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# Open Mathematics

### formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 16, Issue 1

# Regularity of one-sided multilinear fractional maximal functions

Feng Liu
• Corresponding author
• College of Mathematics and Systems Sciences, Shandong University of Science and Technology, Qingdao 266590, China
• Email
• Other articles by this author:
/ Lei Xu
Published Online: 2018-12-31 | DOI: https://doi.org/10.1515/math-2018-0129

## Abstract

In this paper we introduce and investigate the regularity properties of one-sided multilinear fractional maximal operators, both in continuous case and in discrete case. In the continuous setting, we prove that the one-sided multilinear fractional maximal operators${\mathfrak{M}}_{\beta }^{+}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}{\mathfrak{M}}_{\beta }^{-}$map W1,p1 (ℝ)×· · ·×W1,pm (ℝ) into W1,q(ℝ) with 1 < p1, … , pm < ∞, 1 ≤ q < ∞ and $1/q=\sum _{i=1}^{m}1/{p}_{i}-\beta$, boundedly and continuously. In the discrete setting, we show that the discrete one-sided multilinear fractional maximal operators are bounded and continuous from 1(ℤ)×· · ·×1(ℤ) to BV(ℤ). Here BV(ℤ) denotes the set of functions of bounded variation defined on ℤ. Our main results represent significant and natural extensions of what was known previously.

MSC 2010: 42B25; 46E35

## 1 Introduction and the main results

Over the last several years a considerable amount of attention has been given to investigate the behavior of differentiability of maximal function. A good start was due to Kinnunen [1] who showed that the usual centered Hardy-Littlewood maximal function Mis bounded on W1,p(ℝd) for all 1 < p ≤ ∞, where W1,p(ℝd) is the first order Sobolev space, which consists of functions f ϵ Lp(ℝd), whose first weak partial derivatives Dif , i = 1, 2, … , d, belong to Lp(ℝd). We endow W1,p(ℝd) with the norm

$∥f∥1,p=∥f∥Lp(Rd)+∥∇f∥Lp(Rd),$

where ▽f = (D1f , D2f , … , D df ) is the weak gradient of f . Later on, Kinnunen’s result was extended to a local version in [2], to a fractional version in [3], to a multilinear version in [4, 5] and to a one-sided version in [6]. Meanwhile, the continuity of M : W1,pW1,p for 1 < p < ∞ was proved by Luiro in [7] and in [8] for its local version. Since Kinnunen’s result does not hold for p = 1, an important question was posed by Hajłasz and Onninen in [9]: Is the operator f ➝ |ΔMf| bounded from W1,1(ℝd) to L1(ℝd)? Progress on the above problem has been restricted to dimension d = 1. In 2002, Tanaka [10] showed that if fW1,1(ℝ), then the uncentered Hardy-Littlewood maximal function M̃f is weakly differentiable and

$∥(M~f)′∥L1(R)≤2∥f′∥L1(R).$(1.1)

This result was later sharpened by Aldaz and Pérez Lázaro [11] who proved that if f is of bounded variation on ℝ, then M̃f is absolutely continuous and its total variation satisfies

$Var(M~f)≤Var(f).$(1.2)

The above result implies directly (1.1) with constant C = 1 (also see [12] for a simple proof). In remarkable work [13], Kurka obtained that (1.1) and (1.2) hold for M(with constant C = 240, 004). Recently, Carneiro and Madrid [14] extended (1.1) and (1.2) to a fractional setting. Very recently, Liu and Wu [15] extended the partial result of [14] to a multilinear setting. For other interesting works related to this theory, we refer the reader to [16, 17, 18, 19, 20, 21, 22, 23, 24, 25], among others.

In this paper we focus on the regularity properties of the one-sided multilinear fractional maximal operators. More precisely, let m be a positive integer. For 0 ≤ β < m, we define the one-sided multilinear fractional maximal operators by ${\mathfrak{M}}_{\beta }^{+}\text{\hspace{0.17em}and\hspace{0.17em}}\phantom{\rule{thinmathspace}{0ex}}{\mathfrak{M}}_{\beta }^{-}$

$Mβ+(f→)(x)=sups>01sm−β∏i=1m∫xx+s|fi(y)|dyandMβ−(f→)(x)=supr>01rm−β∏i=1m∫x−rx|fi(y)|dy,$

where $\stackrel{\to }{f}=\left({f}_{1},\dots ,{f}_{m}\right)\phantom{\rule{thickmathspace}{0ex}}\text{with each}\phantom{\rule{thickmathspace}{0ex}}{f}_{i}\in {L}_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1}\left(\mathbb{R}\right)$When β = 0, the operator ${\mathfrak{M}}_{\beta }^{+}\phantom{\rule{thinmathspace}{0ex}}\left(\text{resp}.,{\mathfrak{M}}_{\beta }^{-}$reduces to the one-sided multilinear Hardy-Littlewood maximal operator M+ (resp., M). When m = 1, the operator ${\mathfrak{M}}_{\beta }^{+}$$\left(\text{resp}.,\phantom{\rule{thinmathspace}{0ex}}{\mathfrak{M}}_{\beta }^{-}\right)$reduces to the one-sided fractional maximal operator ${\mathcal{M}}_{\beta }^{+}\phantom{\rule{thinmathspace}{0ex}}\left(\text{resp}.,\phantom{\rule{thinmathspace}{0ex}}{\mathcal{M}}_{\beta }^{-}\right).$Especially, the one-sided Hardy-Littlewood maximal operator M+ (resp., M) corresponds to the operator ${\mathcal{M}}_{\beta }^{+}\phantom{\rule{thinmathspace}{0ex}}\left(\text{resp}.,\phantom{\rule{thinmathspace}{0ex}}{\mathcal{M}}_{\beta }^{-}\right)$in this case β = 0.

As we all known, the reasons to study one-sided operators involve not only the generalization of the theory of the two-sided operators but also the close connection between the one-sided operators and two-sided operators. The one-sided Hardy-Littlewood maximal operator M+ can be seen as the special case of the ergodic maximal operator. Furthermore, there is a close connection between the one-sided fractional maximal functions and the well-known Riemann-Liourille fractional integral that can be viewed as the one-sided version of Riesz potential and the Weyl fractional integral (see [26]). It was known that both ${\mathcal{M}}_{\beta }^{+}$and ${\mathcal{M}}_{\beta }^{-}$are of type (p, q) for 1 < p < 1, 0 ≤ β < 1/p and q = p/(1 − ). Moreover, both${\mathcal{M}}_{\beta }^{+}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}{\mathcal{M}}_{\beta }^{-}$are of weak type (1, q) for 0 ≤ β < 1 and q = 1/(1 − β). Observing that the following inequalities are valid:

$Mβ+(f→)(x)≤∏i=1mMβi+fi(x),∀x∈R,$(1.3)

where = (f1, … , fm) and $\beta =\sum _{i=1}^{m}{\beta }_{i}\phantom{\rule{thinmathspace}{0ex}}\text{with}\phantom{\rule{thinmathspace}{0ex}}{\beta }_{i}\ge 0\phantom{\rule{thinmathspace}{0ex}}\left(i=1,2,\dots ,m\right)$By (1.3), the Lp bounds for ${\mathcal{M}}_{\beta }^{+}$and Hölder’s inequality, one has

$∥Mβ+(f→)∥Lq(R)≤C(β,p1,…,pm)∏i=1m∥fi∥Lpi(R)$(1.4)

for $1/q=\sum _{i=1}^{m}1/{p}_{i}-\beta ,$provided that (i) β = 0, i1 mq ≤ ∞ and 1 < p1, … , pm ≤ ∞; (ii) 0 < β < m, 1 ≤ q < ∞ and 1 < p1, … , pm < ∞. The same result holds for${\mathfrak{M}}_{\beta }^{-}.$

The investigation on the regularity of one-sided maximal operator began with Tanaka [10] in 2002 when he observed that if f ∈ W1,1(ℝ), then the distributional derivatives of M+f and Mf are integrable functions, and

$∥(M+f)′∥L1(R)≤∥f′∥L1(R)and∥(M−f)′∥L1(R)≤∥f′∥L1(R).$

By a combination of arguments in [10, 12], both M +f and M f are absolutely continuous on ℝ. Recently, Liu and Mao [6] proved that both M+ and M are bounded and continuous on W1,p(ℝ) for 1 < p < ∞. Very recently, Liu [27] extended the main results of [6] to the fractional case. More precisely, Liu proved the following result.

