such that M R ( ~f )( x ) > u x, ~f ( ~ ~ ∈ R d . Since f j ∈ L ( R d ) and M R ( ~f )( x ) > 0, there exists a sequence { ~r k } k ≥ with ~r k = ( r − ,k , . . . , r − d,k ; r +1 ,k , . . . , r + d,k ) ∈ R d + , all r − i,k + r + i,k are bounded such thatlim k →∞ u x, ~f ( ~r k ) = M R ( ~f )( x ) . Hence there exists a subsequence { ~r ′ k } k ≥ ⊂ { ~r k } k ≥ and ~r = ( r − , . . . , r − d ; r +1 , . . . , r + d ) ∈ R d + with r − i + r + i > ≤ i ≤ d such that lim k →∞ ~r ′ k = ~r . It follows that M R ( ~f )( x ) = u x, ~f ( ~r ) . This, of course, yields that R d = E ∪ { x ∈ R d : M R ( ~f )( x ) = u x, ~f ( ~ } ∪ E, EGULARITY AND CONTINUITY OF THE STRONG MAXIMAL OPERATORS 37 where E = S ∞ k =1 . . . S ∞ k d =1 E k ,...,k d and E k ,...,k d = { x ∈ R d : M R ( ~f )( x ) = M /k ,..., /k d R ( ~f )( x ) } .By Lemma 5.1, Q mj =1 | f j | is approximately differentiable a.e. Then M R ( ~f ) is approximately dif-ferentiable a.e. in the set { x ∈ R : M R ( ~f )( x ) = u x, ~f ( ~ } . By Lemma 5.2 we have that M /k ,..., /k d R ( ~f ) is Lipschitz continuous for any k i ≥ ≤ i ≤ d . Then, for any k i ≥ ≤ i ≤ d , the function M /k ,..., /k d R ( ~f ) is approximately differentiable a.e.. It followsthat M R ( ~f ) χ E is approximately differentiable a.e. Note that | E | = 0. Therefore, M R ( ~f ) isapproximately differentiable a.e. This completes the proof of Theorem 1.6. (cid:3) Properties of discrete strong maximal functions This section is devoted to proving Theorem 1.7. For a ∈ R and r > 0, we define g ( a ; r ) = |{ k ∈ Z ; | k − a | < r }| . If a ∈ Z , then g ( a ; r ) ≥ χ (0 , ( r ) + (2[ r − 1] + 1) χ (1 , ∞ ) , where [ x ] = max { k ∈ Z ; k ≤ x } . If a ∈ R \ Z , then there exists an integer n ∈ Z such that | n − a | ≤ / { k ∈ Z ; | k − n | < r − / } ⊂ { k ∈ Z ; | k − a | < r } . It follows that g ( a ; r ) ≥ χ ( , ] + (2[ r − / 2] + 1) χ ( , ∞ ) ( r ) for r > / 2. Specially, if there existsan integer n such that | n − a | < r , then(6.1) g ( a ; r ) ≥ F ( r ) := χ (0 , ] ( r ) + (2[ r − / 2] + 1) χ ( , ∞ ) ( r ) ∀ r > a ∈ R . For ~r = ( r , . . . , r d ) ∈ R d + and ~x = ( x , . . . , x d ) ∈ R d , it is easy to see that N ( R ~r ( ~x )) = Q di =1 g ( x i ; r i ). Furthermore, if there exists ~n ∈ R ~r ( ~x ) ∩ Z d , then by (6.1), it holds that(6.2) N ( R ~r ( ~x )) ≥ d Y i =1 F ( r i ) . We now divide the proof of Theorem 1.7 into two parts.6.1. The boundedness part. Without loss of generality we may assume all f j ≥ k∇| f |k ℓ ( Z d ) ≤ k∇ f k ℓ ( Z d ) . For all 1 ≤ l ≤ d , it suffices to show that(6.3) k D l M R ( ~f ) k ℓ ( Z d ) . d,m m X i =1 k D l f i k ℓ ( Z d ) Y j = i, ≤ j ≤ m k f j k ℓ ( Z d ) We only prove (6.3) for l = d , since the other cases are analogous. In what follows, we set ~n = ( n ′ , n d ) ∈ Z d with n ′ = ( n , . . . , n d − ) ∈ Z d − . For each n ′ ∈ Z d − , let X + n ′ = { n d ∈ Z : M R ( ~f )( n ′ , n d + 1) ≤ M R ( ~f )( n ′ , n d ) } ,X − n ′ = { n