Regularity and continuity of the multilinear strong maximal operators

JID:MATPUR

AID:3193 /FLA

[m3L; v1.283; Prn:20/02/2020; 11:41] P.1 (1-38) J. Math. Pures Appl. ••• (••••) •••–•••

Contents lists available at ScienceDirect

Journal de Mathématiques Pures et Appliquées www.elsevier.com/locate/matpur

Regularity and continuity of the multilinear strong maximal operators ✩ Feng Liu a , Qingying Xue b,∗ , Kôzô Yabuta c a

College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, Shandong 266590, People’s Republic of China b School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’s Republic of China c Research Center for Mathematical Sciences, Kwansei Gakuin University, Gakuen 2-1, Sanda 669-1337, Japan

a r t i c l e

i n f o

Article history: Received 26 March 2019 Available online xxxx

a b s t r a c t Let m ≥ 1, in this paper, our object of investigation is the regularity and continuity properties of the following multilinear strong maximal operator

MSC: 42B25 47G10 Keywords: Multilinear strong maximal operators Sobolev spaces Triebel-Lizorkin spaces and Besov spaces Sobolev capacity

MR (f)(x) = sup

Rx R∈R

ˆ m  1 |fi (y)|dy, |R| i=1 R

where x ∈ Rd and R denotes the family of all rectangles in Rd with sides parallel to the axes. When m = 1, denote MR by MR . Then, MR coincides with the classical strong maximal function initially studied by Jessen, Marcinkiewicz and Zygmund. We showed that MR is bounded and continuous from the product Sobolev spaces W 1,p1 (Rd ) × · · · × W 1,pm (Rd ) to W 1,p (Rd ), from the product Besov spaces Bsp1 ,q (Rd ) × · · · × Bspm ,q (Rd ) to Bsp,q (Rd ), from the product Triebel-Lizorkin spaces Fsp1 ,q (Rd ) × · · · × Fspm ,q (Rd ) to Fsp,q (Rd ). As a consequence, we further showed that MR is bounded and continuous from the product fractional Sobolev spaces to fractional Sobolev space. As an application, we obtain a weak type inequality for the Sobolev capacity, which can be used to prove the p-quasicontinuity of MR . In addition, we proved that MR (f) is approximately differentiable a.e. when f = (f1 , · · · , fm ) with each fj ∈ L1 (Rd ) being approximately differentiable a.e. © 2020 Elsevier Masson SAS. All rights reserved.

r é s u m é Soit que m ≥ 1. L’objectif de cet article est d’étudier les propriétés de la régularité et la continuité du opérateur multilinéaire forte maximal MR (f)(x) = sup

Rx R∈R

ˆ m  1 |fi (y)|dy, |R| i=1 R

✩ 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. 11671039, 11871101) and NSFC-DFG (No. 11761131002). The third named author was supported partly by Grant-in-Aid for Scientific Research (C) No. 15K04942, Japan Society for the Promotion of Science. * Corresponding author. E-mail addresses: [email protected] (F. Liu), [email protected] (Q. Xue), [email protected] (K. Yabuta).

https://doi.org/10.1016/j.matpur.2020.02.006 0021-7824/© 2020 Elsevier Masson SAS. All rights reserved.

JID:MATPUR

AID:3193 /FLA

2

[m3L; v1.283; Prn:20/02/2020; 11:41] P.2 (1-38) F. Liu et al. / J. Math. Pures Appl. ••• (••••) •••–•••

où x ∈ Rd et R désigne le famille de tous les rectangles dans Rd avec les côtés parallèle aux axes. Quand m = 1, désigne MR par MR . Alors, MR coïncide avec la fonction forte maximale classique initialement étudiée par Jessen, Marcinkiewicz et Zygmund. Nous avons montré que MR est bornée et continue du produit Sobolev espaces W 1,p1 (Rd ) × · · · × W 1,pm (Rd ) à W 1,p (Rd ), du produit Besov espaces Bsp1 ,q (Rd ) ×· · ·×Bspm ,q (Rd ) à Bsp,q (Rd ), du produit Triebel-Lizorkin espaces Fsp1 ,q (Rd ) × · · · × Fspm ,q (Rd ) to Fsp,q (Rd ). En tant qu’application, nous obtenons une inégalité de faible type pour la Sobolev capacité, qui peut être utilisée pour prouver la p-quasicontinuité de MR . De plus, nous avons prouvé que MR (f) est approximativement différenciable p.p. lorsque f = (f1 , · · · , fm ) avec chaque fj dans L1 (Rd ) étant approximativement différenciable p.p. © 2020 Elsevier Masson SAS. All rights reserved.

1. Introduction 1.1. Hardy-Littlewood maximal functions Let f ∈ L1loc (Rd ) with d ≥ 1 and M be the well-known Hardy-Littlewood maximal operator defined on Rn as follows. ˆ 1 Mf (x) = sup |f (y)|dy, r>0 |Br (x)| Br (x)

where Br (x) is the open ball in Rd centered at x with radius r and |Br (x)| denotes the volume of Br (x).  at a point x is defined by taking the supremum of Analogously, the uncentered maximal function Mf averages over open balls that contain the point. It was well known that the maximal functions and their purpose in differentiation on R were first introduced by Hardy and Littlewood [19], and on Rd were treated by Wiener [44]. The celebrated theorem of Hardy-Littlewood-Wiener states that the operator M is of type (p, p) for 1 < p ≤ ∞ and weak type (1, 1). As a basic and important tool in Harmonic analysis and other fields, such as PDE, the maximal functions and their variants are often used to control some other important operators and give some good absolute size estimates (see [5], [30] and [31]). There is a basic question in the theory of Hardy-Littlewood maximal operators: How does the HardyLittlewood maximal operator preserve the smoothness properties of a function? Achievements have been made in this direction in the past few years. Among them is the nice work of Kinnunen [24] in 1997, where the regularity properties of maximal operators on the W 1,p spaces has been studied. Recall that the Sobolev spaces W 1,p (Rd ), 1 ≤ p ≤ ∞, are defined by W 1,p (Rd ) := {f : Rd → R : f 1,p = f Lp (Rd ) + ∇f Lp (Rd ) < ∞}, where ∇f = (D1 f, . . . , Dd f ) is the weak gradient of f . Kinnunen showed that M is bounded from W 1,p (Rd )  also holds by a simple modification to W 1,p (Rd ) for 1 < p ≤ ∞. It was noticed that the W 1,p -bound for M of Kinnunen’s arguments or Theorem 1 of [18]. Later on, the result of Kinnunen has been extended to a local version in [25], to a fractional version in [26], to a multisublinear version in [8,35] and to a one-sided version in [34]. Whether the continuity for M on W 1,p (Rd ) space holds or not is another certainly nontrivial problem, since the maximal operator is not necessarily sublinear at the derivative level. This problem was first posed by Hajłasz and Onninen [18] and was later settled affirmatively by Luiro [38]. Due to the lack of reflexivity of L1 , it makes the understanding of the W 1,1 (Rd ) regularity more subtler. One interesting question was raised by Hajłasz and Onninen in [18]: Is the operator f → |∇Mf | bounded from W 1,1 (Rd ) to L1 (Rd )? A complete answer was addressed only in dimension d = 1 in [2,29,33,42] and

JID:MATPUR

AID:3193 /FLA

[m3L; v1.283; Prn:20/02/2020; 11:41] P.3 (1-38) F. Liu et al. / J. Math. Pures Appl. ••• (••••) •••–•••

3

partial progress on the general case d ≥ 2 was given by Hajłasz and Malý [17] and Luiro [40]. For more previous works or related topic we refer the readers to consult [3,6,7,9,29,32,36], and the references therein. Now we know that M is bounded on Lp (Rd ) = W 0,p (Rd ) and W 1,p (Rd ) for p > 1. Therefore a natural question arises: what is the properties of M on the fractional Sobolev spaces W s,p (Rd ) defined by the Bessel potentials when 0 < s < 1? This question was first studied by Korry [28] who observed that M : W s,p (Rd ) → W s,p (Rd ) is bounded for all 0 < s < 1 and 1 < p < ∞. Notice that Fsp,2 (Rd ) = W s,p (Rd ) for any s > 0 and 1 < p < ∞ (see [15]). It may be further expected that M still enjoys the boundedness on Triebel-Lizorkin spaces Fsp,q (Rd ). This was done by Korry [27], who indeed proved that M is bounded on the inhomogeneous Triebel-Lizorkin spaces Fsp,q (Rd ) and Besov spaces Bsp,q (Rd ) for all 0 < s < 1 and 1 < p, q < ∞. Recently, Luiro [39] established the continuity of M on Fsp,q (Rd ) for all 0 < s < 1 and 1 < p, q < ∞. Still more recently, Liu and Wu [37] extended the above results to the maximal operators associated with polynomial mappings. 1.2. Multilinear strong maximal operators Over the past few decades, many celebrated works have been done in the study of the maximal functions associated with different kinds of basis. These bases mainly including: some differentiation bases (balls or cubes, rectangles with some restrictions see [20], [46] and [47]), translation in-variant basis of rectangles [10], basis formed by convex sets, using rectangles with a side parallel to some direction (lacunary parabolic set of directions in [41], Cantor set of directions in [21], arbitrary set of directions [1], [22]). In this paper, we will focus on the translation in-variant basis of rectangles studied by Córdoba and Fefferman [10]. Let f = (f1 , . . . , fm ) be an m-dimensional vector of locally integrable functions and R denotes the collection of all open rectangles R ⊂ Rd with sides parallel to the coordinate axes. In 2011, Grafakos, Liu, Pérez and Torres [16] introduced and studied the weighted strong and endpoint estimates for the multilinear strong maximal function MR , which is defined by ˆ m  1 |fi (yi )|dyi , Rx i=1 |R|

MR (f)(x) = sup

R∈R

x ∈ Rd .

(1.1)

R

Whenever m = 1, we simply denote MR by MR . Then MR coincides with the classical strong maximal operator. As the most prototypical representative of the multi-parameter operators, MR can be looked as a geometric maximal operator which commutes with full d-parameter group of dilations (x1, x2 , . . . , xd ) → (δ1 x1 , δ2 x2 , . . . , δd xd ). It was proved by García-Cuerva and Rubio de Francia that MR is bounded on Lp (Rd ) for all 1 < p < ∞ (see [12, p. 452]). In 1935, a maximal theorem was given by Jessen, Marcinkiewicz and Zygmund in [20]. They pointed out that unlike the classical Hardy-Littlewood maximal operator, the strong maximal function is not of weak type (1, 1). As a replacement, they showed that it is bounded from L(log+ L)(Rd ) to L1 (Rd ). Subsequently, an additional proof of the maximal theorem was given by Córdoba and Fefferman in 1975, using an alternative geometric method [10]. The basis of the work of Córdoba and Fefferman is a selection theorem for families of rectangles in Rd. Some delicate properties of rectangles in Rd were also quantified in that study.  It was known that M  is bounded and Furthermore, if m = 1 and d = 1, the operator MR = M. 1,p 1,1  is absolutely continuous on W (R) for 1 < p < ∞. It follows from [2,33] that if f ∈ W (R), then Mf  ) L1 (R) ≤ f  L1 (R) . For d ≥ 1, Aldaz and Pérez Lázaro [3] continuous on R and it holds that (Mf considered a class of local strong maximal operator and proved that it maps BV(U ) into L1 (U ), where U is an open set of Rd and BV(U ) is a subclass of L1 (U ) functions. See [14, Definition 1.3] and [4, Definition 3.4] for instance.

JID:MATPUR

AID:3193 /FLA

[m3L; v1.283; Prn:20/02/2020; 11:41] P.4 (1-38) F. Liu et al. / J. Math. Pures Appl. ••• (••••) •••–•••

4

The results in [16] indicate that MR is bounded from Lp1 (Rd ) × · · · × Lpm (Rd ) to Lp (Rd ) for all 1 < m pi d  p1 , . . . , pm , p ≤ ∞ and 1/p = i=1 1/pi . Moreover, for f = (f1 , . . . , fm ) with each fi ∈ L (R ), the following norm inequality holds MR (f)Lp (Rd ) p1 ,...,pm

m 

fi Lpi (Rd ) .

(1.2)

i=1

It is well known that the geometry of rectangles in Rd is more intricate than that of cubes or balls, even when both classes of sets are restricted to have sides parallel to the axes. Even for m = 1, a basic observation is that Mf (x) d MR f (x) for all x ∈ Rd . However, there does not exist any constant C > 0 such that MR f (x) ≤ CMf (x) for all x ∈ Rd . This indicates fully that the strong maximal functions are uncontrollable. For these reasons, this makes the investigation of the strong maximal functions very complex, but also quite interesting. Based on the facts concerning the previous results on the Hardy-Littlewood maximal operators, it is therefore a natural question to ask whether the multilinear strong maximal operators are bounded and continuous on the products of the first order Sobolev spaces W 1,p(Rd ) or the fractional Sobolev spaces W s,p (Rd ) or on its generalizations Fsp,q (Rd ) and Bsp,q (Rd ). This is the main motivation of this work. 1.3. Main results We now state our main results as follows. m Theorem 1.1 (Properties on Sobolev spaces). Let 1 < p1 , . . . , pm , p < ∞ and 1/p = i=1 1/pi . Then MR is bounded and continuous from W 1,p1 (Rd ) × · · · × W 1,pm (Rd ) to W 1,p (Rd ). Moreover, if f = (f1 , . . . , fm ) with each fi ∈ W 1,pi (Rd ), then, for 1 ≤ l ≤ d, it holds that |Dl MR (f)(x)| m,d,p1 ,...,pm

m 

MR (fμl )(x), a.e. x ∈ Rd ,

μ=1

where fμl = (f1 , . . . , fμ−1 , Dl fμ , fμ+1 , . . . , fm ). Remark 1.1. The case p = ∞ is also valid in Theorem 1.1, which follows from the similar arguments to those used in [24, Remark (iii)]. m Theorem 1.2 (Properties on Besov spaces). Let 1 < p1 , . . . , pm , p, q < ∞, 1/p = i=1 1/pi and 0 < s < 1. Then MR is bounded and continuous from Bsp1 ,q (Rd ) × · · · × Bspm ,q (Rd ) to Bsp,q (Rd ). m Theorem 1.3 (Properties on Triebel-Lizorkin spaces). Let 1 < p1 , . . . , pm , p, q < ∞, 1/p = i=1 1/pi and 0 < s < 1. Then MR is bounded and continuous from Fsp1 ,q (Rd ) × · · · × Fspm ,q (Rd ) to Fsp,q (Rd ). Theorem 1.3 implies the following result immediately. m Corollary 1.4 (Properties on fractional Sobolev spaces). Let 1 < p1 , . . . , pm , p < ∞, 1/p = i=1 1/pi and 0 < s < 1. Then MR is bounded and continuous from the fractional Sobolev spaces W s,p1 (Rd ) × · · · × W s,pm (Rd ) to W s,p (Rd ). Theorem 1.1 can be used to obtain a weak type inequality for the Sobolev capacity, which can be further employed to prove the quasicontinuity of the strong maximal function of a Sobolev function. We first need to give the definition of Sobolev p-capacity.

JID:MATPUR

AID:3193 /FLA

[m3L; v1.283; Prn:20/02/2020; 11:41] P.5 (1-38) F. Liu et al. / J. Math. Pures Appl. ••• (••••) •••–•••

5

Definition 1.2 (Sobolev p-capacity, [23]). For 1 < p < ∞, the Sobolev p-capacity of the set E ⊂ Rd is defined by ˆ Cp (E) :=

inf

f ∈A(E) Rd

(|f (y)|p + |∇f (y)|p )dy,

(1.3)

where A(E) = {f ∈ W 1,p (Rd ) : f ≥ 1 on a neighbourhood of E}. We set Cp (E) = ∞ if A(E) = ∅. It was shown in [11] that the Sobolev p-capacity is a monotone and a countably subadditive set function. Also, it is an outer measure over Rd . Definition 1.3 (p-quasicontinuous and p-quasieverywhere, [11]). A function f is said to be p-quasicontinuous in Rd if for every  > 0, there exists a set F ⊂ Rd such that Cp (F ) <  and the restriction of f to Rd \ F is continuous and finite. A property holds p-quasieverywhere if it holds outside a set of the Sobolev p-capacity zero. Remark 1.4. It was known that each Sobolev function has a quasicontinuous representative, that is, for each u ∈ W 1,p (Rd ), there is a p-quasicontinuous function v ∈ W 1,p (Rd ) such that u = v a.e. in Rd . This representative is unique in the sense that if v and w are p-quasicontinuous and v = w a.e. in Rd , then w = v p-quasieverywhere in Rd , see [11] for more details. In 1997, Kinnunen proved that Mf is p-quasicontinuous if f ∈ W 1,p (Rd ) for any 1 < p < ∞. Motivated by Kinnunen’s work [24], we shall prove the following result: Theorem 1.5 (p-quasicontinuity). Let 1 < p1 , . . . , pm < ∞, and 1/p = (f1 , . . . , fm ) with each fi ∈ W 1,pi (Rd ), then MR (f) is p-quasicontinuous.

m i=1

1/pi . Suppose that f =

In 2010, Hajłasz and Malý [17] proved that Mf is approximately differentiable a.e. provided that f ∈ L (Rd ). Motivated by Hajłasz and Malý’s work, we shall establish the following result: 1

Theorem 1.6. Let f = (f1 , . . . , fm ) with each fj ∈ L1 (Rd ) being approximately differentiable a.e., then MR (f) is approximately differentiable a.e. Remark 1.5. Since every function in W 1,1 (Rd ) space is approximately differentiable a.e., thus Theorem 1.6 yields that if f = (f1 , · · · , fm ) with each fj ∈ W 1,1 (Rd ), then MR (f) is approximately differentiable a.e. However, it is unknown that whether MR (f) is weak differentiable when each fj ∈ W 1,1 (Rd ), even in the case m = 1 and d ≥ 2. This paper will be organized as follows. Section 2 will be devoted to presenting the proof of Theorem 1.1. Section 3 will be devoted to giving the proofs of Theorems 1.2 and 1.3. The proofs of Theorems 1.5 and 1.6 will be given in Sections 4 and 5, respectively. We would like to remark that the main ideas employed in the proofs of Theorem 1.1 is greatly motivated by [24] and [38], but our methods and techniques are more delicate and complex than those in [24]. It should be pointed out that the main ideas in the proofs of Theorems 1.2 and 1.3 are motivated by [37]. Throughout this paper, if there exists a constant c > 0 depending only on 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.