#### Theorem A

([27]). Let 1 < p < ∞, 0 ≤ β < 1/p and q = p/(1 − ). Then both ${\mathcal{M}}_{\beta }^{+}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}{\mathcal{M}}_{\beta }^{-}$map W1,p(ℝ) into W1,q(ℝ) boundedly and continuously. Moreover, if f ϵ W1,p(ℝ), then

$|(Mβ+f)′(x)|≤Mβ+f′(x)and|(Mβ−f)′(x)|≤Mβ−f′(x)$

for almost every x ∈ ℝ.

In this paper we shall extended Theorem A to the multilinear case. We now formulate our main results as follows.

#### Theorem 1.1

Let $1<{p}_{1},\dots ,{p}_{m}<\mathrm{\infty },0\le \beta <\sum _{i=1}^{m}1/{p}_{i},1/q=\sum _{i=1}^{m}1/{p}_{i}-\beta \phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}1\le q<\mathrm{\infty }.\phantom{\rule{thinmathspace}{0ex}}Then\phantom{\rule{thinmathspace}{0ex}}{\mathfrak{M}}_{\beta }^{+}$maps W1,p1 (ℝ) ×· ··× W1,pm (ℝ) into W1,q(ℝ) boundedly and continuously. Especially, if f ➝= (f1, … , fm) with each fi ∈ W1,pi (ℝ), then the weak derivative $\left({\mathfrak{M}}_{\beta }^{+}\left(\stackrel{\to }{f}\right){\right)}^{\prime }$exists almost everywhere. More precisely,

$|(Mβ+(f→))′(x)|≤∑j=1mMβ+(f→j)(x)$

for almost every x ϵ, where ${\stackrel{\to }{f}}^{j}=\left({f}_{1},\dots ,{f}_{j-1},{f}_{j}^{\prime },{f}_{j+1},\dots ,{f}_{m}\right).$Moreover,

$∥Mβ+(f→)∥1,q≤C(β,p1,…,pm)∏i=1m∥fi∥1,pi.$

The same results hold for${\mathfrak{M}}_{\beta }^{-}.$

#### Theorem 1.2

Let $\stackrel{\to }{f}=\left({f}_{1},\dots ,{f}_{m}\right)$with each ${f}_{i}\in {L}^{{p}_{i}}\left(\mathbb{R}\right)\phantom{\rule{thinmathspace}{0ex}}for\phantom{\rule{thinmathspace}{0ex}}1<{p}_{1},\dots ,{p}_{m}<\mathrm{\infty }\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}1\le \beta <\sum _{i=1}^{m}1/{p}_{i}.$

• (i)

Then the weak derivative $\left({\mathfrak{M}}_{\beta }^{+}\left(\stackrel{\to }{f}\right){\right)}^{\prime }$exists almost everywhere. Precisely,

$|(Mβ+(f→))′(x)|≤C(m,β)Mβ−1+(f→)(x)$

for almost every x ϵ ℝ.

• (ii)

Let $1/q=\sum _{i=1}^{m}1/{p}_{i}-\beta +1.$Then

$∥(Mβ+(f→))′∥Lq(R)≤C(m,β,p1,…,pm)∏i=1m∥fi∥Lpi(R).$

The same results hold for ${\mathfrak{M}}_{\beta }^{-}.$

#### Remark 1.1

Theorem 1.1 extends Theorems 1.1-1.2 in [6], which correspond to the case m = 1 and β = 0. Theorem 1.1 also extends Theorem A, which corresponds to the case m = 1.

On the other hand, the investigation of the regularity properties of discrete maximal operators has also attracted the attention of many authors (see [6, 14, 16, 27, 28, 29, 30, 31, 32, 33] for example). Let us recall some notation and relevant results. For 1 ≤ p < ∞ and a discrete function f : ℤ ➝ ℝ, we define the p-norm and the -norm of f by ∥f∥ℓp(ℤ) = (ΣnϵZ |f(n)|p)1/p and $\parallel f{\parallel }_{{\ell }^{\mathrm{\infty }}\left(\mathbb{Z}\right)}=\underset{n\in \mathbb{Z}}{sup}|f\left(n\right)|.$We also define the first derivative of f by f '(n) = f(n + 1) − f(n) for any n ϵ ℤ. For f : ℤ ➝ ℝ, we define the total variation of f by

$Var(f)=∥f′∥ℓ1(Z).$

We denote by BV(ℤ) the set of all functions f : ℤ ➝ ℝ satisfying Var(f) < ∞.

In 2011, Bober et al. [28] first studied the regularity properties of discrete Hardy-Littlewood maximal operators and proved that

$Var(M~f)≤Var(f)$(1.5)

and

$Var(Mf)≤(2+146315)∥f∥ℓ1(Z).$(1.6)

Here M $\left(\text{resp}.,\phantom{\rule{thinmathspace}{0ex}}\stackrel{~}{M}\right)$denotes the discrete centered (resp., uncentered) Hardy-Littlewood maximal operator, which are defined by

$Mf(n)=supr∈N12r+1∑k=−rr|f(n+k)|andM~f(n)=supr,s∈N1r+s+1∑k=−rs|f(n+k)|,$

where ℕ = {0, 1, 2, 3, … , }. We note that inequality (1.5) is sharp. It was known that inequality Var(Mf)≤ 294, 912, 004Var(f ) was established by Temur in [32]. Inequality (1.6) is not optimal, and it was asked in [28] whether the sharp constant for (1.6) is in fact C = 2, which was addressed by Madrid in [31]. Recently, Carneiro and Madrid [14] extended (1.5) to the fractional setting. They also pointed out that the discrete fractional maximal operators ${M}_{\beta }\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}{\stackrel{~}{M}}_{\beta }$are bounded and continuous from 1(ℤ) to BV(ℤ) (also see [29, 34]). Here M β and ${\stackrel{~}{M}}_{\beta }$are the discrete centered and uncentered fractional maximal operators, which are defined by

$Mβf(n)=supr∈N1(2r+1)1−β∑k=−rr|f(n+k)|andM~βf(n)=supr,s∈N1(r+s+1)1−β∑k=−rs|f(n+k)|.$

Our second aim of this paper is to consider the discrete one-sided multilinear fractional maximal operators

$Mβ+(f→)(n)=sups∈N1(s+1)m−β∏i=1m∑k=0s|fi(n+k)|,Mβ−(f→)(n)=supr∈N1(r+1)m−β∏i=1m∑k=−r0|fi(n+k)|,$

where $0\le \beta with each ${f}_{i}\in {L}_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1}\left(\mathbb{Z}\right).$When β = 0, the operators ${\mathrm{M}}_{\beta }^{+}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}{\mathrm{M}}_{\beta }^{-}$reduce to the discrete one-sided multilinear Hardy-Littlewood maximal operators M+ and M, respectively. When m = 1, the operators ${\mathrm{M}}_{\beta }^{+}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}{\mathrm{M}}_{\beta }^{-}$reduce to the discrete one-sided fractional maximal operators ${\mathrm{M}}_{\beta }^{+}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}{\mathrm{M}}_{\beta }^{-}$respectively. Particularly, the discrete one-sided Hardy-Littlewood maximal operators M+ and M correspond to the special case of ${\mathrm{M}}_{\beta }^{+}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}{\mathrm{M}}_{\beta }^{-}$when β = 0, respectively. Recently, Liu and Mao [6] proved that both +−M+ and M are bounded and continuous from 1(ℤ) to BV(ℤ). Moreover, if f ∈ BV(ℤ), then

$max{Var(M+f),Var(M−f)}≤Var(f).$(1.7)

We notice that the constant C = 1 in inequality (1.7) is sharp. Very recently, Liu [27] pointed out that ${M}_{\beta }^{+}$and ${M}_{\beta }^{-}$are not bounded from BV(ℤ) to BV(ℤ) when 0 < β < 1. However, Liu established the following result.

#### Theorem B

([27]). Let 0 ≤ β < 1. Then${\mathrm{M}}_{\beta }^{+}$is bounded and continuous from ℓ1(ℤ) to BV(ℤ).Moreover, if f ϵ ℓ1(ℤ), then

$Var(Mβ+f)≤2∥f∥ℓ1(Z),$

and the constant C = 2 is the best possible. The same results hold for ${M}_{\beta }^{-}.$

In this paper we shall extended Theorem B to the following.

