# Regularity of Commutators of the One-Sided Hardy-Littlewood Maximal Functions

Authors: Daiqing Zhang
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this paper, the regularity properties of two classes of commutators of the one-sided Hardy-Littlewood maximal functions and their fractional variants are investigated. Some new bounds for the derivatives of the above commutators and the boundedness and continuity for the above commutators on the Sobolev spaces will be presented. The corresponding results for the discrete analogues are also considered.


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## Body

## 1. Introduction

The regularity theory of maximal operators has been the subject of many recent articles in harmonic analysis. One of the driving questions in this theory is whether a given maximal operator improves, preserves, or destroys a priori regularity of an initial datumf. The question was first studied by Kinnunen [1], who showed that the usual centered Hardy-Littlewood maximal function M is bounded on the first order Sobolev spaces W1,pℝd for all 1<p≤∞. Recall that the Sobolev spaces W1,pℝd, 1≤p≤∞, are defined by
(1)W1,pℝd≔f:ℝd⟶ℝ:f1,p=fLpℝd+∇fLpℝd<∞,where ∇f is the weak gradient of f. It was noted that the W1,p-bound for the uncentered maximal operator M~ also holds by a simple modification of Kinnunen’s arguments or Theorem 1 of [12]. Later on, Kinnunen’s result was extended to a local version in [2], to a fractional version in [3], to a multisublinear version in [4, 5], and to a one-sided version in [6]. Due to the lack of sublinearity for M at the derivative level, the continuity of M:W1,pℝd⟶W1,pℝd for 1<p<∞ is certainly a nontrivial issue. This problem was addressed by Luiro [7] in the affirmative and was later extended to the local version in [8] and the multisublinear version in [4, 9]. Other works on the regularity of maximal operators can be consulted in [10, 11]. Since the map M:L1ℝd⟶L1ℝd is not bounded, the W1,1-regularity for the maximal operator seems to be a deeper issue. A crucial question was posed by Hajłasz and Onninen in [12]: Is the mapf↦∇Mfbounded fromW1,1ℝdtoL1ℝd? A complete solution was obtained only in dimension d=1 (see [13–16] for an example), and partial progress on the general dimension d≥2 was given by Hajłasz and Malý [17] and Luiro [18]. For other interesting works related to this theory, we suggest the readers to consult [19–22], among others.Very recently, Liu et al. [23] investigated the regularity of commutators of the Hardy-Littlewood maximal function. Precisely, let b be a locally integrable function defined on ℝn, we define the commutator of the Hardy-Littlewood maximal function b,M by
(2)b,Mfx=bxMfx−Mbfx,x∈ℝn.The maximal commutator ofM with b is defined by
(3)Mbfx=supr>01Bx,r∫Bx,rbx−byfydy,where Bx,r is the open ball in ℝd centered at x with radius r and volume Bx,r.We now list the main result of [23] as follows:Theorem 1 (see [23]).
Let1<p1,p2,p<∞ and 1/p=1/p1+1/p2. If b∈W1,p2ℝd, then
(i)
The mapb,M:W1,p1ℝd⟶W1,pℝd is bounded and continuous. In particular, if f∈W1,p1ℝd, then
(4)∇b,Mfx≤∇bxMfx+bxM∇fx+M∇bfx,for almost every x∈ℝd. Moreover,
(5)b,Mf1,p≤Cp1,p2,db1,p2f1,p1.(ii)
The mapMb:W1,p1ℝd⟶W1,pℝd is bounded. Moreover, if f∈W1,p1ℝd, then
(6)Mbf1,p≤Cp1,p2,db1,p2f1,p1.The main motivations of this work not only extend Theorem1 to a one-sided setting but also investigate the regularity properties of the discrete analogue for commutators of the one-sided Hardy-Littlewood maximal functions and their fractional variants. Let us recall some definitions and backgrounds. For 0≤β<1, the one-sided fractional maximal operators Mβ+ and Mβ− are defined by
(7)Mβ+fx=sups>01s1−β∫xx+sfydy,Mβ−fx=supr>01r1−β∫x−rxfydy.Whenβ=0, the operators Mβ+ (resp., Mβ−) reduce to the one-sided Hardy-Littlewood maximal functions M+ (resp., M−). The study of the one-sided maximal operators originated ergodic maximal operator (see [24]). The one-sided fractional maximal operators have a close connection with the well-known Riemann-Liouville fractional integral operator and the Weyl fractional integral operator (see [25]). It was known that Mβ+ is of type p,q for 1<p<∞, 0≤β<1/p and q=p/1−pβ. For p=1 we have Mβ+:L1ℝ⟶L1/1−β,∞ℝ bounded. The same conclusions hold for Mβ−.In order to establish theW1,1-regularity for the one-dimensional uncentered Hardy-Littlewood maximal function, Tanaka [16] first studied the regularity of M+ and M−. Precisely, Tanaka proved that if f∈W1,1ℝ, then the distributional derivatives of M+f and M−f are integrable functions, and
(8)M+f′L1ℝ≤f′L1ℝ,M−f′L1ℝ≤f′L1ℝ.A combination of arguments in [15, 16] yields that both M+f and M−f are absolutely continuous on ℝ. Later on, Liu and Mao [6] proved that both M+ and M− map W1,pℝ⟶W1,pℝ boundedly and continuously for 1<p<∞. Similar arguments to those in Remark (iii) in [1] can be used to conclude that both M+ and M− map W1,∞ℝ⟶W1,∞ℝ boundedly. Recently, the main result of [6] was extended to the fractional version in [26] and to the multilinear case in [27]. We now introduce the partial result of [26] as follows:Theorem 2 (see [26]).