JID:MATPUR

AID:3193 /FLA

[m3L; v1.283; Prn:20/02/2020; 11:41] P.6 (1-38) F. Liu et al. / J. Math. Pures Appl. ••• (••••) •••–•••

6

2. Properties on Sobolev spaces 2.1. Preliminary lemmas We first present several preliminary lemmas, which play important roles in the proof of Theorem 1.1. Some basic ideas will be taken from [38], where the proof for the continuity in W 1,p (Rd ) of the Hardy-Littlewood maximal operator has been given. We only consider the case d = 2 and other cases are analogous and more complex. For A ⊂ R2 and x ∈ R2 , define d(x, A) := inf |x − a| and A(λ) := {x ∈ R2 ; d(x, A) ≤ λ} for λ ≥ 0. a∈A

m We denote by f p,A the Lp -norm of f χA for all measurable sets A ⊂ R2 . Let 1/p = j=1 1/pj and pj 2  1 < p1 , p2 , . . . , pm , p < ∞. Let f = (f1 , . . . , fm ) with each fj ∈ L (R ). For convenience, we set R+ = (0, ∞) and R+ = [0, ∞). We also set 2

(R+ )41 = {(r1 , r2 , 0, 0) : (r1 , r2 ) ∈ R+ , r1 + r2 > 0}, 2

(R+ )42 = {(0, 0, r3 , r4 ) : (r3 , r4 ) ∈ R+ , r3 + r4 > 0}, 4

(R+ )41,2 = {(r1 , r2 , r3 , r4 ) : (r1 , r2 , r3 , r4 ) ∈ R+ , r1 + r2 > 0, r3 + r4 > 0}. 4

Define the function u(x1 ,x2 ),f : R+ → R by m  1 u(x1 ,x2 ),f(r1,1 , r1,2 , 0, 0) := (r1,1 + r1,2 )m j=1

x1ˆ +r1,2

|fj (y1 , x2 )|dy1

for (r1,1 , r1,2 , 0, 0) ∈ (R+ )41 ;

|fj (x1 , y2 )|dy2

for (0, 0, r2,1 , r2,2 ) ∈ (R+ )42 ;

x1 −r1,1

m  1 u(x1 ,x2 ),f(0, 0, r2,1 , r2,2 ) := (r2,1 + r2,2 )m j=1

x2ˆ +r2,2

x2 −r2,1

2 

m  1 u(x1 ,x2 ),f(r1,1 , r1,2 , r2,1 , r2,2 ) := (ri,1 + ri,2 )m j=1 i=1

x1ˆ +r1,2 x2ˆ +r2,2

|fj (y1 , y2 )|dy1 dy2 ,

x1 −r1,1 x2 −r2,1

for (r1,1 , r1,2 , r2,1 , r2,2 ) ∈ (R+ )41,2 . In particular, we denote u(x1 ,x2 ),f(0, 0, 0, 0) = MR (f)(x) =

m j=1

|fj (x1 , x2 )|. We can write

sup r1,1 ,r1,2 ,r2,1 ,r2,2 >0

u(x1 ,x2 ),f(r1,1 , r1,2 , r2,1 , r2,2 ).

For a fixed point x = (x1 , x2 ) ∈ R2 , we define the sets Bi (f)(x1 , x2 ) (i = 1, 2, 3) by  4 B1 (f)(x1 , x2 ) := (r1,1 , r1,2 , r2,1 , r2,2 ) ∈ R+ : MR (f)(x1 , x2 ) = lim sup (r1,1,k ,r1,2,k ,r2,1,k ,r2,2,k ) →(r1,1 ,r1,2 ,r2,1 ,r2,2 )

u(x1 ,x2 ),f(r1,1,k , r1,2,k , r2,1,k , r2,2,k )  for some r1,1,k , r1,2,k , r2,1,k , r2,2,k > 0 .

JID:MATPUR

AID:3193 /FLA

[m3L; v1.283; Prn:20/02/2020; 11:41] P.7 (1-38) F. Liu et al. / J. Math. Pures Appl. ••• (••••) •••–•••

7

 2 B2 (f)(x1 , x2 ) := (r1,1 , r1,2 , r2,1 , r2,2 ) ∈ R+ × {(0, 0)} : MR (f)(x1 , x2 ) = lim sup (r1,1,k ,r1,2,k )→(r1,1 ,r1,2 )

 u(x1 ,x2 ),f(r1,1,k , r1,2,k , 0, 0) for some r1,1,k , r1,2,k > 0 .

 2 B3 (f)(x1 , x2 ) := (r1 , r2 ) ∈ {(0, 0)} × R+ : MR (f)(x1 , x2 ) = lim sup (r2,1,k ,r2,2,k )→(r2,1 ,r2,2 )

 u(x1 ,x2 ),f(0, 0, r2,1,k , r2,2,k ) for some r2,1,k , r2,2,k > 0 .

The function u(x1 ,x2 ),f enjoys the following basic properties, we omit the proof here. Lemma 2.1. Let f = (f1 , . . . , fm ) with each fj ∈ Lpj (R2 ) for 1 < pj < ∞, (j = 1, 2, . . . , m). Then the following statements hold: 4

(i) u(x1 ,x2 ),f are continuous on (R+ )4+ := {(r1,1 , r1,2 , r2,1 , r2,2 ) ∈ R+ : r1,1 + r1,2 , r2,1 + r2,2 > 0} for all 4

(x1 , x2 ) ∈ R2 , and continuous on R+ for a.e. (x1 , x2 ) ∈ R2 ; lim

4

(r1,1 ,r1,2 ,r2,1 ,r2,2 )∈R+ r1,1 +r1,2 ,r2,1 +r2,2 →∞

u(x1 ,x2 ),f(r1,1 , r1,2 , r2,1 , r2,2 ) = 0,

for a.e. (x1 , x2 ) ∈ R2 ;

B1 (f)(x1 , x2 ) are nonempty and closed for every (x1 , x2 ) ∈ R2 ; 2 (ii) u(x1 ,x2 ),f are continuous on {(r1,1 , r1,2 ) ∈ R+ : r1,1 + r1,2 > 0} × {(0, 0)} for all x1 ∈ R and a.e. x2 ∈ R, and continuous at (0, 0, 0, 0) for a.e. (x1 , x2 ) ∈ R2 ; lim

2

(r1,1 ,r1,2 )∈R+ r1,1 +r1,2 →∞

u(x1 ,x2 ),f(r1,1 , r1,2 , 0, 0) = 0,

for all x1 ∈ R

and a.e. x2 ∈ R;

B2 (f)(x1 , x2 ) are nonempty and closed for a.e. (x1 , x2 ) ∈ R2 ; 2 (iii) u(x1 ,x2 ),f are continuous on {(0, 0)} × {(r2,1 , r2,2 ) ∈ R+ : r2,1 +r2,2 > 0} for all x2 ∈ R and a.e. x1 ∈ R and continuous at (0, 0, 0, 0) for a.e. (x1 , x2 ) ∈ R2 ; lim

2

(r2,1 ,r2,2 )∈R+ r2,1 +r2,2 →∞

u(x1 ,x2 ),f(0, 0, r2,1 , r2,2 ) = 0,

for all x2 ∈ R

and a.e. x1 ∈ R;

B3 (f)(x1 , x2 ) are nonempty and closed for a.e. (x1 , x2 ) ∈ R2 . Lemma 2.2. The following relationships between MR (f) and u(x1 ,x2 ),f are valid. (iv) MR (f)(x1 , x2 ) =

m 

|fj (x1 , x2 )| for a.e. (x1 , x2 ) ∈ R2 such that 0 ∈

j=1

(v) MR (f)(x1 , x2 ) = u(x1 ,x2 ),f(r) for a.e. (x1 , x2 ) ∈ R2 (vi) MR (f)(x1 , x2 ) = u(x1 ,x2 ),f(r) for a.e. (x1 , x2 ) ∈ R2 (vii) MR (f)(x1 , x2 ) = u(x1 ,x2 ),f(r) for a.e. (x1 , x2 ) ∈ R2 r) for a.e. (x1 , x2 ) ∈ R2 (viii) MR (f)(x1 , x2 ) = u ( (x1 ,x2 ),f

3

Bi (f )(x1 , x2 );

i=1

such that r ∈ B1 (f)(x1 , x2 ) ∩ (R+ )41 ; such that r ∈ B1 (f)(x1 , x2 ) ∩ (R+ )4 ; 2

such that r ∈ B2 (f)(x1 , x2 ); such that r ∈ B3 (f)(x1 , x2 );

(ix) MR (f)(x1 , x2 ) = u(x1 ,x2 ),f(r) if r ∈ B1 (f)(x) ∩ (R+ )41,2 for all x ∈ R2 .

JID:MATPUR 8

AID:3193 /FLA

[m3L; v1.283; Prn:20/02/2020; 11:41] P.8 (1-38) F. Liu et al. / J. Math. Pures Appl. ••• (••••) •••–•••

For convenience, for any r = (r1 , r2 , . . . , rd ) ∈ Rd+ and x = (x1 , x2 , . . . , xd ) ∈ Rd , we let Rr (x) = {y = (y1 , y2 , . . . , yd ) ∈ Rd ; |yi − xi | < ri , i = 1, 2, . . . , d}.  = (Λ, Λ) and 0 = (0, 0), 1 < p, pi < ∞ and 1/p = m 1/pi , and f = (f1 , . . . , fm ) Lemma 2.3. Let Λ > 0, Λ i=1 with each fi ∈ Lpi (R2 ). Let fj = (f1,j , . . . , fm,j ) such that fi,j → fi in Lpi (R2 ) when j → ∞ for all i = 1, 2, . . . , m. Then for all λ > 0 and i = 1, 2, 3, we have    lim |{x ∈ RΛ  (0); Bi (fj )(x)  Bi (f )(x)(λ) }| = 0.

j→∞

(2.1)

Proof. Without loss of generality we may assume all fi,j ≥ 0 and fi ≥ 0. We shall prove (2.1) for the case i = 1 and the other cases are analogous. Let λ > 0 and Λ > 0. We first conclude that the set {x ∈ R2 ; B1 (fj )(x)  B1 (f)(x)(λ) } is measurable for all j ≥ 1. To see this, let E be the set of all points which are not Lebesgue points of any of the functions fi,j and fi . Obviously, |E| = 0. We denote by Q+ the set of positive rationals. Fix j ≥ 1, we can write {x ∈ R2 \ E : B1 (fj )(x)  B1 (f)(x)λ } ∞  ∞ 

1 x ∈ R2 : ∃r ∈ R4+ s.t. d(r, B1 (f)(x)) > λ + = i i=1 k=1  1 and MR (fj )(x) < ux,fj (r) + k ∞  ∞ 

1  x ∈ R2 : d(t, B1 (f)(x)) > λ + = i i=1 k=1  t∈Q4+   1 . x ∈ R2 : MR (fj )(x) < ux,fj (t) + k On the other hand, for any fixed t ∈ Q4+ , we have {x : d(t, B1 (f)(x)) > λ} =

∞



l=1  s∈Q4+ ∩{ s:| s− t|≤λ}

 1 x ∈ R2 : MR (f)(x) > ux,f(s) + . l

Therefore, we get the measurability of {x ∈ R2 ; B1 (fj )(x)  B1 (f)(x)(λ) } for any j ≥ 1.  Now, we claim that for a.e. x ∈ RΛ  (0), there exists γ(x) ∈ N \ {0} such that ux,f(r) < MR (f)(x) −

1 , when d(r, B1 (f)(x)) > λ. γ(x)

(2.2)

∞  Actually, if (2.2) does not hold, then for a.e. x ∈ RΛ  (0), there exists a bounded sequence of {rk }k=1 such that

lim ux,f(rk ) = MR (f)(x) and d(rk , B1 (f)(x)) > λ.

k→∞

∞ Hence, we may choose a subsequence {sk }∞ r as k → ∞. It follows that k=1 of {rk }k=1 such that sk →    r ∈ B1 (f )(x) and d(r, B1 (f )(x)) ≥ λ, which is a contradiction. Therefore, (2.2) holds. Let

A1,j := {x ∈ R2 : |MR (fj )(x) − MR (f)(x)| ≥ (4γ)−1 }, A2,j := {x ∈ R2 : |ux,fj (r) − ux,f(r)| ≥ (2γ)−1 if d(r, B1 (f)(x)) > λ}, A3,j := {x ∈ R2 : ux,fj (r) < MR (fj )(x) − (4γ)−1 , if d(r, B1 (f)(x)) > λ}.

JID:MATPUR

AID:3193 /FLA

[m3L; v1.283; Prn:20/02/2020; 11:41] P.9 (1-38) F. Liu et al. / J. Math. Pures Appl. ••• (••••) •••–•••

9

Given  ∈ (0, 1), we get from (2.2) that there exists γ = γ(Λ, λ, ) ∈ N \ {0} and a measurable set E0 with |E0 | <  such that   2  RΛ r) < MR (f)(x) − γ −1 , if d(r, B1 (f)(x)) > λ ∪ E0  (0) ⊂ x ∈ R : ux,f( ⊂ A1,j ∪ A2,j ∪ A3,j ∪ E0 .

(2.3)

¯ = 0. One Let A¯ be the set of all points x such that x is a Lebesgue point of all fj . Note that |R2 \ A| 2 can easily check that A3,j ∩ A¯ ⊂ {x ∈ R : B1 (fj )(x) ⊂ B1 (f)(x)(λ) }. This together with (2.3) yields that 2   ¯  {x ∈ RΛ  (0); B1 (fj )(x)  B1 (f )(x)(λ) } ⊂ A1,j ∪ A2,j ∪ E0 ∪ (R \ A).

(2.4)

Since fi,j → fi in Lpi (R2 ) when j → ∞, then there exists Ni = Ni (, γ) ∈ N such that fi,j − fi Lpi (R2 ) <

 and fi,j Lpi (R2 ) ≤ fi Lpi (R2 ) + 1 ∀j ≥ Ni . γ

(2.5)

Moreover, it holds that |MR (fj )(x) − MR (f)(x)| m ˆ m ˆ   1    ≤ sup f (y)dy − f (y)dy   i,j i m Rx |R| i=1 i=1 ≤

≤

R∈R

R

m 

l−1 

sup

Rx l=1 R∈R m 

1 |R|m

R

ˆ fμ (y)dy

μ=1 R

ˆ

ˆ

m 

|fl,j (y) − fl (y)|dy

fν,j (y)dy

ν=l+1 R

(2.6)

R

MR (Fjl )(x),

l=1

where Fjl = (f1 , . . . , fl−1 , fl,j − fl , fl+1,j , . . . , fm,j ). Let N0 = max1≤j≤m Nj . Then, for any j ≥ N0 , we get from (2.5) and (2.6) that |A1,j | ≤ (4γ)p MR (fj ) − MR (f)pLp (R2 ) ≤ (4γm)

p

m l−1  

fμ pLpμ (R2 )

l=1 μ=1

m 

fν,j pLpν (R2 ) fl,j − fl pLpl (R2 )

(2.7)

ν=l+1

m,p1 ,...,pm ,p . Since |ux,fj (r) − ux,f(r)| ≤

m 

MR (Fjl )(x).

l=1

Similarly, we can obtain that |A2,j | m,p1 ,...,pm ,p  for any j ≥ N0 . This together with (2.4) and (2.7) yields (2.1).  For any fixed h > 0 and fi ∈ Lpi (R2 ) with 1 < pi < ∞, define (fi )lh (x) =

(fi )lτ (h) (x) − fi (x) h

and (fi )lτ (h) (x) = fi (x + hel ), l = 1, 2.