#### Theorem 1.3

Let 0 ≤ β < m. Then ${\mathrm{M}}_{\beta }^{+}$is bounded and continuous from ℓ1(ℤ) ×· ··× 1(ℤ) to BV(ℤ). Moreover, if $\stackrel{\to }{f}=\left({f}_{1},\dots ,{f}_{m}\right)$with each fi ϵ ℓ1(ℤ), then

$Var(Mβ+(f→))≤2m∏i=1m∥fi∥ℓ1(Z).$

The same results hold for ${\mathrm{M}}_{\beta }^{-}.$

#### Remark 1.2

When m = 1, Theorem 1.3 implies Theorem B.

The rest of this paper is organized as follows. Section 2 contains some notation and preliminary lemmas, which can be used to prove the continuity part in Theorem 1.1. Motivated by the ideas in [5, 7], we give the proofs of Theorems 1.1-1.2 in Section 3. Finally, we prove Theorem 1.3 in Section 4. It should be pointed out that the proof of the boundedness part in Theorem 1.3 is based on the method of [31]. The proof of the continuity part in Theorem 1.3 relies on the previous boundedness result and a useful application of the Brezis-Lieb lemma in [35]. Throughout this paper, the letter C, sometimes with additional parameters, will stand for positive constants, not necessarily the same one at each occurrence but independent of the essential variables.

## 2 Preliminary notation and lemmas

In this section we shall introduce some notation and lemmas, which play key roles in the proof of the continuity part in Theorem 1.1. Let A ⊂ ℝ and r ϵ ℝ. We define

$d(r,A):=infa∈A|r−a|andA(λ):={x∈R:d(x,A)≤λ}forλ≥0.$

Denote ∥f ∥p,A by the Lp-norm of A for all measurable sets A ⊂ ℝ. Let $\stackrel{\to }{f}=\left({f}_{1},\dots ,{f}_{m}\right)$with each fi ϵ Lpi (ℝ) for 1 < pi < ∞ and 1 ≤ < ∞ with $1/q=\sum _{i=1}^{m}1/{p}_{i}-\beta .$In what follows, we only consider the operator${\mathfrak{M}}_{\beta }^{+}$and the other case is analogous. Fix x ∈ ℝ, we define the set ${\mathfrak{R}}_{\beta }^{+}\left(\stackrel{\to }{f}\right)\left(x\right)$by

$Rβ+(f→)(x):={s≥0:Mβ+(f→)(x)=lim supk→∞1skm−β∏i=1m∫xx+sk|fi(y)|dyforsomesk>0,sk→s}.$

We also define the function ${u}_{x,\stackrel{\to }{f},\beta }^{+}:\left[0,\mathrm{\infty }\right)↦\mathbb{R}$by

$ux,f→,β+(0)={∏i=1m|fi(x)|, if β=0; 0, if0<β

We notice that the followings are valid.

1. ${u}_{x,\stackrel{\to }{f},\beta }^{+}$is continuous on (0,∞) for all x ∈ ℝ and at r = 0 for almost everywhere x ∈ ℝ ;

2. $\underset{s\to \mathrm{\infty }}{lim}{u}_{x,\stackrel{\to }{f},\beta }^{+}\left(s\right)=0\phantom{\rule{thinmathspace}{0ex}}\text{since}\phantom{\rule{thinmathspace}{0ex}}{u}_{x,\stackrel{\to }{f},\beta }^{+}\left(s\right)\le \prod _{i=1}^{m}\parallel {f}_{i}{\parallel }_{{L}^{{p}_{i}}\left(\mathbb{R}\right)}{s}^{-1/q}$

3. The set ${\mathfrak{R}}_{\beta }^{+}\left(\stackrel{\to }{f}\right)\left(x\right)$is nonempty and closed for any x ∈ ℝ;

4. Almost every point is a Lebesgue point.

From the above observations we have

$Mβ+(f→)(x)=ux,f→,β+(s)if0

$Mβ+(f→)(x)=ux,f→,β+(0)foralmosteveryx∈Rsuchthat0∈Rβ+(f→)(x).$

#### Lemma 2.1

Let 1 < p1, … , pm < ∞ and 1 ≤ q < ∞ with $1/q=\sum _{i=1}^{m}1/{p}_{i}-\beta .\phantom{\rule{thinmathspace}{0ex}}Let\phantom{\rule{thinmathspace}{0ex}}{\stackrel{\to }{f}}_{j}=\left({f}_{1,j},\dots ,{f}_{m,j}\right)$and f̄ = (f1, … , fm) such that fi,jfi in Lpi (ℝ) when j ➝ ∞. Then, for all ℝ > 0 and λ > 0, it holds that

$limj→∞|{x∈(−R,R):Rβ+(f→j)(x)⊈Rβ+(f→)(x)(λ)}|=0.$(2.1)

Proof. Without loss of generality, we may assume that all fi,j ≥ 0 and fi ≥ 0. By the similar argument as in the proof of Lemma 2.2 in [7], we can conclude that the set $\left\{x\in \mathbb{R}:\phantom{\rule{thinmathspace}{0ex}}{\mathfrak{R}}_{\beta }^{+}\left({\stackrel{\to }{f}}_{j}\right)\left(x\right)⊈{\mathfrak{R}}_{\beta }^{+}\left(\stackrel{\to }{f}\right)\left(x{\right)}_{\left(\lambda \right)}\right\}$is measurable for any j ϵ ℤ. Let λ > 0 and R > 0. We first claim that for almost every x ∈ (−R, R), there exists γ(x) ϵ ℕ \ {0} such that

$ux,f→,β+(s)λ.$(2.2)

Otherwise, for almost every x ϵ (−R, R), there exists a bounded sequence of radii $\left\{{s}_{k}{\right\}}_{k=1}^{\mathrm{\infty }}$such that

$limk→∞ux,f→,β+(sk)=Mβ+(f→)(x)andd(sk,Rβ+(f→)(x))>λ.$

We can choose a subsequence $\left\{{r}_{k}{\right\}}_{k=1}^{\mathrm{\infty }}\phantom{\rule{thinmathspace}{0ex}}of\phantom{\rule{thinmathspace}{0ex}}\left\{{s}_{k}{\right\}}_{k=1}^{\mathrm{\infty }}$such that rks as k ➝ ∞. Then we have $s\in {\mathfrak{R}}_{\beta }^{+}\left(\stackrel{\to }{f}\right)\left(x\right)$and $d\left(s,{\mathfrak{R}}_{\beta }^{+}\left(\stackrel{\to }{f}\right)\left(x\right)\right)\ge \lambda ,$which is a contradiction. Thus (2.2) holds. Given ϵ ∈ (0, 1), (2.2) yields that there exists γ = γ(R, λ, ϵ) ∈ ℕ \ {0} and a measurable set E with |E| < ϵ such that

$(−R,R)⊂{x∈R:ux,f→,β+(s)λ}∪E.$

Notice that

$Mβ+(f→)(x)−ux,f→,β+(s)≤|Mβ+(f→j)(x)−Mβ+(f→)(x)|+|ux,f→j,β+(s)−ux,f→,β+(s)|+Mβ+(f→j)(x)−ux,f→j,β+(s).$

It yields that

${x∈R:ux,f→,β+(s)λ}⊂A1,j∪A2,j∪A3,j,$

where

$A1,j:={x∈R:|Mβ+(f→j)(x)−Mβ+(f→)(x)|≥(4γ)−1},A2,j:={x∈R:|ux,f→j,β+(s)−ux,f→,β+(s)|≥(2γ)−1forsomessuchthatd(s,Rβ+(f→)(x))>λ},A3,j:={x∈R:ux,f→j,β+(s)λ}.$

Hence,

$(−R,R)⊂A1,j∪A2,j∪A3,j∪E.$(2.3)

Let Ā be the set of all points x such that x is a Lebesgue point of all fj. Note that |R \ Ā| = 0 and A3,j Ā $\left\{x\in \mathbb{R}:{\mathfrak{R}}_{\beta }^{+}\left({\stackrel{\to }{f}}_{j}\right)\left(x\right)\subset {\mathfrak{R}}_{\beta }^{+}\left(\stackrel{\to }{f}\right)\left(x{\right)}_{\left(\lambda \right)}\right\}.$This together with (2.3) implies