Let1<p≤∞, 0≤β<1/p, and 1/q=1/p−β. Then, the map Mβ+:W1,pℝ⟶W1,qℝ is bounded and continuous. Moreover, if f∈W1,pℝ, then
(9)Mβ+f′x≤Mβ+f′x,for almost every x∈ℝ. The same conclusions hold for the operator Mβ−.Now we introduce two classes of commutators of the one-sided fractional maximal functions.Definition 3.
Letb:ℝ⟶ℝ be a locally integrable function and β∈0,1. The commutators of the one-sided fractional maximal function b,Mβ+ and b,Mβ− are defined by
(10)b,Mβ+fx=bxMβ+fx−Mβ+bfx,x∈ℝ,b,Mβ−fx=bxMβ−fx−Mβ−bfx,x∈ℝ.Definition 4.
Letb:ℝ⟶ℝ be a locally integrable function and β∈0,1. The maximal commutators of Mβ+ and Mβ− with b are defined, respectively, by
(11)Mb,β+fx=supr>01r1−β∫xx+rbx−byfydy,x∈ℝ,Mb,β−fx=supr>01r1−β∫x−rxbx−byfydy,x∈ℝ.
It should be pointed out that the following facts are useful in proving our main results.Remark 5.
(i) The operatorb,Mβ+ is neither positive nor sublinear. By Hölder’s inequality and the Lp-bounds and continuity for Mβ+, we have that the map b,Mβ+:Lp1ℝ⟶Lqℝ is bounded and continuous, provided that 1<p1,p2,q<∞, 0≤β<1/p1, 1/q=1/p1+1/p2−β, 1/p1+1/p2<1, and b∈Lp2ℝ. Moreover,
(12)b,Mβ+fLqℝ≤Cp1,p2,βfLp1ℝbLp2ℝ.
The same conclusions also hold forb,Mβ−.
(ii) The operatorMb,β+ is positive and sublinear. Clearly
(13)Mb,β+fx≤bxMβ+fx+Mβ+bfx,forallx∈ℝ.
Inequality (13) together with Hölder’s inequality, the bounds, and sublinearity of Mβ+ yields that the map Mb,β+:Lp1ℝ⟶Lqℝ is bounded if 1<p1,p2,q<∞, 0≤β<1/p1, 1/p1+1/p2<1, 1/q=1/p1+1/p2−β, and b∈Lp2ℝ. Moreover,
(14)Mb,β+fLqℝ≤Cp1,p2,βfLp1ℝbLp2ℝ.
The same conclusions also hold forMb,β−.
Based on the above, it is a natural question to ask whether the commutatorsb,Mβ+, b,Mβ−, Mb,β+, and Mb,β− have somewhat regularity properties. This is one main motivation of this paper, which can be addressed by the following results.Theorem 6.
Let1<p1,p2,q<∞, 0≤β<1/p1, 1/p1+1/p2<1, and 1/q=1/p1+1/p2−β. If b∈W1,p2ℝ, then the map b,Mβ+:W1,p1ℝ⟶W1,qℝ is bounded and continuous. In particular, if f∈W1,p1ℝ, it holds that
(15)b,Mβ+f′x≤bxMβ+f′x+b′xMβ+fx+Mβ+bf′x+Mβ+b′fx,for almost every x∈ℝ. Moreover,
(16)b,Mβ+f1,q≤Cp1,p2,βb1,p2f1,p1.
The same conclusions also hold for the operatorb,Mβ−.Theorem 7.
Let1<p1,p2,q<∞, 0≤β<1/p1, 1/p1+1/p2<1, and 1/q=1/p1+1/p2−β. If b∈W1,p2ℝ and f∈W1,p1ℝ, then
(17)Mb,β+f′x≤bxMβ+f′x+b′xMβ+fx+Mβ+bf′x+Mβ+b′fx,for almost every x∈ℝ. Moreover,
(18)Mb,β+f1,q≤Cp1,p2,βb1,p2f1,p1.