JID:MATPUR

AID:3193 /FLA

[m3L; v1.283; Prn:20/02/2020; 11:41] P.10 (1-38) F. Liu et al. / J. Math. Pures Appl. ••• (••••) •••–•••

10

It is well known that for l = 1, 2 and 1 < pi < ∞, (fi )lτ (h) → fi in Lpi (R2 ) when h → 0, and if fi ∈ W 1,pi (R2 ) we have (fi )lh → Dl fi in Lpi (R2 ) when h → 0 (see [13]). Let A, B be two subsets of R2 , we define the Hausdorff distance of A and B by π(A, B) := inf{δ > 0 : A ⊂ B(δ) and B ⊂ A(δ) }. Applying Lemma 2.3, we can get the following corollary. Corollary 2.4. Let fi ∈ Lpi (R2 ) with 1 < pi < ∞. Then for all Λ > 0, λ > 0, i = 1, 2, 3 and l = 1, 2, we have    lim |{x ∈ RΛ el )) > λ}| = 0.  (0); π(Bi (f )(x), Bi (f )(x + h

h→0

Proof. Fix i ∈ {1, 2, 3}. It suffices to show that    lim |{x ∈ RΛ el )(λ) or Bi (f)(x + hel )  Bi (f)(x)(λ) }| = 0.  (0) : Bi (f )(x)  Bi (f )(x + h

h→0

(2.8)

One can easily check that Bi (f)(x + hel ) = Bi (fτl (h) )(x) and Bi (f)(x) = Bi (fτl (−h) )(x + hel ). Here fτl (h) = (f1 (x + hel ), . . . , fm (x + hel )). It follows that    {x ∈ RΛ el )(λ) }  (0) : Bi (f )(x)  Bi (f )(x + h = {x ∈ R (0) : Bi (fl )(x + hel )  Bi (f)(x + hel )(λ) } Λ

τ (−h)

(2.9)

l   ⊂ {x ∈ RΛ+1 el  (0) : Bi (fτ (−h) )(x)  Bi (f )(x)(λ) } − h where Λ + 1 = (Λ + 1, Λ + 1) and |h| ≤ 1. Moreover,   l    {x ∈ RΛ  (0) : Bi (f )(x + hel )  Bi (f )(x)(λ) } = {x ∈ RΛ  (0) : Bi (fτ (h) )(x)(λ)  Bi (f )(x)(λ) }.

(2.10)

We note that (fi )lτ (h) → fi in Lpi (R2 ) when h → 0. By Lemma 2.3, it yields that l   lim |{x ∈ RΛ+1  (0) : Bi (fτ (−h) )(x)  Bi (f )(x)(λ) }| = 0

(2.11)

l   lim |{x ∈ RΛ  (0) : Bi (fτ (h) )(x)(λ)  Bi (f )(x)(λ) }| = 0.

(2.12)

h→0

and

h→0

Now, it is easy to see that (2.8) follows from (2.9)-(2.12).  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.

JID:MATPUR

AID:3193 /FLA

[m3L; v1.283; Prn:20/02/2020; 11:41] P.11 (1-38) F. Liu et al. / J. Math. Pures Appl. ••• (••••) •••–•••

11

Lemma 2.5. Let fi ∈ W 1,pi (R2 ) with 1 < pi < ∞. Then for any l = 1, 2 and a.e. (x1 , x2 ) ∈ R2 , we have (i) For all r ∈ B1 (f)(x1 , x2 ) with r1,1 + r1,2 > 0, r2,1 + r2,2 > 0, it holds that Dl MR (f)(x1 , x2 ) m 

1 = m (r m (r + r ) 1,1 1,2 2,1 + r2,2 ) μ=1

+r1,2 x2ˆ +r2,2 x1ˆ

m 

|fj (y1 , y2 )|dy1 dy2 (2.13)

j =μ,1≤j≤mx −r 1 1,1 x2 −r2,1 x1ˆ +r1,2 x2ˆ +r2,2

×

Dl |fμ (y1 , y2 )|dy1 dy2

x1 −r1,1 x2 −r2,1

(ii) For all r ∈ B1 (f)(x1 , x2 ) ∪ B2 (f)(x1 , x2 ) with r1,1 + r1,2 > 0, r2,1 = r2,2 = 0, we have Dl MR (f)(x1 , x2 ) m 

1 = (r + r1,2 )m 1,1 μ=1

m 

x1ˆ +r1,2

x1ˆ +r1,2

|fj (y1 , x2 )|dy1

j =μ,1≤j≤mx −r 1 1,1

Dl |fμ (y1 , x2 )|dy1

(2.14)

x1 −r1,1

(iii) For all r ∈ B1 (f)(x1 , x2 ) ∪ B3 (f)(x1 , x2 ) with r1,1 = r1,2 = 0, r2,1 + r2,2 > 0, it holds Dl MR (f)(x1 , x2 ) m 

1 = (r2,1 + r2,2 )m μ=1

m 

x2ˆ +r2,2

+r2,2 x2ˆ

|fj (x1 , y2 )|dy2

j =μ,1≤j≤mx −r 2 2,1

Dl |fμ (x1 , y2 )|dy2

(2.15)

x2 −r2,1

(iv) If 0 ∈ Bi (f)(x1 , x2 ) for i = 1, 2, 3, then, Dl MR (f)(x1 , x2 ) = Dl |f |(x1 , x2 ).

(2.16)

Proof. We may assume without loss of generality that all fi ≥ 0, since |fi | ∈ W 1,pi (R2 ) if fi ∈ W 1,pi (R2 ). Fix Λ > 0 and l ∈ {1, 2}. Invoking Corollary 2.4, for any i ∈ {1, 2, 3}, we can choose a sequence {si,k }∞ k=1 ,    si,k > 0 and si,k → 0 such that limk→∞ π(Bi (f )(x), Bi (f )(x + si,k el )) = 0 for a.e. x ∈ RΛ  (0). Then, Step 1 in the proof of Theorem 1.1 yields that MR (f) ∈ W 1,p (R2 ) and (MR (f))lsi,k − Dl MR (f)Lp (R2 ) → 0 as k → ∞. We also see that (fμ )lsi,k − Dl fμ Lp (R2 ) → 0 as k → ∞, MR ((fμ )lsi,k − Dl fμ )Lp (R2 ) → 0 as k → ∞,

(fμ )lτ (si,k ) − fμ Lp (R2 ) → 0 as k → ∞, MR ((fμ )lτ (si,k ) − fμ )Lp (R2 ) → 0 as k → ∞,

j ((fμ )l − Dl fμ )Lp (R2 ) → 0 as k → ∞ M si,k

(j = 1, 2),

j ((fμ )l M τ (si,k ) − fμ )Lp (R2 ) → 0 as k → ∞

(j = 1, 2),

j is the one dimensional uncentered Hardy-Littlewood maximal operator with respect to the where M ∞ variable xj (j = 1, 2). Furthermore, there exists a subsequence {hi,k }∞ k=1 of {si,k }k=1 and a measurable set   Ai,1 ⊂ RΛ  (0) such that |RΛ  (0)\Ai,1 | = 0 and

JID:MATPUR

AID:3193 /FLA

[m3L; v1.283; Prn:20/02/2020; 11:41] P.12 (1-38) F. Liu et al. / J. Math. Pures Appl. ••• (••••) •••–•••

12

(i) (fμ )lhi,k (x) → Dl fμ (x), (fμ )lτ (hi,k ) (x) → fμ (x), MR ((fμ )lhi,k − Dl fμ ) → 0, MR ((fμ )lτ (hi,k ) − fμ ) → 0,  l j ((fμ )l − Dl fμ )(x) → 0 (j = 1, 2), M j ((fμ )l M hi,k τ (hi,k ) − fμ ) → 0 (j = 1, 2) and (MR (f ))hi,k (x) → Dl MR (f)(x) when k → ∞ for any x ∈ Ai,1 ; (ii) limk→∞ π(Bi (f))(x), Bi (f)(x + hi,kel )) = 0 for any x ∈ Ai,1 .

Let Ai,2 :=

∞  k=1

{x ∈ R2 : MR (f)(x + hi,kel ) ≥ ux+hi,k el ,f(0, 0, 0, 0)},

Ai,3 := {x ∈ R2 : MR (f)(x) = ux,f(0, 0, 0, 0) if (0, 0, 0, 0) ∈ Bi (f(x)}, ∞  {x ∈ R2 : MR (f)(x + hi,kel ) = ux+hi,k el ,f(0, 0, 0, 0) Ai,4 := k=1

Ai,5

if (0, 0, 0, 0) ∈ Bi (f)(x + hi,kel )}, ∞  := {x ∈ R2 : MR (f)(x + hi,kel ) = ux+hi,k el ,f(r1,1 , r1,2 , 0, 0) k=1

Ai,6

if (r1,1 , r1,2 , 0, 0) ∈ B1 (f)(x + hi,kel ) and r1,1 + r1,2 > 0}, : = {x ∈ R2 : MR (f)(x) = ux,f(r1,1 , r1,2 , 0, 0) if (r1,1 , r1,2 , 0, 0) ∈ B1 (f)(x)

Ai,7

and r1,1 + r1,2 > 0}, ∞  := {x ∈ R2 : MR (f)(x + hi,kel ) = ux+hi,k el ,f(0, 0, r2,1 , r2,2 ) k=1

Ai,8

if (0, 0, r2,1 , r2,2 ) ∈ B1 (f)(x + hi,kel ) and r2,1 + r2,2 > 0}, : = {x ∈ R2 : MR (f)(x) = ux,f(0, 0, r2,1 , r2,2 ) if (0, 0, r2,1 , r2,2 ) ∈ B1 (f)(x)

Ai,9

and r2,1 + r2,2 > 0}, ∞  := {x ∈ R2 : MR (f)(x + hi,kel ) = ux+hi,k el ,f(r) if r ∈ B2 (f)(x + hi,kel )}; k=1

Ai,10 := {x ∈ R2 : MR (f)(x) = ux,f(r) if r ∈ B2 (f)(x)}; ∞  {x ∈ R2 : MR (f)(x + hi,kel ) = ux+hi,k el ,f(r) if r ∈ B3 (f)(x + hi,kel )}; Ai,11 := k=1

Ai,12 := {x ∈ R2 : MR (f)(x) = ux,f(r) if r ∈ B3 (f )(x)}. Let Ai =

12 k=1

 Ai,k and A = A1 ∪ A2 ∪ A3 . Note that |RΛ  (0) \ A| = 0. Let x = (x1 , x2 ) ∈ A

be a Lebesgue point of all fμ and Dl fμ and r = (r1,1 , r1,2 , r2,1 , r2,2 ) ∈ Bi (f)(x). There exists rki = (r1,1,i,k , r1,2,i,k , r2,1,i,k , r2,2,i,k ) ∈ Bi (f)(x + hi,kel ) such that limk→∞ (r1,1,i,k , r1,2,i,k , r2,1,i,k , r2,2,i,k ) = (r1,1 , r1,2 , r2,1 , r2,2 ). Furthermore, we assume that x1 is a Lebesgue point of Dl fμ (·, x2 ) for i = 2, x2 is a Lebesgue point of Dl fμ (x1 , ·) for i = 3, and Dl fμ (x1 , ·)Lpμ (R) , Dl fμ (·, x2 )Lpμ (R) < ∞. Case A (r1,1 + r1,2 > 0 and r2,1 + r2,2 > 0). In this case r ∈ B1 (f)(x) and this happens when x ∈ A1 . Without loss of generality we may assume that all r1,1,1,k > 0, r1,2,1,k > 0, r2,1,1,k > 0 and r2,2,1,k > 0. Denote [x1 − r1,1,1,k , x1 + r1,2,1,k ] × [x2 − r2,1,1,k , x2 + r2,2,1,k ] by Rk and dy1 dy2 = dy . Then, noting rk ∈ B1 (f)(x + h1,kel ) and using Lemma 2.2 (ix), we have

JID:MATPUR

AID:3193 /FLA

[m3L; v1.283; Prn:20/02/2020; 11:41] P.13 (1-38) F. Liu et al. / J. Math. Pures Appl. ••• (••••) •••–•••

Dl MR (f)(x) = lim

k→∞

≤ lim

k→∞

= lim

k→∞

×

(fν )lτ (h1,k ) (y1 , y2 )dy1 dy2

k

1 m (r m (r + r ) 1,1 1,2 2,1 + r2,2 ) μ=1 μ−1 

¨ m 

w=μ+1 R

x1ˆ +r1,2 x2ˆ +r2,2

(fμ )lh1,k (y1 , y2 )dy1 dy2 Rk

fw (y1 , y2 )dy1 dy2

k

Dl fμ (y1 , y2 )dy x1 −r1,1 x2 −r2,1 x1ˆ +r1,2 x2ˆ +r2,2 m 

fν (y1 , y2 )dy

ν=1x −r 1 1,1 x2 −r2,1

(2.17)

x1ˆ +r1,2 x2ˆ +r2,2

m 

×

¨

(r1,1,1,k + r1,2,1,k )m (r2,1,1,k + r2,2,1,k )m μ=1

ν=1 R

=

1 (MR (f)(x + h1,kel ) − MR (f)(x)) h1,k

1 (u r1 ) − ux,f(rk1 )) ( h1,k x+h1,k el ,f k m  1

μ−1 ¨

13

fw (y1 , y2 )dy .

w=μ+1x −r 1 1,1 x2 −r2,1

Here, we used the fact that limk→∞ rk1 = r and (fμ )lτ (h1,k ) χRk → fμ χ[x1 −r1,1 ,x1 +r1,2 ]×[x2 −r2,1 ,x2 +r2,2 ] and (fμ )lh1,k χRk → Dl fμ χ[x1 −r1,1 ,x1 +r1,2 ]×[x2 −r2,1 ,x2 +r2,2 ] in L1 (R2 ) as k → ∞. Then

m 

1 Dl MR (f)(x) ≤ m (r m (r + r ) 1,1 1,2 2,1 + r2,2 ) μ=1 

×

x1ˆ +r1,2 x2ˆ +r2,2

Dl fμ (y1 , y2 )dy1 dy2 x1 −r1,1 x2 −r2,1

x1ˆ +r1,2 x2ˆ +r2,2

(2.18)

fν ((y1 , y2 ))dy1 dy2 .

ν =μ,1≤ν≤mx −r 1 1,1 x2 −r2,1

On the other hand, using Lemma 2.2 (ix), we have

Dl MR (f)(x) = lim

k→∞

≥ lim

1

k→∞

h1,k

1 (MR f(x + h1,kel ) − MR (f)(x)) h1,k

(ux+h1,k el ,f(r) − ux,f(r))

m 

1 = m (r1,1 + r1,2 ) (r2,1 + r2,2 )m μ=1 ×



+r1,2 x2ˆ +r2,2 x1ˆ

x1ˆ +r1,2 x2ˆ +r2,2

Dl fμ (y1 , y2 )dy1 dy2

(2.19)

x1 −r1,1 x2 −r2,1

fν ((y1 , y2 ))dy1 dy2 .

1≤ν =μ≤mx −r 1 1,1 x2 −r2,1

 Combining (2.19) with (2.18) yields (2.13) for a.e. x ∈ RΛ  (0) ∩ A1 . Case B (r1,1 + r1,2 > 0 and r2,1 = r2,2 = 0). We consider the following two cases. (i) (r1,1 , r1,2 , 0, 0) ∈ B2 (f)(x). This happens in the case x ∈ A2 . Without loss of generality we may assume that all r1,1,2,k , r1,2,2,k > 0. We notice that r2,1,2,k = r2,2,2,k = 0 for all k ≥ 1. Then, noting rk ∈ B2 (f)(x + h2,kel ) and using Lemma 2.2 (vii), we have

JID:MATPUR

AID:3193 /FLA

[m3L; v1.283; Prn:20/02/2020; 11:41] P.14 (1-38) F. Liu et al. / J. Math. Pures Appl. ••• (••••) •••–•••

14

1 (MR (f)(x + h2,kel ) − MR (f)(x)) h2,k

Dl MR (f)(x) = lim

k→∞

1 (u (r1,1,2,k , r1,2,2,k , 0, 0) − ux,f(r1,1,2,k , r1,2,2,k , 0, 0)) h2,k x+h2,k el ,f x1 +r ˆ1,2,2,k m 1  1 (fμ )lh2,k (y1 , x2 )dy1 ≤ lim m k→∞ h2,k (r + r ) 1,1,2,k 1,2,2,k μ=1

≤ lim

k→∞

×

μ−1 

x1 −r1,1,2,k

x1 +r ˆ1,2,2,k

(fν )lτ (h2,k ) (y1 , x2 )dy1

ν=1x −r 1 1,1,2,k m 

1 ≤ (r1,1 + r1,2 )m μ=1

m 

x1 +r ˆ1,2,2,k

fw (y1 , x2 )dy1

w=μ+1x −r 1 1,1,2,k

+r1,2 x1ˆ

(2.20)



Dl fμ (y1 , x2 )dy1

x1ˆ +r1,2

fν (y1 , x2 )dy1 .