${x∈(−R,R):Rβ+(f→j)(x)⊈Rβ+(f→)(x)(λ)}⊂A1,j∪A2,j∪E∪(R∖A¯).$

It follows that

$|{x∈(−R,R):Rβ+(f→j)(x)⊈Rβ+(f→)(x)(λ)}|≤|A1,j|+|A2,j|+ϵ.$(2.4)

We can write

$|Mβ+(f→j)(x)−Mβ+(f→)(x)|≤sups>01sm−β|∏i=1m∫xx+sfi,j(y)dy−∏i=1m∫xx+sfi(y)dy|≤∑l=1msups>01sm−β∏μ=1l−1∫xx+sfμ(y)dy∏ν=l+1m∫xx+sfν,j(y)dy∫xx+s|fl,j(y)−fl(y)|dy≤∑l=1mMβ+(f→jl)(x)$(2.5)

for any x ϵ ℝ, where ${\stackrel{\to }{f}}_{j}^{l}=\left({f}_{1},\dots ,{f}_{l-1},{f}_{l,j}-{f}_{l},{f}_{l+1,j},\dots ,{f}_{m,j}\right).$From (2.5) we have

$|A1,j|≤|{x∈R:∑l=1mMβ+(f→jl)(x)≥(4γ)−1}|≤∑l=1m|{x∈R:Mβ+(f→jl)(x)≥(4mγ)−1}|≤(4mγ)q∑l=1m∥Mβ+(f→jl)∥Lq(R)q.$(2.6)

Since fi,jfi in Lpi (ℝ) as j ➝ ∞, then there exists N0 = N0(ϵ, γ ) ∈ ℕ \ {0} such that

$∥fi,j−fi∥Lpi(R)<ϵγand∥fi,j∥Lpi(R)≤∥fi∥Lpi(R)+1,∀j≥N0.$(2.7)

(2.7) together with (2.6) and (1.4) yields that

$|A1,j|≤C(m,q,β,p1,…,pm,f→)ϵ,∀j≥N0.$(2.8)

On the other hand, one can easily check that

$|ux,f→j,β+(s)−ux,f→,β+(s)|≤∑l=1mMβ+(f→jl)(x),∀s>0.$

This together with the argument similar to those used in deriving (2.8) implies

$|A2,j|≤C(m,q,β,p1,…,pm,f→)ϵ,∀j≥N0.$(2.9)

It follows from (2.4), (2.8) and (2.9) that

$|{x∈(−R,R):Rβ+(f→j)(x)⊈Rβ+(f→)(x)(λ)}|≤C(m,q,β,p1,…,pm,f→)ϵ,∀j≥N0,$

which gives (2.1) and completes the proof of Lemma 2.1.

We now define the Hausdorff distance between two sets A and B by

$π(A,B):=inf{δ>0:A⊂B(δ)andB⊂A(δ)}.$

The following result can be obtained by Lemma 2.1 and a similar argument as in the proof of Corollary 2.3 in [7], we omit the details.

#### Lemma 2.2

Let $\stackrel{\to }{f}=\left({f}_{1},\dots ,{f}_{m}\right)$with each fi ∈ Lpi (ℝ) for 1 < p1, … , pm < ∞. Let 1 ≤ q < ∞ and 1/q = $\sum _{i=1}^{m}1/{p}_{i}-\beta .$Then, for all ℝ > 0 and λ > 0, we have

$limh→0|{x∈(−R,R):π(Rβ+(f→)(x),Rβ+(f→)(x+h))>λ}|=0.$

The following result presents some formulas for the derivatives of the one-sided multilinear fractional maximal functions, which play the key roles in the proof of the continuity part in Theorem 1.1.

#### Lemma 2.3

Let f̄ = (f1, … , fm) with each fi ϵ W1,pi (ℝ) for 1 < pi < ∞. Let 1 ≤ q < ∞ and $1/q=\sum _{i=1}^{m}1/{p}_{i}-\beta .$

Then, for almost every x ∈, we have

$(Mβ+(f→))′(x)=∑l=1m1sm−β∏1≤j≤mj≠l∫xx+s|fj(y)|dy∫xx+s|fl|′(y)dyforall0(2.10)

$(Mβ+(f→))′(x)=∑l=1m|fl|′(x)∏1≤j≤mj≠l|fj(x)|,ifβ=0and0∈Rβ+(f→)(x),0,if0<β(2.11)

Proof. We may assume that all fi ≥ 0 since |f| ϵ W1,p(ℝ) if f ∈ W1,p(ℝ) with 1 < p < ∞. By the boundedness part in Theorem 1.1 we see that ${\mathfrak{M}}_{\beta }^{+}\left(\stackrel{\to }{f}\right)\in {W}^{1,q}\left(\mathbb{R}\right).$Invoking Lemma 2.2, we can choose a sequence $\left\{{s}_{k}{\right\}}_{k=1}^{\mathrm{\infty }}\phantom{\rule{thinmathspace}{0ex}}{s}_{k}>0$such that limk➝∞ sk = 0 and $\underset{k\to \mathrm{\infty }}{lim}\pi \left({\mathfrak{R}}_{\beta }^{+}\left(\stackrel{\to }{f}\right)\left(x\right),{\mathfrak{R}}_{\beta }^{+}\left(\stackrel{\to }{f}\right)\left(x+{s}_{k}\right)\right)=0$for almost every x ϵ (−R, R). For 1 ≤ im and h ϵ ℝ, we set

$fhi(x)=fτ(h)i(x)−fi(x)handfτ(h)i(x)=fi(x+h).$

It was known that

$∥fτ(sk)i−fi∥Lpi(R)→0ask→∞,∥fski−(fi)′∥Lpi(R)→0ask→∞,$

$∥M+(fτ(sk)i−fi)∥Lpi(R)→0ask→∞,∥M+(fski−fi′)∥Lpi(R)→0ask→∞,∥(Mβ+(f→))sk−(Mβ+(f→))′∥Lq(R)→0ask→∞.$

Here $\left({\mathfrak{M}}_{\beta }^{+}\left(\stackrel{\to }{f}\right){\right)}_{{s}_{k}}\left(x\right)=\frac{1}{{s}_{k}}\left({\mathfrak{M}}_{\beta }^{+}\left(\stackrel{\to }{f}\right)\left(x+{s}_{k}\right)-{\mathfrak{M}}_{\beta }^{+}\left(\stackrel{\to }{f}\right)\left(x\right)\right).$Furthermore, there exists a subsequence $\left\{{h}_{k}{\right\}}_{k=1}^{\mathrm{\infty }}\phantom{\rule{thinmathspace}{0ex}}of\phantom{\rule{thinmathspace}{0ex}}\left\{{s}_{k}{\right\}}_{k=1}^{\mathrm{\infty }}$and a measurable set A1 (−R, R) with |(−R, R)\A1| = 0 such that

1. ${f}_{\tau \left({h}_{k}\right)}^{i}\left(x\right)\to {f}_{i}\left(x\right),{f}_{{h}_{k}}^{i}\left(x\right)\to {f}_{i}^{\prime }\left(x\right),{\mathcal{M}}^{+}\left({f}_{\tau \left({h}_{k}\right)}^{i}-{f}_{i}\right)\left(x\right)\to 0,{\mathcal{M}}^{+}\left({f}_{{h}_{k}}^{i}-{f}_{i}^{\prime }\right)\left(x\right)\to 0\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\left({\mathfrak{M}}_{\beta }^{+}\left(\stackrel{\to }{f}\right){\right)}_{{h}_{k}}\left(x\right)\to \left({\mathfrak{M}}_{\beta }^{+}\left(\stackrel{\to }{f}\right){\right)}^{\prime }\left(x\right)$when k ➝ ∞ for any x ϵ A1 and 1 ≤ im;

2. $\underset{k\to \mathrm{\infty }}{lim}\pi \left({\mathfrak{R}}_{\beta }^{+}\left(\stackrel{\to }{f}\right)\left(x\right),{\mathfrak{R}}_{\beta }^{+}\left(\stackrel{\to }{f}\right)\left(x+{h}_{k}\right)\right)=0$for any x ∈ A1.