d ∈ Z : M R ( ~f )( n ′ , n d + 1) > M R ( ~f )( n ′ , n d ) } . Then we can write k D l M R ( ~f ) k ℓ ( Z d ) = X n ′ ∈ Z d − X n d ∈ X + n ′ ( M R ( ~f )( n ′ , n d ) − M R ( ~f )( n ′ , n d + 1))+ X n ′ ∈ Z d − X n d ∈ X − n ′ ( M R ( ~f )( n ′ , n d + 1) − M R ( ~f )( n ′ , n d )) . ∗ , AND K ˆOZ ˆO YABUTA Therefore, to prove (6.3) with l = d , it suffices to show that(6.4) X n ′ ∈ Z d − X n d ∈ X + n ′ ( M R ( ~f )( n ′ , n d ) − M R ( ~f )( n ′ , n d + 1)) . d m X i =1 k D l f i k ℓ ( Z d ) Y j = i, ≤ j ≤ m k f j k ℓ ( Z d ) ;(6.5) X n ′ ∈ Z d − X n d ∈ X − n ′ ( M R ( ~f )( n ′ , n d + 1) − M R ( ~f )( n ′ , n d )) . d m X i =1 k D l f i k ℓ ( Z d ) Y j = i, ≤ j ≤ m k f j k ℓ ( Z d ) . We only prove (6.4), since (6.5) is analogous. For ~r ∈ R d + , define A ~r ( ~f ) : R d → R by A ~r ( ~f )( ~x ) = 1 N ( R ~r ( ~x )) m m Y j =1 X ~k ∈ R ~r ( ~x ) ∩ Z d f j ( ~k ) , ∀ ~x ∈ R d . We can write M R ( ~f )( ~n ) = sup ~r ∈ R d + ~x ∈ R d, ~n ∈ R~r ( ~x ) A ~r ( ~f )( ~x ) , ∀ ~n ∈ Z d . Lemma 6.1. Let ~f = ( f , . . . , f m ) with each f j ∈ ℓ ( Z d ) . Then for any ~n ∈ Z d , M R ( ~f )( ~n ) isattained for some R with ~n ∈ R ∈ R .Proof. Fix ~n ∈ Z d . If M R ( ~f )( ~n ) = 0, then all f j ≡ 0. For any R with ~n ∈ R ∈ R , it suffices toshow that M R ( ~f )( ~n ) = 1 N ( R ) m m Y i =1 X ~k ∈ R ∩ Z d | f i ( ~k ) | = 0 . If M R ( ~f )( ~n ) > 0. Suppose that M R ( ~f )( ~n ) is not attained for R with ~n ∈ R ∈ R . Let { r k } k ≥ be an increasing sequence of positive numbers with lim k →∞ r k = ∞ . By the definition of M R ( ~f )and our assumption, we have M R ( ~f )( ~n ) = sup ~n ∈ R ∈R N ( R ) ≥ rk N ( R ) m m Y i =1 X ~k ∈ R ∩ Z d | f i ( ~k ) | , ∀ k ≥ . It follows that M R ( ~f )( ~n ) ≤ r mk m Y i =1 k f i k ℓ ( Z ) , ∀ k ≥ . Let k → ∞ , we obtain e M R ( ~f )( ~n ) = 0, which is a contradiction. Thus, M R ( ~f )( ~n ) is attained forsome R with ~n ∈ R ∈ R . (cid:3) Since all f j ∈ ℓ ( Z d ), by Lemma 6.1, for any ( n ′ , n d ) ∈ Z d , there exist ~x ∈ R d and ~r ( n ′ , n d ) ∈ R d + such that ( n ′ , n d ) ∈ R ~r ( n ′ ,n d ) ( ~x ) and M R ( ~f )( n ′ , n d ) = A ~r ( n ′ ,n d ) ( ~f )( ~x ). Then M R ( ~f )( n ′ , n d ) − M R ( ~f )( n ′ , n d + 1) ≤ A ~r ( n ′ ,n d ) ( ~f )( ~x ) − A ~r ( n ′ ,n d ) ( ~f )( ~x + ~e d ) . For convenience, we set ~r ( n ′ , n d ) = ( r ( n ′ , n d ) , . . . , r d ( n ′ , n d )). Note that ( n ′ , n d ) ∈ R ~r ( n ′ ,n d ) ( ~x )and R ~r ( n ′ ,n d ) ( ~x ) ⊂ R ~r ( n ′ ,n d ) ( n ′ , n d ). These facts together with (6.2) yields that EGULARITY AND CONTINUITY OF THE STRONG MAXIMAL OPERATORS 39 A ~r ( n ′ ,n d ) ( ~f )( ~x ) − A ~r ( n ′ ,n d ) ( ~f )( ~x + ~e d ) ≤ d Y i =1 F ( r i ( n ′ , n d )) m m X µ =1 (cid:12)(cid:12)(cid:12) X ~k ∈ R ~r ( n ′ ,nd ) ( ~x ) ∩ Z d f µ ( ~k ) − X ~k ∈ R ~r ( n ′ ,nd ) ( ~x + ~e d ) ∩ Z d f µ ( ~k ) (cid:12)(cid:12)(cid:12) × (cid:16) µ − Y i =1 X ~k ∈ R ~r ( n ′ ,nd ) ( ~x ) ∩ Z d f i ( ~k ) (cid:17)(cid:16) m