ν =μ,1≤ν≤mx −r 1 1,1

x1 −r1,1

Here we used the fact that limk→∞ r1,1,2,k = r1,1 , limk→∞ r1,2,2,k = r1,2 and (fμ )lh2,k (·, x2 )χ[x1 −r1,1,2,k ,x1 +r1,2,2,k ] → Dl fμ (·, x2 )χ[x1 −r1,1 ,x1 +r1,2 ] in L1 (R) as k → ∞. Further more, using Lemma 2.2 (vii), we have Dl MR (f)(x) 1 ≥ lim (ux+h2,k el ,f(r1,1 , r1,2 , 0, 0) − ux,f(r1,1 , r1,2 , 0, 0)) k→∞ h2,k x1 +r ˆ1,2,2,k m  1 (fμ )lh2,k (y1 , x2 )dy1 ≥ lim m k→∞ (r + r ) 1,1,2,k 1,2,2,k μ=1 ×

μ−1 

x1 −r1,1,2,k

x1 +r ˆ1,2,2,k

ν=1x −r 1 1,1,2,k m 

x1 +r ˆ1,2,2,k

m 

(fν )lτ (h2,k ) ((y1 , x2 ))dy1

(2.21)

fw ((y1 , x2 ))dy1

w=μ+1x −r 1 1,1,2,k

1 ≥ (r1,1 + r1,2 )m μ=1

x1ˆ +r1,2

Dl fμ (y1 , x2 )dy1

x1ˆ +r1,2



fν (y1 , x2 )dy1 .

ν =μ,1≤ν≤mx −r 1 1,1

x1 −r1,1

 (2.21) together with (2.20) yields (2.14) for a.e. x ∈ RΛ  (0) ∩ A2 .  (ii) (r1,1 , r1,2 , 0, 0) ∈ B1 (f )(x). This happens in the case x ∈ A1 . Assume that r1,1,1,k , r1,2,1,k > 0. As in the case A, noting x ∈ A1 ⊂ A1,5 , we have m 

1 k→∞ (r1,1,1,k + r1,2,1,k )m μ=1

Dl MR (f)(x) ≤ lim

×

μ−1 

x+r ˆ1,2,1,k

(fμ )lh1,k (y1 , x2 )dy1 x−r1,1,1,k

x+r ˆ1,2,1,k

m 

(fν )lτ (h1,k) (y1 , x2 )dy1

ν=1x−r 1,1,1,k

(2.22)

x+r ˆ1,2,1,k

fw (y1 , x2 )dy1 .

w=μ+1x−r 1,1,1,k

We claim that the limits of the right side will tend to m 

1 (r1,1 + r1,2 )m μ=1

x1ˆ +r1,2

Dl fμ (y1 , x2 )dy1 x1 −r1,1



+r1,2 x1ˆ

ν =μ,1≤ν≤mx −r 1 1,1

fν (y1 , x2 )dy1 .

JID:MATPUR

AID:3193 /FLA

[m3L; v1.283; Prn:20/02/2020; 11:41] P.15 (1-38) F. Liu et al. / J. Math. Pures Appl. ••• (••••) •••–•••

15

To see this, we only consider the limit of the following parts, since the same reasoning applies to the other terms.

1 r1,1,1,k + r1,2,1,k

x+r ˆ1,2,1,k

(fμ )lh1,k (y1 , x2 )dy1 . x−r1,1,1,k

Now, we know from the property (i) for x ∈ A1 that   1 lim  k→∞ r1,1,1,k + r1,2,1,k

x+r ˆ1,2,1,k

(fμ )lh1,k (y1 , x2 )

  − Dl fμ (y1 , x2 ))dy1 

x−r1,1,1,k

1 ((fμ )l − Dl fμ )(x1 , x2 ) ≤ lim M h1,k

(2.23)

k→∞

  1 + lim  k→∞ r1,1,1,k + r1,2,1,k

x+r ˆ1,2,1,k

  Dl fμ (y1 , x2 ) − Dl fμ (y1 , x2 ))dy1 

x−r1,1,1,k

= 0. We see moreover that x +r  

1 ˆ1,2,1,k   1 1  lim  − Dl fμ (y1 , x2 )dy1  k→∞ r1,1,1,k + r1,2,1,k r1,1 + r1,2 x1 −r1,1,1,k   1 1   ≤ lim (r1,1,1,k + r1,2,1,k ) − M1 (Dl fμ )(x1 , x2 ) = 0. k→∞ r1,1,1,k + r1,2,1,k r1,1 + r1,2

(2.24)

Noting that D fμ (·, x2 )Lp (R)) < ∞, we get   1 lim  k→∞ r1,1 + r1,2

x1 +r ˆ1,2,1,k

x1 −r1,1,1,k

1 Dl fμ (y1 , x2 )dy1 − r1,1 + r1,2 

+r1,2 x1ˆ

  Dl fμ (y1 , x2 )dy1 

x1 −r1,1

(|r1,1,1,k − r1,1 | + |r1,2,1,k − r1,2 |)1/p ≤ lim k→∞ r1,1 + r1,2 1/p  x1 −min{rˆ1,1 ,r1,1,1,k } x1 +max{r ˆ1,2 ,r1,2,1,k } × + |Dl fμ (y1 , x2 ))|p dy1 x1 −max{r1,1 ,r1,1,1,k }

x1 +min{r1,2 ,r1,2,1,k }

(2.25)



(|r1,1,1,k − r1,1 | + |r1,2,1,k − r1,2 |)1/p ≤ C lim Dl fμ (·, x2 )Lp (R)) = 0. k→∞ r1,1 + r1,2 From (2.22) to (2.25), it follows that

1 lim k→∞ r1,1,1,k + r1,2,1,k

x1 +r ˆ1,2,1,k

(fμ )lh1,k (y1 , x2 )dy1 x1 −r1,1,1,k

and hence we verified the claim.

1 = r1,1 + r1,2

+r1,2 x1ˆ

Dl fμ (y1 , x2 )dy1 , x1 −r1,1

(2.26)

JID:MATPUR

AID:3193 /FLA

[m3L; v1.283; Prn:20/02/2020; 11:41] P.16 (1-38) F. Liu et al. / J. Math. Pures Appl. ••• (••••) •••–•••

16

On the other hand, noting x ∈ A1 ⊂ A1,6 , by the same reasoning as in the case A, we get m 

1 Dl MR (f)(x) ≥ (r + r1,2 )m 1,1 μ=1 m 

×

x1ˆ +r1,2

Dl fμ (y1 , x2 )dy1

μ−1 

+r1,2 x1ˆ

fν (y1 , x2 )dy1

ν=1x −r 1 1,1

x1 −r1,1,

(2.27)

x1ˆ +r1,2

fw (y1 , x2 )dy1

w=μ+1x −r 1 1,1

 The above claim and (2.27) yield (2.14) for a.e. x ∈ RΛ  (0) ∩ A1 . Case C (r1,1 = r1,2 = 0 and r2,1 + r2,2 > 0). Similar argument as in Case B gives (2.15) for a.e.  x ∈ RΛ  (0) ∩ (A1 ∪ A2 ). Case D (r = (0, 0, 0, 0)). We consider the following three cases: (i) Assume that (0, 0, 0, 0) ∈ B2 (f)(x). Then x ∈ A2 . The lower bound of Dl MR (f)(x) follows from Dl MR (f)(x) 1

= lim

k→∞

≥ lim

≥

μ=1

≥

m 

(MR (f)(x + h2,k el ) − MR (f)(x))

m 1 

k→∞ m 

h2,k h2,k

lim

k→∞

fμ (x + h2,kel ) −

μ=1

1 h2,k

fμ (x)

μ=1

(fμ (x + h2,k el ) − fμ (x))

(2.28) μ−1 

m



fν (x)

ν=1



fj (x + h2,kel )

j=μ+1

fi (x) .



Dl fμ (x)

μ=1

m 

i =μ,1≤i≤m

To get the upper bound of Dl MR (f)(x), note that limk→∞ r1,1,2,k = 0, limk→∞ r1,2,2,k = 0 and r2,1,2,k = r2,2,2,k = 0 for all k ≥ 1. If r1,1,k + r1,2,k = 0 for infinitely many k, then by Lemma 2.2 (iv), one obtains that 1 (MR (f)(x + h2,kel ) − MR (f)(x)) h2,k m m

 1  ≤ lim fμ (x + h2,kel ) − fμ (x) k→∞ h2,k μ=1 μ=1

Dl MR (f)(x) = lim

k→∞

≤

m 

Dl fμ (x)

μ=1

(2.29)

fν (x) .

 ν =μ,1≤ν≤m

If there exists k0 ∈ N such that r1,1,2,k + r1,2,2,k > 0 when k ≥ k0 . Then (2.20) gives that m 

1 Dl MR (f)(x) ≤ lim k→∞ r1,1,2,k + r1,2,2,k μ=1 x1 +r ˆ1,2,2,k

× x1 −r1,2,2,k

fν (y1 , x2 )dy1

m



x1 +r ˆ1,2,2,k

(fμ )lh2,k (y1 , x2 )dy1

ν=1

x1 −r1,1,2,k

1 r + r1,2,2,k j=ν+1 1,1,2,k

μ−1 

1 r1,1,2,k + r1,2,2,k

x1 +r ˆ1,2,2,k

(fj )lτ (h2,k ) (y1 , x2 )dy1 .

x1 −r1,1,2,k

(2.30)

JID:MATPUR

AID:3193 /FLA

[m3L; v1.283; Prn:20/02/2020; 11:41] P.17 (1-38) F. Liu et al. / J. Math. Pures Appl. ••• (••••) •••–•••

17

Since x1 is a Lebesgue point for Dl fμ (·, x2 ), we have   lim 

k→∞

1 r1,1,2,k + r1,2,2,k

x1 +r ˆ1,2,2,k

  (fμ )lh2,k (y1 , x2 )dy1 − Dl fμ (x1 , x2 )

x1 −r1,1,2,k x1 +r ˆ1,2,2,k

≤ lim

1 r1,1,2,k + r1,2,2,k

≤ lim

1 ((fμ )l M h2,k

k→∞

k→∞

|(fμ )lh2,k (y1 , x2 ) − Dl fμ (y1 , x2 ))|dy1

(2.31)

x1 −r1,1,2,k

− Dl fμ )(x) x1 +r ˆ1,2,2,k

1 + lim k→∞ r1,1,2,k + r1,2,2,k

|Dl fμ (y1 , x2 ) − Dl fμ (y1 , x2 ))|dy1 = 0. x1 −r1,1,2,k

´ x1 +r1,2,2,k

1 Similarly, it holds that limk→∞ r1,1,2,k +r 1,2,2,k We get from (2.30) and (2.31) that

Dl MR (f)(x) ≤

x1 −r1,1,2,k

(fμ )lτ (h2,k ) (y1 , x2 )dy1 = fμ (x1 , x2 ).

m  μ=1



Dl fμ (x1 , x2 )

fν (x1 , x2 )

(2.32)

1≤ν =μ≤m

 (2.32) together with (2.28)-(2.29) yields (2.16) in the case 0 ∈ B2 (f)(x) for a.e. x ∈ RΛ  (0).  (ii) Assume that (0, 0, 0, 0) ∈ B3 (f)(x). We can get (2.16) for a.e. x ∈ RΛ  (0) similarly.  (iii) Assume that (0, 0, 0, 0) ∈ B1 (f )(x). In the case x ∈ A1 . Note that Dl MR (f)(x) = lim ≥

k→∞ m 

1 h1,k

(MR (f)(x + h1,k el ) − MR (f)(x))

μ=1

fν (x) .



Dl fμ (x)

(2.33)

ν =μ,1≤ν≤m

Below we estimate the upper bound of Dl MR (f)(x). We consider the following four cases: (a) If (r1,1,k , r1,2,k , r2,1,k , r2,2,k ) = (0, 0, 0, 0) for infinitely many k, then Dl MR (f)(x) = lim

k→∞

≤

m  μ=1

≤

m  μ=1

1 (MR (f)(x + h1,kel ) − MR (f)(x)) h1,k

lim

k→∞

1 h1,k

(fμ (x + h1,kel ) − fμ (x))

μ−1 

m



fν (x)

ν=1

Dl fμ (x1 , x2 )



fj (x + h1,k el )

j=μ+1

fi (x1 , x2 ) .

i =μ,1≤i≤m

This leads to the desired results. (b) Denote [x1 − r1,1,1,k , x1 + r1,2,1,k ] × [x2 − r2,1,1,k , x2 + r2,2,2,k ] by Rk . If there exists k0 ∈ N such that r1,1,1,k + r1,2,1,k > 0 and r2,1,1,k + r2,2,1,k > 0 when k ≥ k0 . Then (2.17) gives that Dl MR (f)(x) ≤

×

μ−1  ν=1

m 

1 lim k→∞ (r1,1,1,k + r1,2,1,k )(r2,1,1,k + r2,2,1,k ) μ=1

1 (r1,1,1,k + r1,2,1,k )(r2,1,1,k + r2,2,1,k )

¨ (fμ )lh1,k (y1 , y2 )dy1 dy2 Rk

¨

fν (y1 , y2 )dy1 dy2

Rk

(2.34)

JID:MATPUR

AID:3193 /FLA

[m3L; v1.283; Prn:20/02/2020; 11:41] P.18 (1-38) F. Liu et al. / J. Math. Pures Appl. ••• (••••) •••–•••

18 m 

×

1 (r + r )(r 1,1,1,k 1,2,1,k 2,1,1,k + r2,2,1,k ) j=μ+1

¨ (fj )lτ (h1,k ) (y1 , y2 )dy1 dy2 . Rk

Since (x1 , x2 ) is a Lebesgue point for Dl fμ , then   lim 

k→∞

1 (r1,1,1,k + r1,2,1,k )(r2,1,1,k + r2,2,1,k )

¨

  (fμ )lh1,k (y1 , y2 )dy1 dy2 − Dl fμ (x1 , x2 )

Rk

≤ lim MR ((fμ )lh1,k − Dl fμ )(x1 , x2 ) k→∞

  + lim  k→∞

1 (r1,1,1,k + r1,2,1,k )(r2,1,1,k + r2,2,1,k )

¨

  Dl fμ (y1 , y2 ) − Dl fμ (x1 , x2 )dy1 dy2 

(2.35)

Rk

= 0. Similarly, we have   lim 

k→∞

1 (r1,1,1,k + r1,1,2,k )(r1,2,1,k + r1,2,2,k )

¨

  (fμ )lτ (h1,k ) (y1 , y2 )dy1 dy2 − fμ (x1 , x2 ) = 0.

(2.36)

Rk

(2.34) together with (2.35)-(2.36) yields the desired estimate. (c) If there exists k0 ∈ N such that r1,1,1,k + r1,2,1,k > 0 when k ≥ k0 and r2,1,1,k = r2,2,1,k = 0 for infinitely many k. Then we may have

Dl MR (f)(x) ≤

m  μ=1





Dl fμ (x1 , x2 )

fν (x1 , x2 ) .

ν =μ,1≤ν≤m

This shows the desired upper bounds. (d) If there exists k0 ∈ N such that r2,1,1,k + r2,2,1,k > 0 when k ≥ k0 and r1,1,1,k = r1,2,1,k = 0 for infinitely many k, we can get the upper bounds by the arguments similar to those used in the case (c).  (2.33) together with (a)-(d) yields (2.16) for a.e. x ∈ RΛ  (0). Since Λ is arbitrary, this proves Lemma 2.5.  2.2. Proof of Theorem 1.1 We divide the proof into three steps: m Step 1: The boundedness part. Let 1 < p1 , . . . , pm , p < ∞ and 1/p = i=1 1/pi . Let f = (f1 , . . . , fm ) with each fi ∈ W 1,pi (Rd ). For a function u and y ∈ Rd we define uh (x) = u(x +h). According to [13, Section 7.11] we know that u ∈ W 1,p (Rd ) for 1 < p < ∞ if and only if u ∈ Lp (Rd ) and lim suph→0 uh −uLp (Rd ) /|h| < ∞. Therefore, we have

lim sup h→0

(fi )h − fi Lp (Rd ) < ∞, |h|

On the other hand, for any fixed h ∈ Rd and x ∈ Rd , we have

∀1 ≤ i ≤ m.

(2.37)

JID:MATPUR

AID:3193 /FLA

[m3L; v1.283; Prn:20/02/2020; 11:41] P.19 (1-38) F. Liu et al. / J. Math. Pures Appl. ••• (••••) •••–•••

19

m ˆ m ˆ   1    |f (y + h)|dy − |fi (y)|dy   i m |R| Rx i=1 R i=1 R R∈R ˆ m  1 ≤ sup |fi (y + h) − fi (y)|dy m |R| Rx i=1 R∈R R ˆ m



i−1 ˆ |fμ (y)|dy |fi (y + h)|dy ×

|(MR (f))h (x) − MR (f)(x)| ≤ sup

≤

m 

μ=1 R

(2.38)

ν=μ+1 R

MR (fhi )(x),

i=1

where fhi = (f1 , . . . , fi−1 , (fi )h − fi , (fi+1 )h , . . . , (fm )h ). (2.38) together with (1.2) yields that (MR (f))h − MR (f)Lp (Rd ) ≤ m,d,p1 ,...,pm

m 

m 

MR (fhi )Lp (Rd )

i=1

i=1

(2.39)



(fi )h − fi Lpi (Rd )

fμ Lpμ (Rd ) .