Let

$A2:=⋂k=1∞{x∈R:Mβ+(f→)(x+hk)≥ux+hk,f→,β+(0)},A3:=⋂k=1∞{x∈R:Mβ+(f→)(x+hk)=ux+hk,f→,β+(0)if0∈Rβ+(f→)(x+hk)},A4:={x∈R:Mβ+(f→)(x)=ux,f→,β+(0)if0∈Rβ+(f→)(x)}.$

It is obvious that |(−R, R)\Aj| = 0 for j = 2, 3, 4. Let x ϵ A1 ∩ A2 ∩ A3 ∩ A4 be a Lebesgue point of all ${f}_{i}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}{f}_{i}^{\prime }.$Fix $s\in {\mathfrak{R}}_{\beta }^{+}\left(\stackrel{\to }{f}\right)\left(x\right)$, there exists radii ${r}_{k}\in {\mathfrak{R}}_{\beta }^{+}\left(\stackrel{\to }{f}\right)\left(x+{h}_{k}\right)$such that limk➝∞ rk = s. We consider the following two cases:

Case A (s > 0). Without loss of generality we may assume that all rk > 0. Then

$(Mβ+(f→))′(x)=limk→∞1hk(Mβ+(f→)(x+hk)−Mβ+(f→)(x))≤limk→∞1hk1rkm−β(∏i=1m∫x+hkx+hk+rkfi(y)dy−∏i=1m∫xx+rkfi(y)dy)=∑l=1mlimk→∞1rkm−β∏μ=1l−1∫xx+rkfμ(y)dy∏ν=l+1m∫xx+rkfτ(hk)ν(y)dy∫xx+rkfhkl(y)dy.$(2.12)

Since ${f}_{\tau \left({h}_{k}\right)}^{\nu }{\chi }_{\left(x,x+{r}_{k}\right)}\to {f}_{\nu }{\chi }_{\left(x,x+s\right)}$and ${f}_{{h}_{k}}^{l}{\chi }_{\left(x,x+{r}_{k}\right)}\to {f}_{l}^{\prime }{\chi }_{\left(x,x+s\right)}$in L1(ℝ) as k ➝ ∞. Then (2.12) yields that

$(Mβ+(f→))′(x)≤∑l=1m1sm−β∏1≤j≤mj≠l∫xx+sfj(y)dy∫xx+sfl′(y)dy.$(2.13)

On the other hand,

$(Mβ+(f→))′(x)=limk→∞1hk(Mβ+(f→)(x+hk)−Mβ+(f→)(x))≥limk→∞1hk1sm−β(∏i=1m∫x+hkx+hk+sfi(y)dy−∏i=1m∫xx+sfi(y)dy)=∑l=1mlimk→∞1sm−β∏μ=1l−1∫xx+sfμ(y)dy∏ν=l+1m∫xx+sfτ(hk)ν(y)dy∫xx+sfhkl(y)dy=∑l=1m1sm−β∏1≤j≤mj≠l∫xx+sfj(y)dy∫xx+sfl′(y)dy.$(2.14)

Combining (2.14) with (2.13) yields that (2.10) holds for almost every x ∈ (−R, R).

Case B (s = 0). We shall discuss the following two cases:

• (i)

When 0 < β < m. Since${\mathfrak{M}}_{\beta }^{+}\left(\stackrel{\to }{f}\right)\left(x\right)=0,$then all fi(y 0 for almost every y ϵ (x,∞). Thus${\mathfrak{M}}_{\beta }^{+}\left(\stackrel{\to }{f}\right)\left(y\right)\equiv 0$for yx. It follows that

$(Mβ+(f→))′(x)=limk→∞1hk(Mβ+(f→)(x+hk)−Mβ+(f→)(x))=0.$

This yields that (2.11) holds for almost every x ∈ (−R, R) in this case 0 < β < m.

• (ii)

When β = 0. We notice that

$limk→∞1hk(∏i=1mfi(x+hk)−∏i=1mfi(x))=∑l=1mlimk→∞fhkl(x)∏μ=1l−1fμ(x)∏ν=l+1mfν(x+hk)=∑l=1mfl′(x)∏1≤j≤mj≠lfj(x).$(2.15)

It follows that

$(Mβ+(f→))′(x)=limk→∞1hk(Mβ+(f→)(x+hk)−Mβ+(f→)(x))≥limk→∞1hk(∏i=1mfi(x+hk)−∏i=1mfi(x))=∑l=1mfl′(x)∏1≤j≤mj≠lfj(x).$(2.16)

Below we estimate the upper bound of $\left({\mathfrak{M}}_{\beta }^{+}\left(\stackrel{\to }{f}\right){\right)}^{\prime }\left(x\right).$If there exists k0 ϵ ℕ\ {0} such that sk > 0 for any kk0, then, by the argument similar to those used in deriving (2.12),

$(Mβ+(f→))′(x)≤∑l=1mlimk→∞(∏μ=1l−11rk∫xx+rkfμ(y)dy)(∏ν=l+1m1rk∫xx+rkfτ(hk)ν(y)dy)×(1rk∫xx+rkfhkl(y)dy).$(2.17)

Since

$|limk→∞1rk∫xx+rkfτ(hk)ν(y)dy−fν(x)|≤limk→∞1rk∫xx+rk|fτ(hk)ν(y)−fν(y)|dy≤limk→∞M+(fτ(hk)ν−fν)(x)=0.$

It follows that

$limk→∞1rk∫xx+rkfτ(hk)ν(y)dy=fν(x).$(2.18)

Similarly,

$limk→∞1rk∫xx+rkfhkl(y)dy=fl′(x).$(2.19)

It follows from (2.17)-(2.19) that

$(Mβ+(f→))′(x)≤∑l=1mfl′(x)∏1≤j≤mj≠lfj(x).$(2.20)

If sk = 0 for infinitely many k, then, by (2.15) we have

$(Mβ+(f→))′(x)=limk→∞1hk(Mβ+(f→)(x+hk)−Mβ+(f→)(x))=limk→∞1hk(∏i=1mfi(x+hk)−∏i=1mfi(x))=∑l=1mfl′(x)∏1≤j≤mj≠lfj(x).$

This together with (2.16) and (2.20) yields that (2.11) holds for almost every x ∈ (−R, R) in the case β = 0. Since R was arbitrary, this proves Lemma 2.3.

## 3 Proofs of Theorems 1.1-1.2

In this section we shall prove Theorems 1.1-1.2. Let us begin with the proof of Theorem 1.1.

Proof of Theorem 1.1. We only prove Theorem 1.1 for ${\mathfrak{M}}_{\beta }^{+}$and the other case is analogous. Let {sk}k⪰1 be an enumeration of positive rational numbers. Then we can write

$Mβ+(f→)(x)=supk≥11skm−β∏i=1m∫xx+sk|fi(y)|dy.$

Define the family of operators {Tk}k≥1 by

$Tk(f→)(x)=max1≤i≤k1sim−β∏j=1m∫xx+si|fj(y)|dy.$

Fix x, h ϵ ℝ, one has

$|Tk(f→)(x+h)−Tk(f→)(x)|≤max1≤i≤k1sim−β|∏j=1m∫x+hx+h+si|fj(y)|dy−∏j=1m∫xx+si|fj(y)|dy|≤∑l=1mmax1≤i≤k1sim−β∏μ=1l−1∫xx+si|fμ(y)|dy∏ν=l+1m∫xx+si|fτ(h)ν(y)|dy∫xx+si|fτ(h)l(y)−fl(y)|dy.$

It follows that

$(Tk(f→))′(x)≤∑l=1mMβ+(f→l)(x)$(3.1)

for almost every x ∈ ℝ, where ${\stackrel{\to }{f}}^{l}=\left({f}_{1},\dots ,{f}_{l-1},{f}_{l}^{\prime },{f}_{l+1},\dots ,{f}_{m}\right).$Here we used the fact that ||fj|'(x)| = |f'(x)| for almost every x ϵ ℝ. By (3.1), (1.4) and Minkowski’s inequality, we obtain