Y ν = µ +1 X ~k ∈ R ~r ( n ′ ,nd ) ( ~x + ~e d ) ∩ Z d f ν ( ~k ) (cid:17) . ≤ m X µ =1 Y ν = µ, ≤ ν ≤ m k f ν k ℓ ( Z d ) d Y i =1 F ( r i ( n ′ , n d )) m X ~k ∈ R ~r ( n ′ ,nd ) ( n ′ ,n d ) ∩ Z d | D d f µ ( ~k ) | . Therefore, M R ( ~f )( n ′ , n d ) − M R ( ~f )( n ′ , n d + 1) can be controlled by m X µ =1 Y ν = µ, ≤ ν ≤ m k f ν k ℓ ( Z d ) d Y i =1 F ( r i ( n ′ , n d )) m X ~k ∈ R ~r ( n ′ ,nd ) ( n ′ ,n d ) ∩ Z d | D d f µ ( ~k ) | . It follows that(6.6) X n ′ ∈ Z d − X n d ∈ X + n ′ ( M R f ( n ′ , n d ) − M R f ( n ′ , n d + 1)) ≤ m X µ =1 Y ν = µ, ≤ ν ≤ m k f ν k ℓ ( Z d ) × (cid:16) X n ′ ∈ Z d − X n d ∈ X + n ′ d Y i =1 F ( r i ( n ′ , n d ))) m X ~k ∈ R ~r ( n ′ ,nd ) ( n ′ ,n d ) ∩ Z d | D d f µ ( ~k ) | (cid:17) . By direct calculations, we obtain(6.7) X n ′ ∈ Z d − X n d ∈ X + n ′ d Y i =1 F ( r i ( n ′ , n d ))) m X ~k ∈ R ~r ( n ′ ,nd ) ( n ′ ,n d ) ∩ Z d | D d f µ ( ~k ) |≤ X ( k ,...,k d ) ∈ Z d | D d f µ ( k , . . . , k d ) | d Y i =1 (cid:16) X n i ∈ Z F ( r i ( n ′ , n d ))) m χ {| k i − n i | < r i ( n ′ ,n d ) } (cid:17) . For fixed 1 ≤ i ≤ d and k i ∈ Z , we have(6.8) X n i ∈ Z F ( r i ( n ′ , n d ))) m χ {| k i − n i | < r i ( n ′ ,n d ) } ≤ X n i ∈ Z | k i − n i |− ] + 1) m χ {| k i − n i |≥ } . Note that m ≥ 2, then X n i ∈ Z | k i − n i |− ] + 1) m χ {| k i − n i |≥ } ≤ X ni ∈ Z | ni |≥ | n i |− ] + 1) m ≤ . Thistogether with (6.8) yields that(6.9) X n i ∈ Z F ( r i ( n ′ , n d ))) m χ {| k i − n i | < r i ( n ′ ,n d ) } ≤ . ∗ , AND K ˆOZ ˆO YABUTA Combining (6.9) with (6.7) gives that(6.10) X n ′ ∈ Z d − X n d ∈ X + n ′ d Y i =1 F ( r i ( n ′ , n d ))) m X ~k ∈ R ~r ( n ′ ,nd ) ( n ′ ,n d ) ∩ Z d | D d f µ ( ~k ) | . d k D d f µ k ℓ ( Z d ) . Then (6.4) follows immediately from (6.6) and (6.10).6.2. The continuity part. Let g i,j → f j in ℓ ( Z d ) for all 1 ≤ j ≤ m when i → ∞ . Forconvenience, we set ~g i = ( g i, , . . . , g i,m ). It suffices to show that(6.11) lim i →∞ k D l M R ( ~g i ) − D l M R ( ~f ) k ℓ ( Z d ) = 0 , for each 1 ≤ l ≤ d. We only prove (6.11) for the case l = d and the other cases are similar. Since we have k D ℓ | g i,j | − D ℓ | f j |k ℓ ( Z d ) ≤ k| g i,j | − | f j |k ℓ ( Z d ) ≤ k g i,j − f j k ℓ ( Z d ) , we may assume without loss of generalitythat all g i,j ≥ f j ≥ 0. Given ǫ ∈ (0 , N = N ( ǫ, ~f ) ∈ N such that for all1 ≤ j ≤ m and i ≥ N ,(6.12) k g i,j − f j k ℓ ( Z d ) < ǫ and k g i,j k ℓ ( Z d ) ≤ k f j k ℓ ( Z d ) + 1 . For all 1 ≤ j ≤ m and i ≥ N , it then follows that(6.13) k D d g i,j k ℓ ( Z d ) ≤ k g i,j k ℓ ( Z d ) ≤ k f j k ℓ ( Z d ) + 1)On the other hand, by the boundedness part in Theorem 1.7, we obtain D d M R ( ~f ) ∈ ℓ ( Z d ).Hence, for the above ǫ > 0, there exists Λ > {k D d M R ( ~f ) χ ( R ~ Λ1 ( ~ c k ℓ ( Z d ) , sup ≤ j ≤ m k f j χ ( R ~ Λ1 ( ~ c k ℓ ( Z d ) } < ǫ. Here ~ Λ = (Λ , . . . , Λ ). Since m > 1, there exists an integer Λ > − m < ǫ. Step1: Reduction. When i ≥ N , we get from (6.13) that | M R ( ~g i )( ~n ) − M R ( ~f )( ~n ) |≤ sup ~n ∈ R ∈R N ( R ) m (cid:12)(cid:12)(cid:12) m Y i =1 X ~k ∈ R ∩ Z d g i,j ( ~k ) − m Y i =1 X ~k ∈ R ∩ Z d f j ( ~k ) (cid:12)(cid:12)(cid:12) ≤ m X µ =1 X ~k ∈ R ∩ Z d | g i,µ ( ~k ) − f µ ( ~k ) | (cid:16) µ − Y ι =1 X ~k ∈ R ∩ Z d g i,ι ( ~k ) (cid:17)(cid:16) m Y ν = µ +1 X ~k ∈ R ∩ Z d f ν ( ~k ) (cid:17) ≤ m X µ =1 k g i,µ − f µ k ℓ ( Z d ) (cid:16) Y ν = µ, ≤ ν ≤ m ( k f ν k ℓ ( Z d ) + 1) (cid:17) . This implies that M R ( ~g i )( ~n ) → M R ( ~f )( ~n ) as i → ∞ for any ~n ∈ Z d , and(6.16) D d M R ( ~g i )( ~n ) → D d M R ( ~f )( ~n ) as i → ∞ , ∀ ~n ∈ Z d . Let Λ = max { Λ , Λ , } . It follows from (6.16) that there exists N = N ( ǫ, Λ) ∈ N such that(6.17) | D d M R ( ~g i )( ~n ) − D d M R ( ~f )( ~n ) | ≤ ǫ ( N ( R ~ ( ~ , ∀ i ≥ N and ~n ∈ R ~ Λ ( ~ ∩ Z d . EGULARITY AND CONTINUITY OF THE STRONG MAXIMAL OPERATORS 41 Then, by (6.14) and (6.17), we have that for all i ≥ N , k D d M R ( ~g i ) − D d M R ( ~f ) k ℓ ( Z d ) = k ( D d M R ( ~g i ) − D d M R ( ~f )) χ N ( R ~ ( ~ k ℓ ( Z d ) + k ( D d M R ( ~g i ) − D d M R ( ~f )) χ ( N ( R ~ ( ~ c k ℓ ( Z d ) ≤ ǫ + k D d M R ( ~g i ) χ ( N ( R ~ ( ~ c k ℓ ( Z d ) Thus, to prove (6.11) for l = d , it suffices to show that(6.18) k D d M R ( ~g i ) χ ( N ( R ~ ( ~ c k ℓ ( Z d ) . d,m, ~f ǫ, ∀ i ≥ N . Step 2: Proof of (6.18) . Note that( R ~ ( ~ c ∩ Z d ⊂ d [ µ =1 E µ := d [ µ =1 Z d \ { ~n = ( n , . . . , n d ) ∈ Z d : | n µ | ≤ } . Fix j ≥ N . Then we have(6.19) k D d M R ( ~g i ) χ ( N ( R ~ ( ~ c k ℓ ( Z d ) ≤ d X µ =1 A ,µ := d X µ =1 X ~n ∈ E µ | D d M R ( ~g i )( ~n ) | . Step 3: Estimates for A ,d . For each n ′ ∈ Z d − , let Y + n ′ = {| n d | ≥ 2Λ : M R ( ~g i )( n ′ , n d + 1) ≤ M R ( ~g i )( n ′ , n d ) } , and Y − n ′ = {| n d | ≥ 2Λ : M R ( ~g i )( n ′ , n d + 1) > M R ( ~g i )( n ′ , n d ) } . Then, we have(6.20) A ,d ≤ X n ′ ∈ Z d − X n d ∈ Y + n ′ ( M R ( ~g i )( n ′ , n d ) − M R ( ~g i )( n ′ , n d + 1))+ X n ′ ∈ Z d − X n d ∈ Y − n ′ ( M R ( ~g i )( n ′ , n d + 1) − M R ( ~g i )( n ′ , n d )) . We want to show that(6.21) X n ′ ∈ Z d − X n d ∈ Y + n ′ ( M R ( ~g i )( n ′ , n d ) − M R ( ~g i )( n ′ , n d + 1)) . d,m, ~f ǫ, ∀ i ≥ N ;(6.22) X n ′ ∈ Z d − X n d ∈ Y − n ′ ( M R ( ~g i )( n ′ , n d + 1) − M R ( ~g i )( n ′ , n d )) . d,m, ~f ǫ, ∀ i ≥ N . We will only prove (6.21), since (6.22) is analogous. Fix i ≥ N . Since all g i,j ∈ ℓ ( Z d ), thenfor any ( n ′ , n d ) ∈ Z d , there exist ~x ∈ R d and ~r ( n ′ , n d ) ∈ R d + such that ( n ′ , n d ) ∈ R ~r ( n ′ ,n d ) ( ~x )and M R ( ~g i )( n ′ , n d ) = A ~r ( n ′ ,n d ) ( ~g i )( ~x ). Let ~r ( n ′ , n d ) = ( r ( n ′ , n d ) , . . . , r d ( n ′ , n d )). By the similararguments as in getting (6.6) and (6.7), we obtain(6.23) X n ′ ∈ Z d − X n d ∈ Y + n ′ ( M R ( ~g i )( n ′ , n d ) − M R ( ~g i )( n ′ , n d + 1)) ≤ m X µ =1 Y ν = µ, ≤ ν ≤ m k g i,ν k ℓ ( Z d ) × (cid:16) X n ′ ∈ Z d − X n d ∈ Y + n ′ d Y i =1 F ( r i ( n ′ , n d ))) m X ~k ∈ R ~r ( n ′ ,nd ) ( n ′ ,n d ) ∩ Z d | D d g i,µ ( ~k ) | (cid:17) ∗ , AND K ˆOZ ˆO YABUTA ≤ m X µ =1 Y ν = µ, ≤ ν ≤ m k g i,ν k ℓ ( Z d ) × X ( k ,...,k d ) ∈ Z d | D d g i,µ ( k , . . . , k d ) | d Y i =1 (cid:16) X n i ∈ Z F ( r i ( n ′ , n d ))) m χ {| k i − n i | < r i ( n ′ ,n d ) } (cid:17) . When | k d | > Λ, by (6.9), we get(6.24) X nd ∈ Z | nd |≥ F ( r d ( n ′ , n d ))) m χ {| k d − n d | < r d ( n ′ ,n d ) } ≤ . When | k d | ≤ Λ, then | k d − n d | ≥ Λ ≥ | n d | ≥ X nd ∈ Z | nd |≥ F ( r d ( n ′ , n d ))) m χ {| k d − n d | < r d ( n ′ ,n d ) } ≤ X nd ∈ Z | nd |≥ | k d − n d |− ] + 1) m χ {| k d − n d |≥ Λ } . d,m ǫ. By (6.12)-(6.14) and (6.24)-(6.25), we obtain(6.26) X ( k ,...,k d ) ∈ Z d | D d g i,µ ( k , . . . , k d ) | d Y i =1 (cid:16) X n i ∈ Z F ( r i ( n ′ , n d ))) m χ {| k i − n i | < r i ( n ′ ,n d ) } (cid:17) . d,m X k ′ ∈ Z d − X kd ∈ Z | kd | > Λ | D d g i,µ ( k ′ , k d ) | + X k ′ ∈ Z d − X kd ∈ Z | kd |≤ Λ | D d g i,µ ( k ′ , k d ) | ǫ . d,m k g i,µ − f µ k ℓ ( Z d ) + k f µ χ ( R ~ Λ ( ~ c ) k ℓ ( Z d ) + 2( k f µ k ℓ ( Z d ) + 1) ǫ . d,m,f µ ǫ. (6.26) together with (6.12)-(6.13) and (6.23) yields (6.21). It follows from (6.20)-(6.22) that(6.27) A ,d . d,m, ~f ǫ, ∀ i ≥ N . Step 4: Estimates for A ,µ with µ = 1 , , . . . , d − . We first estimates A , . For each n ′ ∈ Z d − , let Z + n ′ = { n d ∈ Z : M R ( ~g i )( n ′ , n d + 1) ≤ M R ( ~g i )( n ′ , n d ) } ,Z − n ′ = { n d ∈ Z : M R ( ~g i )( n ′ , n d + 1) > M R ( ~g i )( n ′ , n d ) } . Then we have(6.28) A , = X | n | > X n ′ ∈ Z d − | M R ( ~g i )( n , . . . , n d − , n d + 1) − M R ( ~g i )( n , . . . , n d ) | = X n ′∈ Z d − | n | > X n d ∈ Z + n ′ ( M R ( ~g i )( n ′ , n d ) − M R ( ~g i )( n ′ , n d + 1))+ X n ′∈ Z d − | n | > X n d ∈ Z − n ′ ( M R ( ~g i )( n ′ , n d + 1) − M R ( ~g i )( n ′ , n d )) . EGULARITY AND CONTINUITY OF THE STRONG MAXIMAL OPERATORS 43 We want to show that(6.29) X n ′∈ Z d − | n | > X n d ∈ Z + n ′ ( M R ( ~g i )( n ′ , n d ) − M R ( ~g i )( n ′ , n d + 1)) . d,m, ~f ǫ, ∀ i ≥ N ;(6.30) X n ′∈ Z d − | n | > X n d ∈ Z − n ′ ( M R ( ~g i )( n ′ , n d + 1) − M R ( ~g i )( n ′ , n d )) . d,m, ~f ǫ, ∀ i ≥ N . We will only prove (6.29), since (6.30) is analogous. By the similar arguments as in getting(6.23), for any ( n ′ , n d ) ∈ Z d , there exists ~r ( n ′ , n d ) = ( r ( n ′ , n d ) , . . . , r d ( n ′ , n d )) ∈ R d + such that(6.31) X n ′∈ Z d − | n | > X n d ∈ Z + n ′ (M R f j ( n ′ , n d ) − M R f j ( n ′ , n d + 1)) . m m X µ =1 Y ν = µ, ≤ ν ≤ m k g i,ν k ℓ ( Z d ) × X ( k ,...,k d ) ∈ Z d | D d g i,µ ( k , . . . , k d ) | (cid:16) X | n | > F ( r ( n ′ , n d ))) m χ {| k − n | < r ( n ′ ,n d ) } (cid:17) × d Y i =2 (cid:16) X n i ∈ Z F ( r i ( n ′ , n d ))) m χ {| k i − n i | < r i ( n ′ ,n d ) } (cid:17) . When | k | ≤ Λ, then | k − n | ≥ Λ ≥ | n | ≥ X | n | > F ( r ( n ′ , n d ))) m χ {| k − n | < r ( n ′ ,n d ) } . d,m ǫ. When | k | > Λ, by (6.9), we get(6.33) X nd ∈ Z | n |≥ F ( r ( n ′ , n d ))) m χ {| k − n | < r ( n ′ ,n d ) } ≤ . Then (6.29) follows from (6.31)-(6.33) and (6.13)-(6.14). By (6.28)-(6.30), it holds that(6.34) A , . d,m, ~f ǫ, ∀ i ≥ N . Similarly, we can get(6.35) A ,µ . d,m, ~f ǫ, ∀ i ≥ N and µ = 2 , . . . , d − . (6.35) together with (6.19), (6.27) and (6.34) yields (6.18). Thus, the proof of the continuitypart is complete. (cid:3) Properties of u x, ~f We summrize the properties of u x, ~f into nine Claims. The proofs of them are thoroughlyelementary. In what follow, we set Q + = Q + ∪ { } . Claim 1. Let 1 < p j < ∞ and f j ∈ L p j ( R ) for 1 ≤ j ≤ m . Then(7.1) lim ( r , ,r , ,r , ,r , ∈ R r , r , r , r , →∞ u x, ~f ( r , , r , , r , , r , ) = 0 for a.e. x ∈ R . ∗ , AND K ˆOZ ˆO YABUTA Proof. Fix 1 ≤ j ≤ m . We note first k f j ( x , · ) k L pj ( R ) , k f M f j ( x , · ) k L pj ( R ) < ∞ a.e x ∈ R , and k f j ( · , x ) k L pj ( R ) , k f M f j ( · , x ) k L pj ( R ) < ∞ a.e x ∈ R . Let ~r = ( r , , r , , r , , r , ) ∈ R . We consider the following three cases.(a) r , + r , > r , + r , > 0. By H¨older’s inequality, one finds that u x, ~f ( ~r ) ≤ m Y j =1 r , + r , Z x + r , x − r , f M f j ( y , x ) dy ≤ m Y j =1 r , + r , ) /p j k f M f j ( · , x ) k L pj ( R ) . Simiarly we get u x, ~f ( ~r ) ≤ m Y j =1 r , + r , ) /p j k f M f j ( x · ) k L pj ( R ) . So, we have u x, ~f ( ~r ) ≤ min (cid:26) m Y j =1 r , + r , ) /p j k f M f j ( · , x ) k L pj ( R ) , m Y j =1 r , + r , ) /p j k f M f j ( x · ) k L pj ( R ) (cid:27) . (b) r , = r , = 0 and r , + r , > 0. By H¨older’s inequality, u x, ~f ( ~r ) = m Y j =1 r , + r , Z x + r , x − r , f j ( x , y ) dy ≤ m Y j =1 r , + r , ) /p j k f ( x , · ) k L pj ( R ) . (c) r , + r , > r , = r , = 0. Similarly to (b) we get u x, ~f ( ~r ) ≤ m Y j =1 r , + r , ) /p j k f j ( · , x ) k L pj ( R ) . From (a), (b) and (c), we see that (7.1) holds for a.e. x ∈ R . (cid:3) Claim 2. Let 1 ≤ j ≤ m and f j ∈ L ( R ). Then the set A j A j := n ( x , x ) ∈ R : lim ( r , ,r , ∈ R r , r , → r , + r , Z x + r , x − r , | f j ( y , x ) − f j ( x , x ) | dy = 0 o is a measurable set in R . Proof. ( x , x ) ∈ A j is equivalent to the following: For any k ∈ N , there exists an ℓ ∈ N suchthat for r , , r , ≥ r , + r , < /ℓ ,1 r , + r , Z x + r , x − r , | f j ( y , x ) − f j ( x , x ) | dy < k . And this is equivalent to: For any k ∈ N , there exists an ℓ ∈ N such that for r , , r , ∈ Q + with r , + r , < /ℓ , 1 r , + r , Z x + r , x − r , | f j ( y , x ) − f j ( x , x ) | dy < k . Thus, A j can be written in the following form: A j = \ k ∈ N [ ℓ ∈ N \ ( r , ,r , ) ∈ Q r , + r , < /ℓ n ( x , x ) ∈ R : 1 r , + r , Z x + r , x − r , | f j ( y , x ) − f j ( x , x ) | dy < k o . EGULARITY AND CONTINUITY OF THE STRONG MAXIMAL OPERATORS 45 Since f j ∈ L ( R ), we see that the function r , + r , R x + r , x − r , | f j ( y , x ) − f j ( x , x ) | dy is mea-surable in R , and