μ =i,1≤μ≤m

(MR (f))h −MR (f)

L (R ) We get from (2.39) and (2.37) that lim suph→0 < ∞. This together with the fact |h| p d 1,p d that MR (f) ∈ L (R ) yields that MR (f) ∈ W (R ). Step 2: Pointwise estimate for MR (f). Let sk (k = 1, 2, . . .) be an enumeration of positive rational numbers. We can write

MR (f)(x) =

sup

2d  r ∈({sk }∞ k=1 )

p

m ˆ  1 |Rr (x)|m i=1

d

|fi (y)|dy,

R r (x)

where r = (r1− , . . . , rd− ; r1+ , . . . , rd+ ) and Rr (x) = (x1 − r1− , x1 + r1+ ) × · · · × (xd − rd− , xd + rd+ ). Fixing k ≥ 1, − + we let Ek = {r = (r1− , . . . , rd− ; r1+ , . . . , rd+ ) ∈ R2d + ; ri , ri ∈ {s1 , . . . , sk }, i = 1, 2, . . . , d}. For k ∈ {1, 2, . . .}, we define the operator Tk by Tk (f)(x) = max  r ∈Ek

m ˆ  1 |Rr (x)|m i=1

|fi (y)|dy.

R r (x)

For any h ∈ Rd , we can write |Tk (f)(x + h) − Tk (f)(x)| ≤

m 

1 max m  r ∈Ek |R r (x)| i=1 i−1  ˆ

ˆ |fi (y + h) − fi (y)|dy R r (x)

|fμ (y)|dy

×

μ=1 R r (x)

m



ˆ

|fi (y + h)|dy .

ν=μ+1 R r (x)

This yields that |Dl (Tk (f))(x)| ≤

m  i=1

Tk (fil )(x) ≤

m 

MR (fil )(x),

for a.e. x ∈ Rd .

i=1

Here fil = (f1 , . . . , fi−1 , Dl fi , fi+1 , . . . , fm ). For all k ≥ 1, by (2.40) and (1.2), it holds that

(2.40)

JID:MATPUR

AID:3193 /FLA

[m3L; v1.283; Prn:20/02/2020; 11:41] P.20 (1-38) F. Liu et al. / J. Math. Pures Appl. ••• (••••) •••–•••

20

Tk (f)1,p ≤ Tk (f)Lp (Rd ) +

d 

Dl Tk (f)Lp (Rd )

l=1 m d  

≤ MR (f)Lp (Rd ) + m 

m,p1 ,...,pm

MR (fil )Lp (Rd )

l=1 i=1

fi 1,pi .

i=1

This yields that {Tk (f)}k is a bounded sequence in W 1,p (Rd ) which converges to MR (f) pointwisely. The weak compactness of Sobolev spaces implies that {Dl (Tk (f))}k converges to Dl (MR (f)) weakly in Lp (Rd ). This together with (2.40) implies that

|Dl MR (f)(x)| ≤

m 

MR (fil )(x),

for a.e. x ∈ Rd .

i=1

Combining this with (1.2) yields that

∇MR (f)Lp (Rd ) ≤

d 

Dl MR (f)Lp (Rd ) ≤

l=1

m,d,p1 ,...,pm

m  d 

m d   l=1 i=1

Dl fi Lpi (Rd )

i=1 l=1

MR (fil )Lp (Rd ) 

fj Lpj (Rd ) .

j =i,1≤j≤m

Therefore, it holds that MR (f)1,p = MR (f)Lp (Rd ) + ∇MR (f)Lp (Rd ) ≤ Cm,d,p1 ,...,pm

m 

fi 1,pi .

(2.41)

i=1

Step 3: The continuity part. For convenience, we only prove the case d = 2 and the case d > 2 is analogous and more complex, we leave the details to the interested reader. Let f = (f1 , . . . , fm ) with each fi ∈ W 1,pi (R2 ) for 1 < pi < ∞. Let fj = (f1,j , . . . , fm,j ) such that fi,j → fi in W 1,pi (R2 ) when j → ∞. Let m 1 < p < ∞ and 1/p = i=1 1/pi . It follows from (2.6) that MR (fj ) − MR (f)Lp (R2 ) → 0 when j → ∞. Thus, it suffices to show that, for any l = 1, 2, . . . , d, it holds that Dl MR (fj ) − Dl MR (f)Lp (R2 ) → 0

when j → ∞.

(2.42)

Without loss of generality we may assume that all fi,j ≥ 0 and fi ≥ 0. Given  > 0 and l = 1, 2, letting fli = (f1 , . . . , fi−1 , Dl fi , fi+1 , . . . , fm ), there exists Λ > 0 such that m  i  <  with B1 = R2 \ RΛ  (0). Here Λ = (Λ, Λ). By the absolute continuity, there exists 1 i=1 MR (fl )p,B m i   η > 0 such that i=1 MR (fl )p,A <  whenever A is a measurable subset of RΛ  (0) such that |A| < η. As 2 we already observed, for a.e. x ∈ R , we notice that: 4 (i) ux,fi is continuous on R+ and lim ux,fi (r1,1 , r1,2 , r2,1 , r2,2 ) = 0; 4 l

l (r1,1 ,r1,2 ,r2,1 ,r2,2 )∈R+ r1,1 +r1,2 +r1,2 +r2,2 →∞ 2 (ii) ux,fi (r1,1 , r1,2 , 0, 0) is continuous on R+ and lim 2 ux,fi (r1,1 , r1,2 , 0, 0) = 0; l l (r1,1 ,r1,2 )∈R+ r1,1 +r1,2 →∞ 2 (iii) ux,fi (0, 0, r2,1 , r2,2 ) is continuous on R+ and lim 2 ux,fi (0, 0, r2,1 , r2,2 ) = 0. l l (r2,1 ,r2,2 )∈R+ r2,1 +r2,2 →∞

JID:MATPUR

AID:3193 /FLA

[m3L; v1.283; Prn:20/02/2020; 11:41] P.21 (1-38) F. Liu et al. / J. Math. Pures Appl. ••• (••••) •••–•••

21

m 4 Then, it follows that for a.e. x ∈ R2 , the function i=1 ux,fi (·, ·, ·, ·) is uniformly continuous on R+ ; the l m m 2 function i=1 ux,fi (·, ·, 0, 0) is uniformly continuous on R+ ; the function i=1 ux,fi (0, 0, ·, ·) is uniformly 2

l

l

continuous on R+ . Hence, we can find δx > 0 such that m m     −1/p ; (iv) If |r1 − r2 | < δx , then  ux,fi (r1 ) − ux,fi (r2 ) < |RΛ  (0)| l

i=1

l

i=1

(v) If |r1,1,1 − r2,1,1 | + |r1,1,2 − r2,1,2 | < δx , then m   u

x,fli

(r1,1,1 , r1,1,2 , 0, 0) −

m 

  −1/p ; ux,fi (r2,1,1 , r2,1,2 , 0, 0) < |RΛ  (0)| l

i=1

i=1

(vi) If |r1,2,1 − r2,2,1 | + |r1,2,2 − r2,2,2 | < δx , then m   u

x,fi (0, 0, r1,2,1 , r1,2,2 ) −

m 

l

  −1/p . ux,fi (0, 0, r2,2,1 , r2,2,2 ) < |RΛ  (0)| l

i=1

i=1

Now we can write  RΛ  (0) =

∞ 

1 

 N, x ∈ RΛ  (0) : δx > i i=1

where |N | = 0. It follows that there exists δ > 0 such that  |{x ∈ RΛ  (0) : |

m i=1

ux,fi (r1 ) − l

m

r2 )| i=1 ux,fli ( η |B2,1 | < 2 ;

 −1/p  for some r1 , r2 ≥ |RΛ  (0)|

with |r1 − r2 | < δ}| =: m m   −1/p  |{x ∈ RΛ  (0) : |  (0)| i=1 ux,fi (r1,1,1 , r1,1,2 , 0, 0) − i=1 ux,fi (r2,1,1 , r2,1,2 , 0, 0)| ≥ |RΛ l

(2.43)

l

for some r1,1,1 , r1,1,2 , r2,1,1 , r2,1,2 with |r1,1,1 − r2,1,1 | + |r1,1,2 − r2,1,2 | < δ}| =: |B2,2 | < η2 ; m m   −1/p  |{x ∈ RΛ  (0) : |  (0)| i=1 ux,fi (0, 0, r1,2,1 , r1,2,2 ) − i=1 ux,fi (0, 0, r2,2,1 , r2,2,2 )| ≥ |RΛ

(2.44)

for some r1,2,1 , r1,2,2 , r2,2,1 , r2,2,2 with |r1,2,1 − r2,2,1 | + |r1,2,2 − r2,2,2 | < δ}| =: |B2,3 | < η2 .

(2.45)

l

l

Applying Lemma 2.3, there exists j1 ∈ N such that for i = 1, 2, 3 η i,j    when j ≥ j1 . |{x ∈ RΛ  (0); Bi (fj )(x)  Bi (f )(x)(δ) }| =: |B | < 2

(2.46)

Let fli,j = (f1,j , . . . , fi−1,j , Dl fi,j , fi+1,j , . . . , fm,j ). Fix i = 1, 2, 3. Invoking Lemma 2.5, for a.e. x ∈ R2 , j ≥ j1 , and for any r1 ∈ Bi (fj )(x) and r2 ∈ Bi (f)(x) with i = 1, 2, 3, we have     Dl MR (fj )(x) − Dl MR (f)(x) m m      = ux,fi,j (r1 ) − ux,fi (r2 ) l

i=1

≤

m 

m m      |ux,fi,j (r1 ) − ux,fi (r1 )| +  ux,fi (r1 ) − ux,fi (r2 ). l

i=1

l

i=1

l

l

i=1

l

i=1

If x ∈ / B1 ∪ B2,i ∪ B i,j , we choose r1 ∈ Bi (fj )(x) and r2 ∈ Bi (f)(x) such that |r1 − r2 | < δ and

(2.47)

JID:MATPUR

AID:3193 /FLA

[m3L; v1.283; Prn:20/02/2020; 11:41] P.22 (1-38) F. Liu et al. / J. Math. Pures Appl. ••• (••••) •••–•••

22

m m       −1/p . u ( r  ) − ux,fi (r2 ) < |RΛ   (0)| x,fi 1 l

(2.48)

l

i=1

i=1

On the other hand, for any r1 ∈ Bi (fj )(x) and r2 ∈ Bi (f)(x), one may obtain that m m m       ux,fi (r1 ) − ux,fi (r2 ) ≤ 2 MR (fli )(x),  l

l

i=1

i=1

for a.e. x ∈ R2 .

(2.49)

i=1

m

|ux,fi,j (r1 ) − ux,fi (r1 )|, we consider the following cases: l ˜ ´ xl +r1,2 ´ x2 +r2,2 4 Case 1. For simplicity, we denote R0 = x11−r1,1 . If r1 = (r1,1 , r1,2 , r2,1 , r2,2 ) ∈ R+ with r1,1 + x2 −r2,1 r1,2 > 0 and r2,1 + r2,2 > 0. Then

To get the estimate of

i=1

|ux,fi,j (r1 ) − ux,fi (r1 )| l

=

l

2 

(rw,1 + rw,2 )

w=1

×

μ=1 R



¨

m 

i−1 ¨ μ=1 R

≤

i−1 

2 

0

¨ fμ (y1 , y2 )dy1 dy2

Dl fi (y1 , y2 )dy1 dy2

ν=i+1 R

(rw,1 + rw,2 )−m

μ−1 ¨

¨ f (y1 , y2 )dy1 y2

=1 R 0

i−1  ¨

R0 fκ,j (y1 , y2 )dy1 dy2

Dl fi,j (y1 , y2 )dy1 dy2



¨ m 

¨ fκ (y1 , y2 )dy1 dy2

fτ,j (y1 , y2 )dy1 dy2 +

i−1  μ=1

l MR (Fμ,j )(x) +

2 

(rw,1 + rw,2 )−m

w=1

|Dl fi,j − Dl fi |(y1 , y2 )dy1 dy2

f (y1 , y2 )dy1 dy2

0

|fν,j − fν |(y1 , y2 )dy1 dy2

R0

0

0

i−1 ¨ =1 R

¨

m ¨ 

i−1 ¨

R0 fκ (y1 , y2 )dy1 dy2

κ=1

R0 fτ,j (y1 , y2 )dy1 dy2

τ =i+1

R0

≤

ν−1 

κ=i+1 R

R0

¨

(rw,1 + rw,2 )−m

ν=i+1 w=1

Dl fi (y1 , y2 )dy1 dy2

×

m 2  

R0 fτ,j (y1 , y2 )dy1 dy2 +

¨

τ =ν+1 R

|fμ,j − fμ |(y1 , y2 )dy1 dy2

R0

τ =i+1

×

0

R0

m ¨ 

×

  fν (y1 , y2 )dy1 dy2 

¨

κ=μ+1

×

m ¨



R0

0

μ=1 w=1

×

R0

0

fν,j (y1 , y2 )dy1 dy2

ν=i+1 R

−

 i−1

¨

¨ fμ,j (y1 , y2 )dy1 dy2 Dl fi,j (y1 , y2 )dy1 dy2 

−m 

m 

 l )(x) + MR (H  l )(x) =: G l (x), MR (G ν,j i,j i,j

ν=i+1

l l ,H  l are defined where Fμ,j = (f1 , . . . , fμ−1 , fμ,j − fμ , fμ+1,j , . . . , fi−1,j , Dl fi,j , fi+1,j , . . . , fm,j ), and G ν,j i,j l l   by G = (f , . . . , f , D f , f , . . . , f , f − f , f , . . . , f ) and = (f , . . . , fi−1 , Dl fi,j − H 1 i−1 l i i+1 ν−1 ν,j ν ν+1,j m,j 1 ν,j i,j Dl fi , fi+1,j , . . . , fm,j ).

JID:MATPUR

AID:3193 /FLA

[m3L; v1.283; Prn:20/02/2020; 11:41] P.23 (1-38) F. Liu et al. / J. Math. Pures Appl. ••• (••••) •••–•••

23

Case 2. If r1 = (0, 0, 0, 0), then |ux,fi,j (r1 ) − ux,fi (r1 )| ≤ l

i−1 μ−1  

l

μ=1

m

i−1



 f (x) (fμ,j − fμ )(x) fκ,j (x) Dl fi,j (x) fτ,j (x) κ=μ+1

=1

m i−1  

+

ν=i+1



i−1 

+

τ =i+1

m

ν−1



 f (x) Dl fi (x) fκ (x) (fν,j − fν )(x) fτ,j (x) τ =ν+1

κ=i+1

=1

fκ (x) (Dl fi,j − Dl fi )(x)

κ=1

m 

fτ,j (x) .

τ =i+1

4

Case 3. If r1 = (0, 0, r2,1 , r2,2 ) ∈ R+ for r2,1 + r2,2 > 0, then |ux,fi,j (r1 ) − ux,fi (r1 )| l

l

  1  =  (r2,1 + r2,2 )m μ=1 i−1

+r2,2 x2ˆ

fμ,j (x1 , y2 )dy2

x2 −r2,1

×



x2ˆ +r2,2

Dl fi,j (x1 , y2 )dy2

m



−

fν,j (x1 , y2 )dy2

ν=i+1x −r 2 2,1

x2 −r2,1

i−1 

x2ˆ +r2,2

x2ˆ +r2,2

fμ (x1 , y2 )dy2

μ=1x −r 2 2,1

×



x2ˆ +r2,2

Dl fi (x1 , y2 )dy2

x2ˆ +r2,2

m



ν=i+1x −r 2 2,1

x2 −r2,1

 1 ≤ (r2,1 + r2,2 )m μ=1 i−1 

μ−1

x2ˆ +r2,2

×

m 

|fμ,j − fμ )|(x1 , y2 )dy2

x2 −r2,1

x2ˆ +r2,2

x2ˆ +r2,2

fκ,j (x1 , y2 )dy2

Dl fi,j (x1 , y2 )dy2

κ=μ+1x −r 2 2,1

×

x2ˆ +r2,2

f (x1 , y2 )dy2

=1 x −r 2 2,1

i−1 

  fν (x1 , y2 )dy2 

x2 −r2,1

x2ˆ +r2,2

fτ,j (x1 , y2 )dy2

τ =i+1x −r 2 2,1

 1 + (r2,1 + r2,2 )m ν=i+1 m 

i−1

x2ˆ +r2,2

f (x1 , y2 )dy2

=1x −r 2 2,1

×

ν−1 

x2ˆ +r2,2

×

m 

τ =ν+1x −r 2 2,1

fτ,j (x1 , y2 )dy2

x2 −r2,1

|fν,j − fν |(x1 , y2 )dy2

x2 −r2,1

x2ˆ +r2,2

Dl fi (x1 , y2 )dy2

x2ˆ +r2,2

fκ (x1 , y2 )dy2

κ=i+1x −r 2 2,1

x2ˆ +r2,2

JID:MATPUR

AID:3193 /FLA

[m3L; v1.283; Prn:20/02/2020; 11:41] P.24 (1-38) F. Liu et al. / J. Math. Pures Appl. ••• (••••) •••–•••

24

x2ˆ +r2,2

 1 + (r2,1 + r2,2 )m κ=1 i−1

×

x2ˆ +r2,2

|Dl fi,j − Dl fi |(x1 , y2 )dy2

fκ (x1 , y2 )dy2

x2 −r2,1

m 

x2ˆ +r2,2

x2 −r2,1

fτ,j (x1 , y2 )dy2 .