$∥Tk(f→)∥1,q≤∥Tk(f→)∥Lq(R)+∥(Tk(f→))′∥Lq(R)≤∥Mβ+(f→)∥Lq(R)+∥∑l=1mMβ+(f→l)∥Lq(R)≤C(β,p1,…,pm)(∏i=1m∥fi∥Lpi(R)+∑l=1m∥fl′∥Lpl(R)∏1≤j≤mj≠l∥fj∥Lpj(R))≤C(m,β,p1,…,pm)∏i=1m∥fi∥1,pi.$

Therefore, $\left\{{T}_{k}\left(\stackrel{\to }{f}\right)\right\}$is a bounded sequence in W1, q(ℝ) which converges to ${\mathfrak{M}}_{\beta }^{+}\left(\stackrel{\to }{f}\right)$pointwise. The weak compactness of Sobolev spaces implies that ${\mathfrak{M}}_{\beta }^{+}\left(\stackrel{\to }{f}\right)\in {W}^{1,q}\left(\mathbb{R}\right),\phantom{\rule{thinmathspace}{0ex}}{T}_{k}\left(\stackrel{\to }{f}\right)$converges to ${\mathfrak{M}}_{\beta }^{+}\left(\stackrel{\to }{f}\right)$weakly in Lq(ℝ) and $\left({T}_{k}\left(\stackrel{\to }{f}\right){\right)}^{\prime }$converges to $\left({\mathfrak{M}}_{\beta }^{+}\left(\stackrel{\to }{f}\right){\right)}^{\prime }$weakly in Lq(ℝ). This together with (3.1) yields that

$|(Mβ+(f→))′(x)|≤∑l=1mMβ+(f→l)(x)$(3.2)

for almost every x ∈ ℝ. It follows from (3.2) and (1.4) that

$∥Mβ+(f→)∥1,q=∥Mβ+(f→)∥Lq(R)+∥(Mβ+(f→))′∥Lq(R)≤C(m,β,p1,…,pm)∏i=1m∥fi∥1,pi.$

This completes the boundedness part of Theorem 1.1.

We now prove the continuity for${\mathfrak{M}}_{\beta }^{+}$by employing the idea in [20]. Let β, m, p1, … , pm, q be given as in

Theorem 1.1. Let = (f1, … , fm) with each fi ϵ W1,pi (ℝ) and j = (f1,j , … , fm,j) such that fi,jfi in W1,pi (ℝ) when j ➝ ∞. We get from (2.5) that

$|Mβ+(f→j)(x)−Mβ+(f→)(x)|≤∑l=1mMβ+(f→jl)(x)$(3.3)

for any x ∈ ℝ, where ${\stackrel{\to }{f}}_{j}^{l}$is given as in (2.5). (3.3) together with (1.4) implies that

$∥Mβ+(f→j)−Mβ+(f→)∥Lq(R)≤∑l=1m∥Mβ+(f→jl)∥Lq(R)≤C(m,β,p1,…,pm)∑l=1m∥fl,j−fl∥Lpl(R)∏μ=1l−1∥fμ∥Lpμ(R)∏ν=l+1m∥fν,j∥Lpν(R).$

It follows that

$∥Mβ+(f→j)−Mβ+(f→)∥Lq(R)→0whenj→∞.$

Hence, to prove the continuity for${\mathfrak{M}}_{\beta }^{+},$it suffices to show that

$∥(Mβ+(f→j))′−(Mβ+(f→))′∥Lq(R)→0whenj→∞.$(3.4)

Below we prove (3.4). We may assume that all fi,j ≥ 0 and fi ≥ 0. For 1 ≤ lm, we set ${\stackrel{\to }{f}}^{l}=$$\left({f}_{1},\dots ,{f}_{l-1},{f}_{l}^{\prime },{f}_{l+1},\dots ,{f}_{m}\right).$Fix ϵ ∈ (0, 1). We can choose R > 0 such that $\sum _{l=1}^{m}\parallel {\mathfrak{M}}_{\beta }^{+}\left({\stackrel{\to }{f}}^{l}\right){\parallel }_{q,{B}_{1}}<ϵ$with B1 = (−1, −R)(R,∞). The absolute continuity implies that there exists η > 0 such that$\sum _{l=1}^{m}\parallel {\mathfrak{M}}_{\beta }^{+}\left({\stackrel{\to }{f}}^{l}\right){\parallel }_{q,B}<\phantom{\rule{thinmathspace}{0ex}}ϵ$for any measurable subset B of (−R, R) with |B| < η. As already observed, for almost every x ∈ ℝ, the function ${u}_{x,{\stackrel{\to }{f}}^{l},\beta }^{+}$is uniformly continuous on [0,∞). Therefore, for almost every x ∈ ℝ, the function $\sum _{l=1}^{m}{u}_{x,{\stackrel{\to }{f}}^{l},\beta }^{+}$is uniformly continuous on [0,∞). We can find δ(x) > 0 such that

$|∑l=1mux,f→l,β+(s1)−∑l=1mux,f→l,β+(s2)|

We can write (−R, R) as

$(−R,R)=(⋃k=1∞{x∈(−R,R):δ(x)>1k})∪N,$

where |N| = 0. We can choose δ > 0 such that

$|{x∈(−R,R):|∑l=1mux,f→l,β+(s1)−∑l=1mux,f→l,β+(s2)|≥R−1/qϵforsomes1,s2with|s1−s2|≤δ}|=:|B2|<η2.$

By Lemma 2.1, there exists N1 ∈ ℕ \ {0} such that

$|{x∈(−R,R):Rβ+(f→j)(x)⊈Rβ+(f→)(x)(δ)}|=:|Bj|<η2∀j≥N1.$

Fix jN1. Let ${\stackrel{\to }{f}}_{l,j}=\left({f}_{1,j},\dots ,{f}_{l-1,j},{f}_{l,j}^{\prime },{f}_{l+1,j},\dots ,{f}_{m,j}\right)$and $s\in {\mathfrak{R}}_{\beta }^{+}\left({\stackrel{\to }{f}}_{j}\right)\left(x\right).$We consider the following two cases:

• (i)

s > 0. We can write

$|ux,f→l,j,β+(s)−ux,f→l,β+(s)|=1sm−β|∏1≤μ≤mμ≠l∫xx+sfμ,j(y)dy∫xx+sfl,j′(y)dy−∏1≤μ≤mμ≠l∫xx+sfμ(y)dy∫xx+sfl′(y)dy|≤∑μ=1l−1Mβ+(F→μ,j)(x)+∑ν=l+1mMβ+(G→ν,j)(x)+Mβ+(H→l,j)(x)=:Gl,j(x),$(3.5)

where

$F→μ,j=(f1,…,fμ−1,fμ,j−fμ,fμ+1,j,…,fl−1,j,fl,j′,fl+1,j,…,fm,j),G→ν,j=(f1,…,fl−1,fl′,fl+1,…,fν−1,fν,j−fν,fν+1,j,…,fm,j),H→l,j=(f1,…,fl−1,fl,j′−fl′,fl+1,j,…,fm,j).$

• (ii)

s = 0. If 0 < β < m, $|{u}_{x,{\stackrel{\to }{f}}_{l,j},\beta }\left(s\right)-{u}_{x,{\stackrel{\to }{f}}^{l},\beta }\left(s\right)|=0.$If β = 0, then we have

$|ux,fl,j→,β+(s)−ux,fl→,β+(s)|≤∑μ=1l−1(∏l1=1μ−1fl1(x))(fμ,j(x)−fμ(x))(∏l2=μ+1l−1fl2,j(x))|fl,j′(x)|(∏l3=l+1mfl3,j(x))+∑ν=l+1m(∏l1=1l−1fl1(x))|fl′(x)|(∏l2=l+1ν−1fl2(x))|fν,j(x)−fν(x)|(∏l3=ν+1mfl3,j(x))+(∏l1=1l−1fl1(x))|fl,j′(x)−fl′(x)|(∏l2=l+1mfl2,j(x)).$

This together with (3.5) and the Lebesgue differentiation theorem leads to

$|ux,f→l,j,β+(s)−ux,f→l,β+(s)|≤Gl,j(x)$(3.6)

for almost every x ∈ ℝ and $s\in {\mathfrak{R}}_{\beta }^{+}\left({\stackrel{\to }{f}}_{j}\right)\left(x\right).$By (3.6) and Lemma 2.3, we obtain

$|(Mβ+(f→j))′(x)−(Mβ+(f→))′(x)|=|∑l=1mux,f→l,j,β(s1)−∑l=1mux,f→l,β(s2)|≤|∑l=1mux,f→l,j,β(s1)−∑l=1mux,f→l,β(s1)|+|∑l=1mux,f→l,β(s1)−∑l=1mux,f→l,β(s2)|≤∑l=1mGl,j(x)+|∑l=1mux,f→l,β(s1)−∑l=1mux,f→l,β(s2)|$(3.7)

for almost every x ∈ ℝ and any ${s}_{1}\in {\mathfrak{R}}_{\beta }^{+}\left({\stackrel{\to }{f}}_{j}\right)\left(x\right)\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}{s}_{2}\in {\mathfrak{R}}_{\beta }^{+}\left(\stackrel{\to }{f}\right)\left(x\right).$On can easily check that

$limj→∞∥Gl,j∥Lq(R)=0,∀1≤l≤m.$

It follows that there exists N2 ∈ ℕ \ {0} such $\sum _{l=1}^{m}\parallel {\mathfrak{G}}_{l,j}{\parallel }_{{L}^{q}\left(\mathbb{R}\right)}<ϵ$for any jN2.