hence n ( x , x ) ∈ R : 1 r , + r , Z x + r , x − r , | f j ( y , x ) − f j ( x , x ) | dy < k o is a measurable set in R . Thus, A j is a measurable set in R . (cid:3) Claim 3. Let 1 ≤ j ≤ m , f j ∈ L ( R ) and r , + r , > 0. Then the set B j B j := n ( x , x ) ∈ R : lim ( r , ,r , ∈ R r , r , → r , + r , Z x + r , x − r , (cid:12)(cid:12)(cid:12)(cid:12) r , + r , Z x + r , x − r , f j ( y , y ) dy − r , + r , Z x + r , x − r , f j ( y , x ) dy (cid:12)(cid:12)(cid:12) dy = 0 o is a measurable set in R . Proof. As in the proof of Claim 2, we can write B j = \ k ∈ N [ ℓ ∈ N \ ( r , ,r , ) ∈ Q r , + r , < /ℓ n ( x , x ) ∈ R : 1 r , + r , Z x + r , x − r , (cid:12)(cid:12)(cid:12)(cid:12) r , + r , Z x + r , x − r , f j ( y , y ) dy − r , + r , Z x + r , x − r , f j ( y , x ) dy (cid:12)(cid:12)(cid:12)(cid:12) dy < k o . Hence as in the proof of Claim 2, we see that B j is a measurable set in R . (cid:3) Claim 4. Let 1 < p j < ∞ and f j ∈ L p j ( R ) for 1 ≤ j ≤ m . Then(7.2) lim ( r , ,r , ∈ R r , r , → u x, ~f ( r , , r , , , 0) = u x, ~f (0 , , , 0) for a . e . x ∈ R . Proof. To prove (7.2), it suffices to show that for fixed 1 ≤ j ≤ m , there exists a null set E in R such that for ( x , x ) ∈ R \ E ,(7.3) lim ( r , ,r , ∈ R r , r , → r , + r , Z x + r , x − r , | f j ( y , x ) − f j ( x , x ) | dy = 0 . From f j ∈ L p j ( R ) it follows that f j ( · , x ) ∈ L p j ( R ) a.e x ∈ R . Hence for these x , byLebesgue’s differentiation theorem (note p j > 1) we see that (7.3) holds for a.e. x ∈ R . ByClaim 2, we see that there exists a null set E in R such that (7.3) holds for x ∈ R \ E . (cid:3) Applying the arguments similar to those used in deriving Claim 4, we can get the followingclaim. The details are omitted. Claim 5. Let 1 < p j < ∞ and f j ∈ L p j ( R ) for 1 ≤ j ≤ m . Thenlim ( r , ,r , ∈ R r , r , → u x, ~f (0 , , r , , r , ) = u x, ~f (0 , , , 0) for a . e . x ∈ R . ∗ , AND K ˆOZ ˆO YABUTA Claim 6. Let 1 < p j < ∞ and f j ∈ L p j ( R ) for 1 ≤ j ≤ m . Then there exists a null set E Q in R such that for ( x , x ) ∈ R \ E Q and ( r , , r , ) ∈ Q with r , + r , > ( r , ,r , ∈ R r , r , → r , + r , Z x + r , x − r , (cid:12)(cid:12)(cid:12)(cid:12) r , + r , Z x + r , x − r , f j ( y , y ) dy − r , + r , Z x + r , x − r , f j ( y , x ) dy (cid:12)(cid:12)(cid:12)(cid:12) dy = 0 . Proof. Let ( r , , r , ) ∈ R with r , + r , > 0. From f j ∈ L p j ( R ) we see that for all x ∈ R , (cid:12)(cid:12)(cid:12)(cid:12) r , + r , Z x + r , x − r , f j ( y , y ) dy (cid:12)(cid:12)(cid:12)(cid:12) ≤ r , + r , ) /p k f j ( · , y ) k L pj ( R ) . and hence (cid:18)Z R (cid:12)(cid:12)(cid:12)(cid:12) r , + r , Z x + r , x − r , f j ( y , y ) dy (cid:12)(cid:12)(cid:12)(cid:12) p dy (cid:19) /p ≤ r , + r , ) /p k f j k L pj ( R ) . Hence, for every x ∈ R , by Lebesgue’s differentiation theorem, (7.4) holds for a.e. x ∈ R . So,by Claim 2, there exists a null set E r , ,r , ∈ R such that (7.4) holds