τ =i+1x −r 2 2,1 4

Case 4. If r1 = (r1,1 , r1,2 , 0, 0) ∈ R+ with r1,1 + r1,2 > 0. Then, similarly as in Case 3, we can obtain |ux,fi,j (r1 ) − ux,fi (r1 )| l

l

 1 ≤ m (r1,1 + r1,2 ) μ=1 i−1 

μ−1

x1ˆ +r1,2

x1 −r1,1

x1ˆ +r1,2

×

x1ˆ +r1,2

fκ,j (y1 , x2 )dy1

Dl fi,j (y1 , x2 )dy1

κ=μ+1x −r 1 1,1 m 

|fμ,j − fμ )|(y1 , x2 )dy1

f (y1 , x2 )dy1

=1 x −r 1 1,1

i−1 

x1ˆ +r1,2

x1 −r1,1

x1ˆ +r1,2

×

fτ,j (y1 , x2 )dy1

τ =i+1x −r 1 1,1

 1 + (r1,1 + r1,2 )m ν=i+1 m 

i−1

x1ˆ +r1,2

f (y1 , x2 )dy1

=1x −r 1 1,1

ν−1 

x1ˆ +r1,2

×

|fν,j − fν |(y1 , x2 )dy1

x1 −r1,1

+r1,2 x1ˆ

×

Dl fi (y1 , x2 )dy1 x1 −r1,1

x1ˆ +r1,2

fκ (y1 , x2 )dy1

κ=i+1x −r 1 1,1 m 

x1ˆ +r1,2

fτ,j (y1 , x2 )dy1

τ =ν+1x −r 1 1,1

 1 + (r1,1 + r1,2 )m κ=1 i−1

+r1,2 x1ˆ

fκ (y1 , x2 )dy1

x1 −r1,1

m 

x1ˆ +r1,2

×

x1ˆ +r1,2

|Dl fi,j − Dl fi |(y1 , x2 )dy1

x1 −r1,1

fτ,j (y1 , x2 )dy1 .

τ =i+1x −r 1 1,1

Together with the above cases, we obtain m 

|ux,fi,j (r1 ) − ux,fi (r1 )| ≤ l

m 

l

i=1

Gli,j (x) =: Glj (x),

i=1

Note that lim Gli,j Lp (Rd ) = 0.

j→∞

It follows that there exists j2 ∈ N such that

for any r1 ∈ [0, ∞)4 .

(2.50)

JID:MATPUR

AID:3193 /FLA

[m3L; v1.283; Prn:20/02/2020; 11:41] P.25 (1-38) F. Liu et al. / J. Math. Pures Appl. ••• (••••) •••–•••

Glj Lp (R2 ) < ,

25

∀j ≥ j2 .

(2.51)

By (2.43)-(2.46), one may obtain that |B2,i ∪ B i,j | < η for all j ≥ j1 and i = 1, 2, 3. These facts together with (2.47)–(2.51) imply that Dl MR (fj ) − Dl MR (f)Lp (R2 ) m      ≤ Glj Lp (R2 ) + 2 MR (fli )

p,B1

i=1

m      +2 MR (fli )

p,B2,3

i=1

∪B 3,j

m      + 2 MR (fli )

p,B2,1 ∪B 1,j

i=1

   −1/p  + |RΛ  (0)| p,(B

m      + 2 MR (fli )

p,B2,2 ∪B 2,j

i=1

3 i,j ))c 1 ∪(∪i=1 B2,i ∪B

≤ 10

for all j ≥ max{j1 , j2 }, which leads to (2.42).  3. Properties on Besov and Triebel-Lizorkin spaces This section will be devoted to presenting the proofs of Theorems 1.2 and 1.3. In what follows, we let Δζ f denote the difference of f , i.e. Δζ f (x) = f (x + ζ) − f (x) for all x, ζ ∈ Rd . We also let Rd = {ζ ∈ Rd ; 1/2 < |ζ| ≤ 1}. To prove Theorems 1.2 and 1.3, we need the following characterizations of homogeneous Triebel-Lizorkin spaces F˙sp,q (Rd ) and homogeneous Besov spaces B˙ sp,q (Rd ). Lemma 3.1. ([45]) (i) Let 0 < s < 1, 1 < p < ∞, 1 < q ≤ ∞ and 1 ≤ r < min(p, q). Then  

q/r 1/q  ˆ   f F˙sp,q (Rd ) ∼  2ksq |Δ2−k ζ f |r dζ  k∈Z

Lp (Rd )

;

Rd

(ii) Let 0 < s < 1, 1 ≤ p < ∞, 1 ≤ q ≤ ∞ and 1 ≤ r ≤ p. Then f B˙ sp,q (Rd ) ∼



 ˆ

1/r q   2ksq  |Δ2−k ζ f |r dζ  p

1/q .

L (Rd )

k∈Z

(3.1)

Rd

Proof of Theorem 1.2. Note that f Bsp,q (Rd ) ∼ f B˙ sp,q (Rd ) + f Lp (Rd ) for s > 0 and 1 < p, q < ∞. For a measurable function g : Rd × Z × Rd → R, we define gp,q :=

 k∈Z

2ksq

ˆ ˆ

|g(x, k, ζ)|p dxdζ

q/p 1/q .

Rd Rd

Using (3.1) with r = p and Fubini’s theorem, we have f B˙ sp,q (Rd ) ∼ Δ2−k ζ f p,q . Let 0 < s < 1 and 1 < p1 , . . . , pm , p, q < ∞ with 1/p = p ,q fj ∈ Bs j (Rd ). Fix ζ ∈ Rd , it is clear that MR (f)(x + ζ) = sup

Rx+ζ R∈R

One can easily check that

m i=1

(3.2) 1/pi . Let f = (f1 , . . . , fm ) with each

m ˆ m ˆ 1  1  |fi (y)|dy = sup |fi (y + ζ)|dy. m |R|m i=1 Rx |R| i=1 R

R∈R

R

JID:MATPUR

AID:3193 /FLA

[m3L; v1.283; Prn:20/02/2020; 11:41] P.26 (1-38) F. Liu et al. / J. Math. Pures Appl. ••• (••••) •••–•••

26

|Δ2−k ζ (MR (f))(x)| ≤

m 

MR (flk,ζ )(x),

(3.3)

l=1 k,ζ k,ζ where flk,ζ = (f1 , . . . , fl−1 , Δ2k ζ fl , fl+1 , . . . , fm ) and fjk,ζ (x) = fj (x + 2−k ζ) for all l + 1 ≤ j ≤ m. Then we get from (3.2)-(3.3) and Minkowski’s inequality that

MR (f)B˙ sp,q (Rd ) 

q/p 1/q ˆ ˆ ksq  2 |Δ2−k ζ MR (f)(x)|p dxdζ k∈Z

 

m   l=1 k∈Z m   l=1

 

m 

Rd Rd

 q   2ksq MR (flk,ζ )Lp (Rd ) 

1/q

Lp (Rd )

m  l−1 q    2ksq  fi Lpi (Rd ) Δ2−k ζ fl Lpl (Rd ) fjk,ζ Lpj (Rd )  i=1

k∈Z



fi Lpi (Rd )

l=1 i =l,1≤i≤m m  



(3.4)

Lp (Rd )

j=l+1

 q   2ksq Δ2−k ζ fl Lpl (Rd )  p L

k∈Z

1/q

1/q

l (Rd )

fi Lpi (Rd ) fl B˙ spl ,q (Rd ) .

l=1 i =l,1≤i≤m

Combining (3.4) with (1.2) implies that MR (f)Bsp,q (Rd ) ≤ C

m 

fi Bspi ,q (Rd ) .

i=1

This completes the proof of the boundedness part. We now prove the continuity part. Let fj = (f1,j , . . . , fm,j ) and fi,j → fi in Bspi ,q (Rd ) as j → ∞. It is known that fi,j → fi in B˙ spi ,q (Rd ) and in Lpi (Rd ) as j → ∞. One can check that |MR (fj ) − MR (f)| ≤

m 

MR (fl ).

(3.5)

l=1

Here fl = (f1 , . . . , fl−1 , fl,j − fl , fl+1,j , . . . , fm,j ). It follows from (3.5) that MR (fj ) → MR (f) in Lp (Rd ) as j → ∞. Therefore, it suffices to show that MR (fj ) → MR (f) in B˙ sp,q (Rd ) as j → ∞. We will prove this claim by contradiction. Without loss of generality, we may assume that there exists c > 0 such that MR (fj ) − MR (f)B˙ sp,q (Rd ) > c,

for every j.

It is obvious that Δ2−k ζ (MR (fj ) − MR (f))Lp (Rd ) → 0 as j → ∞ for every (k, ζ) ∈ Z × Rd . By (3.3), for every (x, k, ζ) ∈ Rd × Z × Rd , we have |Δ2−k ζ (MR (fj ) − MR (f))(x)| ≤ |Δ2−k ζ (MR (fj ))(x)| + |Δ2−k ζ (MR (f))(x)| m m   k,ζ k,ζ   |MR (fl,j )(x) − MR (fl )(x)| + 2 MR (flk,ζ )(x). ≤ l=1

(3.6)

l=1

k,ζ k,ζ k,ζ k,ζ Here flk,ζ is given as in (3.3) and fl,j = (f1,j , . . . , fl−1,j , Δ2−k ζ fl,j , fl+1,j , . . . , fm,j ) with fi,j (x) = fi,j (x + −k 2 ζ) for all l + 1 ≤ i ≤ m. From the third inequality to the last one in (3.4), we obtain

JID:MATPUR

AID:3193 /FLA

[m3L; v1.283; Prn:20/02/2020; 11:41] P.27 (1-38) F. Liu et al. / J. Math. Pures Appl. ••• (••••) •••–•••

m     MR (flk,ζ ) 

p,q

l=1





2ksq

k∈Z

 

m   l=1 m 

27

m p

q/p 1/q ˆ ˆ    MR (flk,ζ )(x) dxdζ  Rd Rd

l=1

 q   2ksq MR (flk,ζ )Lp (Rd )  p

1/q (3.7)

L (Rd )

k∈Z



fi Lpi (Rd ) fl B˙ spl ,q (Rd ) .

l=1 i =l,1≤i≤m

One can also verify that k,ζ |MR (fl,j ) − MR (flk,ζ )| ≤

l−1 

m  k,ζ k,ζ k,ζ MR (Iμ,j )+ MR (Jν,j ) + MR (Ki,j ),

μ=1

(3.8)

ν=l+1

where k,ζ k,ζ k,ζ Iμ,j = (f1 , . . . , fμ−1 , fμ,j − fμ , fμ+1,j , . . . , fl−1,j , Δ2−k ζ fl,j , fl+1,j , . . . , fm,j ), k,ζ k,ζ k,ζ k,ζ k,ζ k,ζ Jν,j = (f1 , . . . , fl−1 , Δ2k ζ fl , fl+1 , . . . , fν−1 , fν,j − fνk,ζ , fν+1,j , . . . , fm,j ), k,ζ k,ζ k,ζ Ki,j = (f1 , . . . , fl−1 , Δ2−k ζ (fl,j − fl ), fl+1,j , . . . , fm,j ). By (3.7) and (3.8), one can deduce that m     k,ζ |MR (fl,j ) − MR (flk,ζ )|  l=1

Thus, we can extract a subsequence such that Let H(x, k, ζ) =

∞ j=1



p,q

m l=1

→ 0 as j → ∞.

k,ζ |MR (fl,j ) − MR (flk,ζ )|p,q < ∞.

∞  m m      k,ζ MR (fl,j )(x) − MR (flk,ζ )(x) + 2 MR (flk,ζ )(x).  j=1

l=1

l=1

It is easily to check that Hp,q < ∞. By (3.6), we get |Δ2−k ζ (MR (fj ) − MR (f))(x)| ≤ H(x, k, ζ) for a.e. (x, k, ζ) ∈ Rd × Z × Rd .

(3.9)

´ Since Hp,q < ∞, we have Rd |H(x, k, ζ)p dx < ∞ for a.e. (k, ζ) ∈ Z × Rd . By (3.9) and the dominated convergence theorem, for a.e. (k, ζ) ∈ Z × Rd , it holds that ˆ lim

j→∞ Rd

|Δ2−k ζ (MR (fj ) − MR (f))(x)|p dx = 0.

(3.10)

Using (3.9) and the fact Hp,q < ∞ again, we have ˆ

ˆ |Δ2−k ζ (MR (fj ) − MR (f))(x)|p dx ≤ Rd

and

H(x, k, ζ)p dx, Rd

for a.e (k, ζ) ∈ Z × Rd

(3.11)

JID:MATPUR

AID:3193 /FLA

[m3L; v1.283; Prn:20/02/2020; 11:41] P.28 (1-38) F. Liu et al. / J. Math. Pures Appl. ••• (••••) •••–•••

28

ˆ ˆ H(x, k, ζ)p dxdζ < ∞

for every k ∈ Z.

(3.12)

Rd Rd

It follows from (3.10)-(3.12) and the dominated convergence theorem that ˆ ˆ lim

j→∞

|Δ2−k ζ (MR (fj ) − MR (f))(x)|p dxdζ

1/p =0

(3.13)

Rd Rd

For every k ∈ Z, by (3.9) and the fact Hp,q < ∞ again, we have ˆ ˆ

|Δ2−k ζ (MR (fj ) − MR (f))(x)|p dxdζ

1/p

≤

ˆ ˆ

Rd Rd

H(x, k, ζ)p dxdζ

1/p (3.14)

Rd Rd

and  k∈Z

2ksq

ˆ ˆ

H(x, k, ζ)p dxdζ

q/p 1/q

< ∞.

(3.15)

Rd Rd

Using (3.14)-(3.15) and the dominated convergence theorem again, one may obtain Δ2−k ζ (MR (fj ) − MR (f))p,q 

q/p 1/q ˆ ˆ ksq = 2 |Δ2−k ζ (MR (fj ) − MR (f))(x)|p dxdζ → 0 as j → ∞. k∈Z

Rd Rd

By (3.2), this yields that MR (fj ) − MR (f)B˙ sp,q (Rd ) → 0 as j → ∞, which gives a contradiction. The proof of Theorem 1.2 is finished.  In order to prove Theorem 1.3, we need to introduce some lemmas. Given an operator T acting on functions in R, we denote by T j , j = 1, 2, . . . , d, the operator defined on functions in Rd by letting T act on the j-th variable while keeping the remaining variables fixed, namely T j f (x) = T (f (x1 , x2 , . . . , xj−1 , ·, xj+1 , . . . , xd ))(xj ) for x ∈ Rd . We also define the operator T by T f (x) = T 1 ◦ T 2 ◦ . . . ◦ T d f (x). We need the following lemma. Lemma 3.2. If T is bounded on Lp (R, q (Lr (Rd ))) for some 1 < p, q, r < ∞, then the operator T is bounded on Lp (Rd , q (Lr (Rd ))). Proof. For all j = 1, . . . , d, we shall prove the following inequality  

1/q    T j fi,ζ qLr (Rd )   i∈Z

Lp (Rd )

 

1/q    ≤ T  fi,ζ qLr (Rd )  i∈Z

Lp (Rd )

.

(3.16)

Here T  represents the operator norm of T on Lp (R, q (Lr (Rd ))). We only prove (3.16) for j = 1 and the other cases are analogous. We may write

JID:MATPUR

AID:3193 /FLA

[m3L; v1.283; Prn:20/02/2020; 11:41] P.29 (1-38) F. Liu et al. / J. Math. Pures Appl. ••• (••••) •••–•••

29

 

1/q p   T 1 fi,ζ qLr (Rd )   p d L (R ) ˆi∈Z 

p/q = T 1 fi,ζ qLr (Rd ) dx i∈Z

Rdˆ

ˆ  ˆ

= ≤ T 

i∈Z

R ˆ

Rd−1

|T (fi,ζ (·, x2 , . . . , xd ))(x1 )|r dζ

d ˆ 

p

R

fi,ζ (x1 , x2 , . . . , xd )qLr (Rd )

p/q

dx1 dx2 . . . dxd

dx1 dx2 . . . dxd

i∈Z

  R

1/q p  = T   fi,ζ qLr (Rd )  p Rd−1

q/r p/q

p

L (Rd )

i∈Z

,

which leads to (3.16) for j = 1. (3.16) together with the definition of T yields that  

1/q    T fi,ζ qLr (Rd )   i∈Z

This proves Lemma 3.2.