If x B1 ∪ B2 ∪ Bj, we can choose ${s}_{1}\in {\mathfrak{R}}_{\beta }^{+}\left({\stackrel{\to }{f}}_{j}\right)\left(x\right)\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}{s}_{2}\in {\mathfrak{R}}_{\beta }^{+}\left(\stackrel{\to }{f}\right)\left(x\right)$such that |s1s2|≤ δ and

$|∑l=1mux,f→l,β(s1)−∑l=1mux,f→l,β(s2)|<|R|−1/qϵ.$

On the other hand, we have that for any $l=1,2,\dots ,m,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{s}_{1}\in {\mathfrak{R}}_{\beta }^{+}\left({\stackrel{\to }{f}}_{j}\right)\left(x\right)\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}{s}_{2}\in {\mathfrak{R}}_{\beta }^{+}\left(\stackrel{\to }{f}\right)\left(x\right),$

$|∑l=1mux,f→l,β(s1)−∑l=1mux,f→l,β(s2)|≤2∑l=1mMβ+(f→l)(x).$

Note that |B2Bj| < η for any jN1. Thus we get from (3.7) that

$∥(Mβ+(f→j))′−(Mβ+(f→))′∥Lq(R)≤∥∑l=1mGl,j∥Lq(R)+∥|R|−1/qϵ∥q,(−R,R)+2∥∑l=1mMβ+(f→l)∥q,B1∪B2∪Bj≤Cϵ,$

for any j ≥ max{N1, N2}, which leads to

$limj→∞∥(Mβ+(f→j))′−(Mβ+(f→))′∥Lq(R)=0.$

This yields (3.4) and completes the proof of Theorem 1.1.

Proof of Theorem 1.2. The proof is similar to the proof of Theorem 2.3 in [5]. We omit the details.

## 4 Proof of Theorem 1.3

We only prove Theorem 1.3 for ${\mathrm{M}}_{\beta }^{+}$and the other case is analogous.

1. Step 1: proof of the boundedness for${\mathrm{M}}_{\beta }^{+}.$We shall adopt the method in [31] to prove the boundedness for${\mathrm{M}}_{\beta }^{+}.$

Let $\stackrel{\to }{f}=\left({f}_{1},\dots ,{f}_{m}\right)$with each fi ∈ ℓ1(ℤ). Without loss of generality, we may assume fi ≥ 0. For convenience, let Г(x) = (x + 1)βm − (x + 2)βm for any x ≥ 0. One can easily check that Г(x) is decreasing on [0,∞) and Σn∈N Г(n) = 1. Since all fi ∈ ℓ1(ℤ), then, for any n ∈ ℤ, there exists sn ∈ ℕ such that ${\mathrm{M}}_{\beta }^{+}\left(\stackrel{\to }{f}\right)\left(n\right)={\mathrm{A}}_{{s}_{n}}\left(\stackrel{\to }{f}\right)\left(n\right),$where

$As(f→)(n)=(s+1)β−m∏i=1m∑k=0sfi(n+k)$

for any s ∈ ℕ and n ∈ ℤ. Let

$X+={n∈Z:Mβ+(f→)(n+1)>Mβ+(f→)(n)}andX−={n∈Z:Mβ+(f→)(n)≥Mβ+(f→)(n+1)}.$

Then we can write

$Var(Mβ+(f→))=∑n∈X+(Mβ+(f→)(n+1)−Mβ+(f→)(n))+∑n∈X−(Mβ+(f→)(n)−Mβ+(f→)(n+1))≤∑n∈X+(Asn+1(f→)(n+1)−Asn+1+1(f→)(n))+∑n∈X−(Asn(f→)(n)−Asn+1(f→)(n+1)).$(4.1)

Fix n ∈ ℤ, by direct computations we obtain

$Asn+1(f→)(n+1)−Asn+1+1(f→)(n)=(sn+1+1)β−m∏i=1m∑k=0sn+1fi(n+1+k)−(sn+1+2)β−m∏i=1m∑k=0sn+1+1fi(n+k)≤∑l=1m((sn+1+1)β−m∑k=0sn+1fl(n+1+k)−(sn+1+2)β−m∑k=0sn+1+1fl(n+k))×∏μ=1l−1∑k=0sn+1+1fμ(n+k)∏ν=lm∑k=0sn+1fν(n+1+k).$(4.2)

Since

$(sn+1+1)β−m∑k=0sn+1fl(n+1+k)−(sn+1+2)β−m∑k=0sn+1+1fl(n+k)≤(sn+1+1)β−m∑k∈Zfl(k)χ[n+1,n+sn+1+1](k)−(sn+1+2)β−m∑k∈Zfl(k)χ[n,n+sn+1+1](k)≤∑k∈Zfl(k)Γ(sn+1)χ[n+1,n+sn+1+1](k)≤∑k∈Zfl(k)Γ(k−n−1)χ(n,∞)(k).$(4.3)

Combining (4.3) with (4.2) yields that

$Asn+1(f→)(n+1)−Asn+1+1(f→)(n)≤∑l=1m∏1≤j≤mj≠l∥fj∥ℓ1(Z)(∑k∈Zfl(k)Γ(k−n−1)χ(n,∞)(k)).$(4.4)

On the other hand, one finds

$Asn(f→)(n)−Asn+1(f→)(n+1)=(sn+1)β−m∏i=1m∑k=0snfi(n+k)−(sn+2)β−m∏i=1m∑k=0sn+1fi(n+1+k)=∑l=1m((sn+1)β−m∑k=0snfl(n+k)−(sn+2)β−m∑k=0sn+1fl(n+1+k))×∏μ=1l−1∑k=0sn+1fμ(n+1+k)∏ν=l+1m∑k=0snfν(n+k).$

It follows that

$Asn(f→)(n)−Asn+1(f→)(n+1)≤∑l=1m((sn+1)β−m∑k∈Zfl(k)χ[n,n+sn](k)−(sn+2)β−m∑k∈Zfl(k)χ[n+1,n+sn+2](k))×∏1≤j≤mj≠l∥fj∥ℓ1(Z)≤∑l=1m∏1≤j≤mj≠l∥fj∥ℓ1(Z)(∑k∈Zfl(k)Γ(sn)χ[n+1,n+sn+1](k)+fl(n))≤∑l=1m∏1≤j≤mj≠l∥fj∥ℓ1(Z)(∑k∈Zfl(k)Γ(k−n−1)χ(n,∞)(k)+fl(n)).$(4.5)

(4.1) and (4.4)-(4.5) imply that

$Var(Mβ+(f→))≤∑l=1m∏1≤j≤mj≠l∥fj∥ℓ1(Z)(∑k∈Zfl(k)(∑n∈X+n

Step 2: proof of the continuity for ${\mathrm{M}}_{\beta }^{+}.\phantom{\rule{thinmathspace}{0ex}}Let\phantom{\rule{thinmathspace}{0ex}}\stackrel{\to }{f}=\left({f}_{1},\dots ,{f}_{m}\right)$with each fj ∈ ℓ1(ℤ) and ${\stackrel{\to }{f}}_{j}=\left({f}_{1,j},\dots ,{f}_{m,j}\right)$such that fi,jfi in 1(ℤ) as j ➝ 1. By the boundedness part in Theorem 1.3, we know that $\left({\mathrm{M}}_{\beta }^{+}\left(\stackrel{\to }{f}\right){\right)}^{\prime }\in {\ell }^{1}\left(\mathbb{Z}\right).$Without loss of generality we may assume that all fi,j ≥ 0 and fi ≥ 0 since |fj| − |f ||≤| fj |. We want to show that