for ( x , x ) ∈ R \ E r , ,r , .Now set E Q = [ r , ,r , ∈ Q + r , r , > E r , ,r , . Then | E Q | = 0 and for ( x , x ) ∈ R \ E Q , (7.4) holds for ( r , , r , ) ∈ Q with r , + r , > (cid:3) Claim 7. Let 1 < p j < ∞ and f j ∈ L p j ( R ) for 1 ≤ j ≤ m . Then there exists a null set E in R such that for ( x , x ) ∈ R \ E and ( r , , r , ) ∈ R with r , + r , > ( r ′ , ,r ′ , ∈ R r ′ , ,r ′ , → ( r , ,r , r ′ , + r ′ , Z x + r ′ , x − r ′ , f j ( y , x ) dy = 1 r , + r , Z x + r , x − r , f j ( y , x ) dy , for 1 ≤ j ≤ m , Proof. Fix ( r ′ , , r ′ , ) ∈ R , we have (cid:12)(cid:12)(cid:12)(cid:12) r ′ , + r ′ , Z x + r ′ , x − r ′ , f j ( y , x ) dy − r , + r , Z x + r , x − r , f j ( y , x ) dy (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) r ′ , + r ′ , − r , + r , (cid:12)(cid:12)(cid:12)(cid:12) Z x + r ′ , x − r ′ , | f j ( y , x ) | dy + 1 r , + r , (cid:18)Z x +max( r ′ , ,r , ) x +min( r ′ , ,r , ) | f j ( y , x ) | dy + Z x − min( r ′ , ,r , ) x − max( r ′ , ,r , ) | f j ( y , x ) | dy (cid:19) ≤ | r , − r ′ , | + | r , − r ′ , | ( r , + r , )( r ′ , + r ′ , ) ( r ′ , + r ′ , ) /p ′ j k f j ( · , x ) k L pj ( R ) + | r , − r ′ , | /p ′ j + | r , − r ′ , | /p ′ j r , + r , k f j ( · , x ) k L pj ( R ) . EGULARITY AND CONTINUITY OF THE STRONG MAXIMAL OPERATORS 47 Now, there exists a null set E , in R such that k f j ( · , x ) k L pj ( R ) < ∞ for x ∈ R \ E , . Set E = R × E , . Then E ia a null set in R . And for ( x , x ) ∈ R \ E , (7.5) holds. (cid:3) Claim 8. Let 1 < p j < ∞ and f j ∈ L p j ( R ) for 1 ≤ j ≤ m . Then there exists a null set E in R such that for ( x , x ) ∈ R \ E and ( r , , r , , r , , r , ) ∈ R with r , + r , > r , + r , > ( r ′ , ,r ′ , ∈ R r ′ , ,r ′ , → ( r , ,r , r , + r , Z x + r , x − r , (cid:12)(cid:12)(cid:12)(cid:12) r , + r , Z x + r , x − r , f j ( y , y ) dy − r ′ , + r ′ , Z x + r ′ , x − r ′ , f j ( y , y ) dy (cid:12)(cid:12)(cid:12) dy = 0 Proof. The left side of (7.6) ≤ r , + r , Z x + r , x − r , (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) r , + r , − r ′ , + r ′ , (cid:12)(cid:12)(cid:12)(cid:12) Z x + r , x − r , | f j ( y , y ) | dy + 1 r ′ , + r ′ , (cid:18)Z x +max( r ′ , ,r , ) x +min( r ′ , ,r , ) | f j ( y , y ) | dy + Z x − min( r ′ , ,r , ) x − max( r ′ , ,r , ) | f j ( y , y ) | dy (cid:19)(cid:19) dy ≤ | r , − r ′ , | + | r , − r ′ , | ( r , + r , ) ( r ′ , + r ′ , ) M R f j ( x , x ) + | r ′ , − r , | /p ′ j + | r ′ , − r , | /p j ( r , + r , ) /p j ( r ′ , + r ′ , ) k f j k L pj ( R ) . Then (7.6) follows from this. (cid:3) Applying Claim 8, we can obtain the following claim immediately. Claim 9. Let 1 < p j < ∞ and f j ∈ L p j ( R ) for 1 ≤ j ≤ m . Then for ( r , , r , , r , , r , ) ∈ R with r , + r , > r , + r , > ( r ′ , ,r ′ , ∈ R r ′ , ,r ′ , → ( r , ,r , u x, ~f ( r ′ , , r ′ , , r , , r , ) = u x, ~f ( r , , r , , r , , r , ) . Acknowledgements. 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