Lp (Rd )

 

1/q    ≤ T d  fi,ζ qLr (Rd )  i∈Z

Lp (Rd )

.



The following vector-valued inequalities of the one dimensional uncentered Hardy-Littlewood maximal function will be very useful in the proof of Theorem 1.3. Lemma 3.3. ([45]) For any 1 < p, q, r < ∞, it holds that  

1/q    Mfj,ζ qLr (Rd )   j∈Z

Lp (R)

 

1/q    p,q,r  fj,ζ qLr (Rd ) 

Lp (R)

j∈Z

.

Applying Lemmas 3.2 and 3.3, we can get the following Lemma 3.4. For any 1 < p, q, r < ∞, it holds that  

1/q    MR fj,ζ qLr (Rd )   j∈Z

Lp (Rd )

 

1/q    p,q,r  fj,ζ qLr (Rd )  j∈Z

Lp (Rd )

.

Proof. For j = 1, . . . , d, we define the operator M j by 1 M f (x1 , x2 , . . . , xd ) = sup a
ˆb |f (x1 , . . . , xj−1 , y, xj+1 , . . . , xd )|dy.

j

a

One can easily check that M j f (x) = M(f (x1 , x2 , . . . , xj−1 , ·, xj+1 , . . . , xd )(xj ),

(3.17)

MR f (x) ≤ M ◦ M ◦ · · · ◦ M f (x).

(3.18)

1

2

d

Using (3.17)-(3.18) and Lemmas 3.2-3.3, for all 1 < p, q, r < ∞, we can get  

1/q    MR fj,ζ qLr (Rd )   j∈Z

Then Lemma 3.4 is proved. 

Lp (Rd )

 

1/q    p,q,r  fj,ζ qLr (Rd )  j∈Z

Lp (Rd )

JID:MATPUR

AID:3193 /FLA

[m3L; v1.283; Prn:20/02/2020; 11:41] P.30 (1-38) F. Liu et al. / J. Math. Pures Appl. ••• (••••) •••–•••

30

m  Proof of Theorem 1.3. Let 0 < s < 1 and 1 < p1 , . . . , pm , p, q < ∞ with 1/p = i=1 1/pi . Let f = pj ,q d (f1 , . . . , fm ) with each fj ∈ Fs (R ). One can easily check that (3.3) also holds. We get from (3.3) that |Δ2−k ζ (MR (f))(x)| m l−1 m    MR (Δ2−k ζ fl )(x) MR fμ (x) MR (fνk,ζ )(x) ≤ l=1

m 

=

MR (Δ2−k ζ fl )(x)

l=1 

≤



μ=1 l−1 

MR fμ (x) 

μ=1

MR (Δ2−k ζ fμ )(x)

ν=l+1 m 

(3.19)

MR (Δ2−k ζ fν + fν )(x)

ν=l+1

MR fν (x),

ν∈τ 

∅ =τ ⊂τm μ∈τ

where τm = {1, 2, . . . , m} and τ  = τm \ τ for τ ⊂ τm . Thus, Lemma 3.1 (i), (3.19) and the Minkowski inequality yield that MR (f)F˙sp,q (Rd )  

q 1/q  ˆ   ksq  2 |Δ2−k ζ (MR (f))|dζ  k∈Z

   2ksq  Rd





∅ =τ ⊂τm

ˆ 

k∈Z

Rd

Lp (Rd )



MR (Δ2−k ζ fμ )

Lp (Rd )

ν∈τ 

μ∈τ

(3.20)

q 1/q   MR fν dζ 

.

We shall prove the following estimate.  

q 1/q  ˆ     2ksq MR (Δ2−k ζ fμ ) MR fν dζ   k∈Z

   = 2ksq k∈Z





Let 1/pτ =

 μ∈τ

Rd

MR (Δ2−k ζ fμ )dζ

q 1/q 



fν Fspν ,q (Rd )

Rd

ν∈τ 

μ∈τ

 

q 1/q  ˆ    ≤ 2ksq MR (Δ2−k ζ fμ )dζ  k∈Z

Rd

Lpτ (Rd )

μ∈τ

  q     ≤ 2ksq MR (Δ2−k ζ fμ ) pμ /pτ μ∈τ

k∈Z

L

   pμ q/pτ    ≤ 2ksq MR (Δ2−k ζ fμ ) p /p

=

k∈Z

L

μ

 pμ q/pτ    ksq   2 MR (Δ2−k ζ fμ ) p /p  μ τ i∈τ

k∈Z

L

(Rd )

τ

(Rd )

(Rd )

k∈Z

Lp (Rd )

    MR fν  

Lpτ  (Rd )

ν∈τ 

1/q   

Lpτ (Rd )

    MR fν  ν∈τ 

pτ /pμ q   

Lpτ (Rd )

pτ /pμ q   

Lpμ (Rd )

 pμ q/pτ    k(pτ s/pμ )(pμ q/pτ )   2  MR (Δ2−k ζ fμ ) pμ /pτ μ∈τ

(3.21)

Lp (Rd )

1/pμ . Then, using Hölder’s inequality and the Lp bounds for MR we have

k∈Z

≤

  MR fν 

ν∈τ 

  

q 1/q  ˆ    2ksq MR (Δ2−k ζ fμ )dζ MR fν  

μ∈τ

Lp (Rd )

ν∈τ 

μ∈τ

fμ Fspμ ,q (Rd )

μ∈τ

ν∈τ 

μ∈τ

ˆ 

Rd

L

(Rd )

    fν  ν∈τ 

Lpν (Rd )

Lpν (Rd )

    fν 

Lpν (Rd )

ν∈τ 

pτ /pμ q   

Lpμ (Rd )

 ν∈τ 

fν Lpν (Rd )

JID:MATPUR

AID:3193 /FLA

[m3L; v1.283; Prn:20/02/2020; 11:41] P.31 (1-38) F. Liu et al. / J. Math. Pures Appl. ••• (••••) •••–•••





pτ /pμ q     k(pτ s/pμ )(pμ q/pτ )  p q/p 2 Δ2−k ζ fμ Lμpμ /pττ (R )   μ∈τ





μ∈τ

≤



μ∈τ

≤



fμ F˙ pμ ,pμ q/pτ (Rd ) pτ s/pμ

fμ F pμ ,pμ q/pτ (Rd ) pτ s/pμ

fμ Fspμ ,q (Rd )







Lpμ (Rd )

d

k∈Z

31

fν Lpν (Rd )

ν∈τ 

fi Lpν (Rd )

ν∈τ 



fν Lpν (Rd )

ν∈τ 

fν Fspν ,q (Rd ) .

ν∈τ 

μ∈τ

In the last estimate, we have used pμ > pτ 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) and (3.21) that MR (f)Fsp,q (Rd ) ≤ C

m 

fi Fspi ,q (Rd ) .

(3.22)

i=1

This completes the proof of the boundedness part. Below we prove the continuity part. Let fi,j → fi in Fspi ,q (Rd ) as j → ∞. It is known that fi,j → fi in F˙ spi ,q (Rd ) and in Lpi (Rd ) as j → ∞. By (3.5), it follows that MR (fj ) → MR (f) in Lp (Rd ) as j → ∞. Therefore, it suffices to show that MR (fj ) → MR (f) in F˙sp,q (Rd ) as j → ∞. Again, we will prove this claim by contradiction. Without loss of generality we may assume that, for every j, there exists c > 0 such that MR (fj ) − MR (f)F˙sp,q (Rd ) > c. For a measurable function g : Rd × Z × Rd → R, we define s gEp,q :=

ˆ  Rd

2ksq

k∈Z

ˆ

|g(x, k, ζ)|dζ

q p/q

1/p dx

.

Rd

s . By (3.6) and (3.8), we By Lemma 3.1, we see that if 1 ≤ r < min(p, q), then f F˙sp,q (Rd ) ∼ Δ2−k ζ f Ep,q get

|Δ2−k ζ (MR (fj ) − MR (f))| m  l−1 m m

   k,ζ k,ζ k,ζ ≤ MR (Iμ,j ) + MR (Jν,j ) + MR (Ki,j ) + 2 MR (flk,ζ ), l=1

μ=1

ν=l+1

l=1

k,ζ k,ζ k,ζ where flk,ζ is given as in (3.19) and Iμ,j , Jν,j and Ki,j are given as in (3.8). Notice that μ−1 l−1 m    k,ζ k,ζ MR (Iμ,j )≤ MR fi MR (fμ,j − fμ ) MR f,j MR (Δ2−k ζ fl,j ) MR fw,j i=1

=

=μ+1

μ−1 

MR fi MR (fμ,j − fμ )

i=1

MR f,j MR (Δ2−k ζ fl,j )

=μ+1 m 

×

l−1 

w=l+1

MR (Δ2−k ζ fw,j + fw,j )

w=l+1

≤

μ−1  i=1

MR fi MR (fμ,j − fμ )

l−1  =μ+1

MR f,j MR (Δ2−k ζ fl,j )

(3.23)

JID:MATPUR

AID:3193 /FLA

[m3L; v1.283; Prn:20/02/2020; 11:41] P.32 (1-38) F. Liu et al. / J. Math. Pures Appl. ••• (••••) •••–•••

32

×

m 

(MR (Δ2−k ζ fw,j ) + MR fw,j ).

w=l+1

This together with the arguments similar to those used in deriving (3.21) yields that μ−1 m   k,ζ s MR (Iμ,j )Ep,q  fμ,j − fμ Fspμ ,q (Rd ) fi Fspi ,q (Rd ) fw,j Fspw ,q (Rd ) .

(3.24)

w=μ+1

i=1

Similarly, we can conclude that ν−1 m   k,ζ s MR (Jν,j )Ep,q  fν,j − fν Fspν ,q (Rd ) fi Fspi ,q (Rd ) fw,j Fspw ,q (Rd ) ; l−1 m   k,ζ s MR (Ki,j )Ep,q  fl,j − fl Fspl ,q (Rd ) f Fsp ,q (Rd ) fw,j Fspw ,q (Rd ) ; =1 s MR (flk,ζ )Ep,q 

(3.25)

w=ν+1

i=1

(3.26)

w=l+1

m 

f Fsp ,q (Rd ) .

(3.27)

=1

It follows from (3.24)-(3.27) that l−1 m     k,ζ k,ζ k,ζ  MR (Iμ,j )+ MR (Jν,j ) + MR (Ki,j )  μ=1

s Ep,q

ν=l+1

→ 0 as j → ∞.

Therefore, one can extract a subsequence, we still denote it by j, such that ∞  l−1 m     k,ζ k,ζ k,ζ  M ( I ) + MR (Jν,j ) + MR (Ki,j )  R μ,j j=1

μ=1

s Ep,q

ν=l+1

< ∞.

(3.28)

Let G(x, k, ζ) =

m  ∞  l−1 

m  k,ζ k,ζ MR (Iμ,j )(x) + MR (Jν,j )(x)

l=1 j=1 μ=1

ν=l+1

 k,ζ +MR (Ki,j )(x) + 2 MR (flk,ζ )(x). m

l=1 s We get from (3.27) and (3.28) that GEp,q < ∞. Furthermore by (3.23), one obtains that

|Δ2−k ζ (MR (fj ) − MR (f))(x)| ≤ G(x, k, ζ) for every (x, k, ζ) ∈ Rd × Z × Rd .

(3.29)

(3.29) together with the dominated convergence theorem leads to ˆ |Δ2−k ζ (MR (fj ) − MR (f))(x)|dζ → 0 as j → ∞ for every (x, k, ζ) ∈ Rd × Z × Rd .

(3.30)

Rd s Since it holds that GEp,q < ∞, we immediately deduce that

 k∈Z

2ksq

ˆ

q 1/q G(x, k, ζ)dζ

Rd

< ∞,

for a.e. x ∈ Rd .

(3.31)

JID:MATPUR

AID:3193 /FLA

[m3L; v1.283; Prn:20/02/2020; 11:41] P.33 (1-38) F. Liu et al. / J. Math. Pures Appl. ••• (••••) •••–•••

33

Using (3.29), we obtain ˆ

ˆ |Δ2−k ζ (MR (fj ) − MR (f))(x)|dζ ≤ Rd

for a.e. x ∈ Rd and k ∈ Z.

G(x, k, ζ)dζ,

(3.32)

Rd

(3.30)-(3.32) and the dominated convergence theorem give 

2

ksq

k∈Z

ˆ

|Δ2−k ζ (MR (fj ) − MR (f))(x)|dζ

q 1/q

→ 0 as j → ∞, for a.e. x ∈ Rd

(3.33)

Rd

By (3.29) again, for a.e. x ∈ Rd , it is true that  k∈Z

2ksq

ˆ

|Δ2−k ζ (MR (fj ) − MR (f))(x)|dζ

q 1/q

≤



2ksq

k∈Z

Rd

ˆ

q 1/q G(x, k, ζ)dζ

(3.34)

Rd

s It follows from (3.33), (3.34), GEp,q < ∞ and the dominated convergence theorem that

s lim Δ2−k ζ (MR (fj ) − MR (f))Ep,q = 0,

j→∞

which yields MR (fj ) − MR (f)F˙sp,q (Rd ) → 0 as j → ∞ and leads to a contradiction.  4. 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 inequality that can be used in studying the pointwise behavior of Sobolev functions by the standard methods (see [11]). Let m f = (f1 , . . . , fm ) with each fi ∈ W 1,pi (Rd ) for 1 < pi < ∞. Let 1 < p < ∞ and 1/p = i=1 1/pi . For λ > 0, we set Oλ = {x ∈ Rd ; MR (f)(x) > λ}. Note that Oλ is an open set. We get from Theorem 1.1 that Cp (Oλ )1/p ≤

1 λ

ˆ (|MR (f)(x)|p + |∇MR (f)(x)|p )dx

Rd

1/p

m 1 1 ≤ MR (f)1,p m,d,p1 ,...,pm fi 1,pi . λ λ i=1

(4.1)

Step 2: The continuity of MR (f). To prove the p-quasicontinuity of MR (f), we first prove that MR (f) ∈ C(Rd ) if f = (f1 , . . . , fm ) with each fi ∈ C0∞ (Rd ). We can write m 

1 |E (x)| 2d  r  r ∈R+ i=1

MR (f)(x) = sup

ˆ |fi (y)|dy, E r (x)

where r = (r1− , . . . , rd− ; r1+ , . . . , rd+ ) and Er (x) = (x − r1− , x + r1+ ) × · · · × (x − rd− , x + rd+ ). For fixed x, h ∈ Rd , we have

JID:MATPUR

AID:3193 /FLA

[m3L; v1.283; Prn:20/02/2020; 11:41] P.34 (1-38) F. Liu et al. / J. Math. Pures Appl. ••• (••••) •••–•••

34

|MR (f)(x + h) − MR (f)(x)| ˆ m  1 sup |fi (y + h) − fi (y)|dy ≤ |Er (x)|m r ∈R2d + i=1  i−1  ˆ ×

E r (x)

|fμ (y)|dy

m



μ=1 E r (x)

ˆ

|fν (y)|dy .

ν=i+1 E r (x+h)

For fixed r ∈ R2d + and i = 1, . . . , m, by Hölder’s inequality, we obtain ˆ

1 |Er (x)|m

|fi (y + h) − fi (y)|dy E r (x)

≤ 2|Er (x)|−1/p

i−1  ˆ

|fμ (y)|dy

μ=1 E r (x)

m 

ˆ

m



|fν (y)|dy

ν=i+1 E r (x+h)

fi Lpi (Rd ) .

i=1

It follows that given  > 0, there exists a constant 0 < δ < +∞ such that ˆ

1 |Er (x)|m

|fi (y + h) − fi (y)|dy

i−1  ˆ

|fμ (y)|dy

m



μ=1 E r (x)

E r (x)

ˆ

|fν (y)|dy < ,

ν=i+1 E r (x+h)

when |Er (x)| > δ . On the other hand, for any x, h ∈ Rd and r ∈ R2d r (x)| ≤ δ , by the mean value + with |E theorem for differentials, we have 1 |Er (x)|

ˆ |fi (y + h) − fi (y)|dy ≤ C(fi )|h| E r (x)

and there exists Mi > 0 such that |fi (x)| ≤ Mi for all x ∈ Rd and i = 1, . . . , m. Then we have 1 |Er (x)|m

ˆ |fi (y + h) − fi (y)|dy

μ=1 E r (x)



E r (x)

≤ C(fi )

i−1  ˆ

|fμ (y)|dy

m



ˆ |fν (y)|dy

ν=i+1 E r (x+h)

Mμ |h|.

μ =i,1≤μ≤m

Therefore, for the above  > 0 and fixed x ∈ Rd , there exists γ = γ() > 0, if |h| < γ, then |MR (f)(x + h) − MR (f)(x)| ≤ C(f). Thus, it holds that MR (f) ∈ C(Rd ). Step 3: The p-quasicontinuity of MR (f). Suppose that fi ∈ W 1,pi (Rd ), we can choose a sequence of functions {fi,k }k≥1 ⊂ C0∞ (Rd ) such that fi,k → fi in W 1,pi (Rd ). This yields that there exists a large K0 ∈ N such that fi,k − fi 1,pi ≤ 2−2k , ∀k ≥ K0 and 1 ≤ i ≤ m. Fix k ≥ K0 . Let fk = (f1,k , . . . , fm,k ) and Ek = {x ∈ Rd : |MR (fk )(x) − MR (f)(x)| > 2−k }.