$limj→∞∥(Mβ+(f→j))′−(Mβ+(f→))′∥ℓ1(Z)=0.$(4.6)

Given ϵ ∈ (0, 1), there exists ${N}_{1}={N}_{1}\left(ϵ,\stackrel{\to }{f}\right)>0$such that

$∥fi,j−fi∥ℓ1(Z)<ϵ$(4.7)

and

$∥fi,j∥ℓ1(Z)≤∥fi,j−fi∥ℓ1(Z)+∥fi∥ℓ1(Z)<∥fi∥ℓ1(Z)+1$(4.8)

for any jN1 and all 1 ≤ im. We get from (4.7)-(4.8) that

$|Mβ+(f→j)(n)−Mβ+(f→)(n)|≤sups∈N(s+1)β−m|∏i=1m∑k=0sfi,j(n+k)−∏i=1m∑k=0sfi(n+k)|≤sups∈N(s+1)β−m∑l=1m∑k=0s|fi,j(n+k)−fi(n+k)|∏μ=1l−1∑k=0sfμ(n+k)∏ν=l+1m∑k=0sfν,j(n+k)≤∑l=1m∥fl,j−fl∥ℓ1(Z)∏μ=1l−1∥fμ∥ℓ1(Z)∏ν=l+1m∥fν,j∥ℓ1(Z)≤c(f→)ϵ$

for any n ∈ ℤ and jN1, which implies that ${\mathrm{M}}_{\beta }^{+}\left({\stackrel{\to }{f}}_{j}\right)\to {\mathrm{M}}_{\beta }^{+}\left(\stackrel{\to }{f}\right)$pointwise as j ➝ ∞ and

$limj→∞(Mβ+(f→j))′(n)=(Mβ+(f→))′(n)$(4.9)

for all n ∈ ℤ. By the fact that $\left({\mathrm{M}}_{\beta }^{+}\left(\stackrel{\to }{f}\right){\right)}^{\prime }\in {\ell }^{1}\left(\mathbb{Z}\right)$and the classical Brezis-Lieb lemma in [35], to prove (4.6), it suffices to show that

$limj→∞∥(Mβ+(f→j))′∥ℓ1(Z)=∥(Mβ+(f→))′∥ℓ1(Z).$(4.10)

By (4.9) and Fatou’s lemma, one finds

$∥(Mβ+(f→))′∥ℓ1(Z)≤lim infj→∞∥(Mβ+(f→j))′∥ℓ1(Z).$

Thus, to prove (4.10), it suffices to show that

$lim supj→∞∥(Mβ+(f→j))′∥ℓ1(Z)≤∥(Mβ+(f→))′∥ℓ1(Z).$(4.11)

We now prove (4.11). Since each fi Є ℓ1(ℤ), then there exists a sufficiently large positive integer R1 = R1(ϵ, f⃗ ) such that

$sup1≤i≤m∑|n|≥R1fi(n)<ϵ.$(4.12)

Note that

$sup1≤i≤m∑|n|≥R1fi(n)<ϵ.$

It follows that there exists an integer R2 = R2(ϵ) > 0 such that ${\mathrm{M}}_{\beta }^{+}\left(\stackrel{\to }{f}\right)\left(n\right)<ϵ$for all |n| ≥ R 2. Moreover, there exists an integer R3 > 0 such that sβm < ∈ if sR3 since β < m. Let R = max{R1, R2, R3}. (4.9) yields that there exists an integer N 2 = N (ϵ, R) > 0 such that

$|(Mβ+(f→j))′(n)−(Mβ+(f→))′(n)|≤ϵ4R+2$(4.13)

for any jN2 and |n| ⪯ 2R. From (4.13) we have

$∥(Mβ+(f→j))′∥ℓ1(Z)≤∑|n|≤2R|(Mβ+(f→j))′(n)−(Mβ+(f→))′(n)|+∥(Mβ+(f→))′∥ℓ1(Z)+∑|n|≥2R|(Mβ+(f→j))′(n)|≤∥(Mβ+(f→))′∥ℓ1(Z)+ϵ+∑|n|≥2R|(Mβ+(f→j))′(n)|$(4.14)

for any jN2. Fix jN2 and set

$Xj+={|n|≥2R:Mβ+(f→j)(n+1)>Mβ+(f→j)(n)},Xj−={|n|≥2R:Mβ+(f→j)(n)≥Mβ+(f→j)(n+1)}.$

Since all fi,j1(ℤ), then, for any n ∈ ℤ, there exists rn ∈ ℕ such that ${\mathrm{M}}_{\beta }^{+}\left({\stackrel{\to }{f}}_{j}\right)\left(n\right)={\mathrm{A}}_{{r}_{n}}\left({\stackrel{\to }{f}}_{j}\right)\left(n\right).$Then we have

$∑|n|≥2R|(Mβ+(f→j)′(n)|=∑n∈Xj+(Mβ+(f→j)(n+1)−Mβ+(f→j)(n))+∑n∈Xj−(Mβ+(f→j)(n)−Mβ+(f→j)(n+1))≤∑n∈Xj+(Arn+1(f→j)(n+1)−Arn+1+1(f→j)(n))+∑n∈Xj−(Arn(f→j)(n)−Arn+1(f→j)(n+1)).$(4.15)

By the arguments similar to those used in deriving (4.4) and (4.5), one has

$Arn+1(f→j)(n+1)−Arn+1+1(f→j)(n)≤∑l=1m∏1≤μ≤mμ≠l∥fμ,j∥ℓ1(Z)(∑k∈Zfl,j(k)Γ(k−n−1)χ(n,∞)(k)),$(4.16)

$Arn(f→j)(n)−Arn+1(f→j)(n+1)≤∑l=1m∏1≤μ≤mμ≠l∥fμ,j∥ℓ1(Z)(∑k∈Zfl,j(k)Γ(k−n−1)χ(n,∞)(k)+fl,j(n)).$(4.17)

It follows from (4.13)-(4.15) that

$∑|n|≥2R|(Mβ+(f→j)′(n)|≤∑l=1m∏1≤μ≤mμ≠l∥fμ,j∥ℓ1(Z)(∑n∈Xj+∑k∈Zfl,j(k)Γ(k−n−1)χ(n,∞)(k))+∑l=1m∏1≤μ≤mμ≠l∥fμ,j∥ℓ1(Z)(∑n∈Xj−∑k∈Zfl,j(k)Γ(k−n−1)χ(n,∞)(k)+∑n∈Xj−fl,j(n))≤∑l=1m∏1≤μ≤mμ≠l∥fμ,j∥ℓ1(Z)(∑|n|≥2R∑k∈Zfl,j(k)Γ(k−n−1)χ(n,∞)(k)+∑|n|≥2Rfl,j(n)).$(4.18)

By (4.7)-(4.8) and (4.12), we obtain

$∑|n|≥2R∑k∈Zfl,j(k)Γ(k−n−1)χ(n,∞)(k)≤∑k∈Zfl,j(k)∑|n|≥2Rn(4.19)

for any jN1. It follows from (4.8), (4.12) and (4.18)-(4.19) that

$∑|n|≥2R|(Mβ+(f→j))′(n)|≤C(f→)ϵ$(4.20)

for any jN1. Combining (4.20) with (4.14) yields that

$∥(Mβ+(f→j))′∥ℓ1(Z)≤∥(Mβ+(f→))′∥ℓ1(Z)+Cϵ$

for any j ≥ max {N1, N 2}. This proves (4.11) and finishes the proof of Theorem 1.3.

## Acknowledgement

This work was supported partly by the National Natural Science Foundation of China (Grant No. 11701333) and the Support Program for Outstanding Young Scientific and Technological Top-notch Talents of College of Mathematics and Systems Science (Grant No. Sxy2016K01).

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Accepted: 2018-11-26

Published Online: 2018-12-31

Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 1556–1572, ISSN (Online) 2391-5455,

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