(4.2)

JID:MATPUR

AID:3193 /FLA

[m3L; v1.283; Prn:20/02/2020; 11:41] P.35 (1-38) F. Liu et al. / J. Math. Pures Appl. ••• (••••) •••–•••

35

By (2.6), we have |MR (fk )(x) − MR (f)(x)| ≤

m 

MR (Fkl )(x),

(4.3)

l=1

where Fkl (x) = (f1 , . . . , fl−1 , fl,k − fl , fl+1,k , . . . , fm,k ). Then, by (4.1)-(4.3), we have (Cp (Ek ))1/p m,d,p1 ,...,pm 2k m,d,p1 ,...,pm 2 Let Gk =

∞ i=k

m l−1  

m 

fμ 1,pμ fl,k − fl 1,pl

l=1 μ=1 −k

fν,k 1,pν

ν=l+1

(4.4)

.

Ei with k ≥ K0 . Then by subadditivity and (4.4), it holds that Cp (Gk ) ≤

∞ 

Cp (Ei ) m,d,p1 ,...,pm

i=k

∞  i=k

2−ip m,d,p1 ,...,pm

2(1−k)p , ∀k ≥ K0 , 2p − 1

which leads to limk→∞ Cp (Gk ) = 0. On the other hand, for x ∈ Rd \ Gk , |MR (fk )(x) − MR (f)(x)| ≤ 2−k ∀k ≥ K0 .

(4.5)

This implies that {MR (fk )} converges to MR (f) uniformly in Rd \ Gk . By Step 2, we see that MR (fk ) ∈ C(Rd ). It follows that MR (f) is continuous in Rd \ Gk . We notice that MR (fK0 )(x) < ∞ for all x ∈ Rd . This together with (4.5) implies that MR (f) is finite in Rd \ Gk . Hence, MR (f) is q-quasicontinuous.  5. Approximate differentiability of MR This section is devoted to proving Theorem 1.6. Let us recall some definitions and present some useful lemmas. Let f be a real-valued function defined on a set E ⊂ Rd . We say that f is approximately differentiable at x0 ∈ E if there is a vector L = (L1 , L2 , . . . , Ld ) ∈ Rd such that for any  > 0 the set   |f (x) − f (x0 ) − L(x − x0 )| < A = x ∈ Rd : |x − x0 | has x0 as a density point. If this is the case, then x0 is a density point of E and L is uniquely determined. The vector L is called the approximate differential of f at x0 and is denoted by ∇f (x0 ). Note that every function f ∈ W 1,1 (Rd ) is approximately differentiable a.e. It was pointed out in [17] that M f is approximately differentiable a.e. under the assumption that f ∈ W 1,1 (Rd ). However, it is unknown that whether f ∈ W 1,1 (Rd ) implies the weak differentiability of M f when d ≥ 2. The relationship between approximate differentiability and weak differentiability is still not clear. To prove Theorem 1.6, we need the following lemma, which provides several characterizations of a.e. approximate differentiability of a function. Lemma 5.1. ([43]) Let f : E → R be measurable, E ⊂ Rd . Then the following conditions are equivalent: (i) f is approximately differentiable a.e. (ii) For any  > 0, there is a closed set F ⊂ E and a locally Lipschitz function g : Rd → R such that f = g on x ∈ F and |E \ F | < .

JID:MATPUR

AID:3193 /FLA

[m3L; v1.283; Prn:20/02/2020; 11:41] P.36 (1-38) F. Liu et al. / J. Math. Pures Appl. ••• (••••) •••–•••

36

(iii) For any  > 0, there is a closed set F ⊂ E and a function g ∈ C 1 (Rd ) such that f = g on x ∈ F and |E \ F | < . Lemma 5.2. Let f = (f1 , . . . , fm ) with each fj ∈ L1 (Rd ). Let ε = (ε1 , . . . , εd ) with each εi > 0. The truncated ε  multilinear strong maximal operator MR is defined by  ε  MR (f )(x)

=

ˆ

m 

sup (r1− ,...,rd− ;r1+ ,...,rd+ )∈R2d + ri+ +ri− ≥εi , i=1,2,...,d

1 |E  r (x)| i=1

|fi (y)|dy, E r (x)

where x = (x1 , . . . , xd ), r = (r1− , . . . , rd− ; r1+ , . . . , rd+ ) and Er (x) = (x1 −r1− , x1 +r1+ ) ×· · ·×(xd −rd− , xd +rd+ ).  ε  Then MR (f ) is Lipschitz continuous for every ε ∈ Rd+ . Proof. Fix ε = (ε1 , . . . , εd ) ∈ Rd+ . We set ε0 = min1≤i≤d εi . Fix r = (r1− , . . . , rd− ; r1+ , . . . , rd+ ) ∈ R2d + with ri+ + ri− ≥ εi for 1 ≤ i ≤ d. It is obvious that ri+ + ri− ≥ ε0 for all 1 ≤ i ≤ d. Notice that for any r ≥ a ≥ 0, b ≥ 0 and δ ≥ 1, it is true that r δ a δ b ≥ ≥1−δ . r+b a+b a

(5.1)

Let x = (x1 , . . . , xd ) ∈ Rd and y = (y1 , . . . , yd ) ∈ Rd . We note that Er (x) ⊂ Er (y),

(5.2)

where r = (r1− + |y1 − x1 |, . . . , rd− + |yd − xd |; r1+ + |y1 − x1 |, . . . , rd+ + |yd − xd |). (5.2) gives that d

md |E (x)| m 

m rj− + rj+ ε0 md  r = ≥ ≥1− |x − y|. − + |Er (y)| ε + |x − y| ε0 r + r + |y − x | 0 j j j j=1 j

(5.3)

We get from (5.2) and (5.3) that  ε  MR (f )(y)

m 

1 ≥ |E  r  (y)| i=1 ≥

ˆ |fi (z)|dz E r (y)

ˆ m  1 |Er (x)| |Er (y)| |Er (x)| i=1

E r (x) m



md |x − y| ≥ 1− ε0

|fi (z)|dz

1 |Er (x)| i=1

(5.4)

ˆ |fi (z)|dz. E r (x)

− + Taking the supremum over r = (r1− , . . . , rd− ; r1+ , . . . , rd+ ) ∈ R2d + with ri + ri ≥ εi for 1 ≤ i ≤ d, we get from (5.4) that



md  ε   ε  MR (f )(y) ≥ 1 − |x − y| MR (f )(x). ε0 It follows that  ε   ε  MR (f )(x) − MR (f )(y) ≤

md  ε  |x − y|MR (f )(x). ε0

(5.5)

JID:MATPUR

AID:3193 /FLA

[m3L; v1.283; Prn:20/02/2020; 11:41] P.37 (1-38) F. Liu et al. / J. Math. Pures Appl. ••• (••••) •••–•••

37

Similarly, we can get md  ε   ε   ε  MR (f )(y) − MR (f )(x) ≤ |x − y|MR (f )(y). ε0

(5.6)

Thus, (5.5) and (5.6) imply that  ε  |MR (f )(x)

−

 ε  MR (f )(y)|

This proves Lemma 5.2.

m md 2md   ε   ε  ≤ |x − y|(MR (f )(x) + MR (f )(y)) ≤ md+1 fj L1 (Rd ) |x − y|. ε0 ε0 j=1



Proof of Theorem 1.6. Let Zj be the set of all Lebesgue points of fj and ux,f(r) defined as in Section 2. We m m set E0 = Rd \ ( j=1 Zj ). Let x ∈ j=1 Zj such that MR (f)(x) > ux,f(0) with 0 ∈ R2d . Since fj ∈ L1 (Rd ) 2d and MR (f)(x) > 0, there exists a sequence {rk }k≥1 with rk = (r− , . . . , r− ; r+ , . . . , r+ ) ∈ R , all 1,k

− + ri,k + ri,k are bounded such that

d,k

1,k

d,k

+

lim ux,f(rk ) = MR (f)(x).

k→∞

2d

Hence there exists a subsequence {rk }k≥1 ⊂ {rk }k≥1 and r = (r1− , . . . , rd− ; r1+ , . . . , rd+ ) ∈ R+ with ri− + ri+ > 0 for all 1 ≤ i ≤ d such that limk→∞ rk = r. It follows that MR (f)(x) = ux,f(r). This, of course, yields that Rd = E0 ∪ {x ∈ Rd : MR (f)(x) = ux,f(0)} ∪ E, ∞ ∞ 1/k ,...,1/kd  where E = k1 =1 . . . kd =1 Ek1 ,...,kd and Ek1 ,...,kd = {x ∈ Rd : MR (f)(x) = MR 1 (f )(x)}. By m Lemma 5.1, j=1 |fj | is approximately differentiable a.e. Then MR (f) is approximately differentiable a.e. 1/k ,...,1/kd  in the set {x ∈ R2 : MR (f)(x) = u (0)}. By Lemma 5.2 we have that M 1 (f ) is Lipschitz x,f

R

1/k ,...,1/k

d  continuous for any ki ≥ 1 and 1 ≤ i ≤ d. Then, for any ki ≥ 1 and 1 ≤ i ≤ d, the function MR 1 (f )  is approximately differentiable a.e.. It follows that MR (f )χE is approximately differentiable a.e. Note that |E0 | = 0. Therefore, MR (f) is approximately differentiable a.e. This completes the proof of Theorem 1.6. 

Acknowledgements The authors want to express their sincere thanks to the referee for his or her valuable remarks and suggestions, which made this paper more readable. This work was completed during the second author was visiting the University of Kansas. The second author is very grateful to both Professor R.H. Torres and the math department for hospitality and stimulation conditions. References [1] A. Alfonseca, F. Soria, A. Vargas, A remark on maximal operators along directions in R2 , Math. Res. Lett. 10 (1) (2003) 41–49. [2] J.M. Aldaz, J. Pérez Lázaro, Functions of bounded variation, the derivative of the one dimensional maximal function, and applications to inequalities, Trans. Am. Math. Soc. 359 (5) (2007) 2443–2461. [3] J.M. Aldaz, J. Pérez Lázaro, Boundedness and unboundedness results for some maximal operators on functions of bounded variation, J. Math. Anal. Appl. 337 (2008) 130–143. [4] L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Math. Monogr., Clarendon Press, Oxford University Press, New York, 2000. [5] B. Bojarski, P. Hajłasz, Pointwise inequalities for Sobolev functions and some applications, Stud. Math. 106 (1993) 77–92. [6] E. Carneiro, J. Mardid, Derivative bounds for fractional maximal functions, Trans. Am. Math. Soc. 369 (2017) 4063–4092.

JID:MATPUR 38

AID:3193 /FLA

[m3L; v1.283; Prn:20/02/2020; 11:41] P.38 (1-38) F. Liu et al. / J. Math. Pures Appl. ••• (••••) •••–•••

[7] E. Carneiro, J. Mardid, L.B. Pierce, Endpoint Sobolev and BV continuity for maximal operators, J. Funct. Anal. 273 (10) (2017) 3262–3294. [8] E. Carneiro, D. Moreira, On the regularity of maximal operators, Proc. Am. Math. Soc. 136 (12) (2008) 4395–4404. [9] E. Carneiro, B.F. Svaiter, On the variation of maximal operators of convolution type, J. Funct. Anal. 265 (2013) 837–865. [10] A. Córdoba, R. Fefferman, A geometric proof of the strong maximal theorem, Ann. Math. (2) 102 (1) (1975) 91–95. [11] H. Federer, W. Ziemer, The Lebesgue set of a function whose distribution derivatives are p-th power summable, Indiana Univ. Math. J. 22 (1972) 139–158. [12] J. García-Cuerva, J.L. Rubio de Francia, Weighted Norm Inequalities and Related Topic, North-Holland Mathematics Studies, vol. 116, North-Holland Publishing Co., Amsterdam, 1985, Mathematical Notes, 104. [13] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer-Verlag, Berlin, 1983. [14] E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Monogr. Math., Birkhäuser, 1984. [15] L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education, Upper Saddle River, New Jersey, 2004. [16] L. Grafakos, L. Liu, C. Pérez, R.H. Torres, The multilinear strong maximal function, J. Geom. Anal. 21 (2011) 118–149. [17] P. Hajłasz, J. Malý, On approximate differentiability of the maximal function, Proc. Am. Math. Soc. 138 (1) (2010) 165–174. [18] P. Hajłasz, J. Onninen, On boundedness of maximal functions in Sobolev spaces, Ann. Acad. Sci. Fenn., Math. 29 (1) (2004) 167–176. [19] G.H. Hardy, J.E. Littlewood, A maximal theorem with function-theoretic applications, Acta Math. 54 (1930) 81–116. [20] B. Jessen, J. Marcinkiewicz, A. Zygmund, Note on the differentiability of multiple integrals, Fundam. Math. 25 (1935) 217–234. [21] N.H. Katz, A counterexample for maximal operators over a Cantor set of directions, Math. Res. Lett. 3 (4) (1996) 527–536. [22] N.H. Katz, Maximal operators over arbitrary sets of directions, Duke Math. J. 97 (1999) 67–79. [23] T. Kilpeläinen, J. Kinnunen, O. Martio, Sobolev spaces with zero boundary values on metric spaces, Potential Anal. 12 (2000) 233–247. [24] J. Kinnunen, The Hardy-Littlewood maximal function of a Sobolev function, Isr. J. Math. 100 (1997) 117–124. [25] J. Kinnunen, P. Lindqvist, The derivative of the maximal function, J. Reine Angew. Math. 503 (1998) 161–167. [26] J. Kinnunen, E. Saksman, Regularity of the fractional maximal function, Bull. Lond. Math. Soc. 35 (4) (2003) 529–535. [27] S. Korry, Boundedness of Hardy-Littlewood maximal operator in the framework of Lizorkin-Triebel spaces, Rev. Mat. Complut. 15 (2) (2002) 401–416. [28] S. Korry, A class of bounded operators on Sobolev spaces, Arch. Math. 82 (1) (2004) 40–50. [29] O. Kurka, On the variation of the Hardy-Littlewood maximal function, Ann. Acad. Sci. Fenn., Math. 40 (2015) 109–133. [30] A.K. Lerner, S. Ombrosi, C. Pérez, R.H. Torres, R. Trujillo-González, New maximal functions and multiple weighted for the multilinear Calderón-Zygmund theory, Adv. Math. 220 (2009) 1222–1264. [31] J.L. Lewis, On very weak solutions of certain elliptic systems, Commun. Partial Differ. Equ. 18 (1993) 1515–1537. [32] F. Liu, Continuity and approximate differentiability of multisublinear fractional maximal functions, Math. Inequal. Appl. 21 (1) (2018) 25–40. [33] F. Liu, T. Chen, H. Wu, A note on the endpoint regularity of the Hardy-Littlewood maximal functions, Bull. Aust. Math. Soc. 94 (2016) 121–130. [34] F. Liu, S. Mao, On the regularity of the one-sided Hardy-Littlewood maximal functions, Czechoslov. Math. J. 67 (1) (2017) 219–234. [35] F. Liu, H. Wu, On the regularity of the multisublinear maximal functions, Can. Math. Bull. 58 (4) (2015) 808–817. [36] F. Liu, H. Wu, Endpoint regularity of multisublinear fractional maximal functions, Can. Math. Bull. 60 (3) (2017) 586–603. [37] F. Liu, H. Wu, On the regularity of maximal operators supported by submanifolds, J. Math. Anal. Appl. 453 (2017) 144–158. [38] H. Luiro, Continuity of the maximal operator in Sobolev spaces, Proc. Am. Math. Soc. 135 (1) (2007) 243–251. [39] H. Luiro, On the regularity of the Hardy-Littlewood maximal operator on subdomains of Rn , Proc. Edinb. Math. Soc. 53 (1) (2010) 211–237. [40] H. Luiro, The variation of the maximal function of a radial function, Ark. Mat. 56 (1) (2018) 147–161. [41] A. Nagel, E.M. Stein, S. Wainger, Differentiation in lacunary directions, Proc. Natl. Acad. Sci. USA 75 (1978) 1060–1062. [42] H. Tanaka, A remark on the derivative of the one-dimensional Hardy-Littlewood maximal function, Bull. Aust. Math. Soc. 65 (2) (2002) 253–258. [43] H. Whitney, On totally differentiable and smooth functions, Pac. J. Math. 1 (1951) 143–159. [44] N. Wiener, The ergodic theorem, Duke Math. J. 5 (1939) 1–18. [45] K. Yabuta, Triebel-Lizorkin space boundedness of Marcinkiewicz integrals associated to surfaces, Appl. Math. J. Chin. Univ. Ser. B 30 (4) (2015) 418–446. [46] A. Zygmund, On the differentiability of multiple integrals, Fundam. Math. 23 (1934) 143–149. [47] A. Zygmund, Trigonometric Series, Cambridge University Press, Cambridge, 1959.