The same conclusions also hold for the operatorMb,β−.On the other hand, the investigation on the regularity of discrete maximal operators has also attracted the attention of many authors (see [6, 19, 28–33]). Let 1≤p<∞ and f:ℤ⟶ℝ be a discrete function, we define the ℓp-norm and the ℓ∞-norm of f by
(19)fℓpℤ=∑n∈ℤfnp1/p,fℓ∞ℤ=supn∈ℤfn.Formally, we define the discrete analogue of the Sobolev spaces by(20)W1,pℤ≔f:ℤ⟶ℝ∣f1,p=fℓpℤ+f′ℓpℤ<∞,where f′n=fn+1−fn is the first derivative of f. It is clear that
(21)fℓpℤ≤f1,p≤3fℓpℤ,forall1≤p≤∞.Estimate (21) implies that the discrete Sobolev space W1,pℤ is just the classical ℓpℤ with an equivalent norm. Hence, the W1,pℤ (1<p≤∞) regularity for discrete maximal operators is trivial. However, the situation p=1 is highly nontrivial. We define the total variation of f by
(22)Varf=f′ℓ1ℤ.We also write(23)Varf;a,b=f′ℓ1a,b−1=∑n=ab−1fn+1−fnfor the variation of f on the interval a,b, where a and b are integers (or possibly a=−∞, or b=∞). It is clear that Varf;−∞,∞=Varf. Denote by BVℤ the set of functions of bounded variation defined on ℤ, which is a Banach space with the norm
(24)fBVℤ≔f−∞+Varf,where f−∞≔limn→−∞fn. Clearly,
(25)fℓ∞ℤ≤fBVℤ≤3fℓ1ℤ.The study of regularity properties of discrete maximal operators began with Bober et al. [28] who studied the endpoint regularity of one dimensional discrete centered and uncentered Hardy-Littlewood maximal operators M and M~, which are defined by
(26)Mfn=supr∈ℕ12r+1∑k=−rrfn+kandM~fn=supr,s∈ℕ1r+s+1∑k=−rsfn+k,where ℕ=0,1,2,3,⋯. It was shown in [28] that
(27)VarM~f≤Varf,ifVarf<∞,(28)VarMf≤2+146315fℓ1ℤ,iff∈ℓ1ℤ.It was noted that inequality (27) is sharp and inequality (27) for M was proven by Temur in [33] (with constant C=294,912,004). Inequality (28) was improved by Madrid [32] who obtained the sharp constant C=2. Recently, Carneiro and Madrid [19] extended (28) to the fractional setting and showed that if 0≤β<1, q=1/1−β, and f:ℤ⟶ℝ is a discrete function such that Varf<∞ and M~βf≡∞, then
(29)M~βf′ℓqℤ≤41/qVarf,where M~β is the discrete uncentered fractional maximal operator defined by
(30)M~βfn=supr,s∈ℕ1r+s+11−β∑k=−rsfn+k.It is currently unknown whether inequality (29) holds for the discrete centered fractional maximal operator
(31)Mβfn=supr∈ℕ12r+11−β∑k=−rrfn+k.It was pointed out in [30] that both the maps f↦Mβf′ and f↦M~βf′ (for 0≤β<1) are bounded and continuous from ℓ1ℤ to ℓ1ℤ. Moreover, if f∈ℓ1ℤ, then
(32)VarMβf≤2fℓ1ℤ,VarM~βf≤2fℓ1ℤ,and the constants C=2 are the best possible. Liu [30] pointed out that the operator f′↦Mβf′ is not bounded from ℓ1ℤ to ℓrℤ for all 1≤r<1/1−β. The continuity of M:BVℤ⟶BVℤ and M~:BVℤ⟶BVℤ was proven by Madrid [34] and Carneiro et al. [20], respectively. Recently, Liu and Mao [6] studied the regularity of the discrete one-sided Hardy-Littlewood maximal operators and proved them.Theorem 8 (see [6]).
Letf:ℤ⟶ℝ be a discrete function such that Varf<∞, then
(33)VarM+f≤Varf.
Moreover, the mapf↦M+f′ is continuous from ℓ1ℤ to ℓ1ℤ. The same results also hold for M−. Here
(34)M+fn=sups∈ℕ1s+1∑k=0sfn+k,M−fn=supr∈ℕ1r+1∑k=−r0fn+k.
Very recently, Liu [26] extended Theorem 8 to the fractional setting.Theorem 9 (see [26]).
Let0≤β<1. Then, the map f↦Mβ+f′ is bounded and continuous from ℓ1ℤtoℓ1ℤ. Moreover, if f∈ℓ1ℤ, then
(35)VarMβ+f≤2fℓ1ℤ,and the constant C=2 is the best possible. The same results hold for Mβ−. Here
(36)Mβ+fn=sups∈ℕ1s+11−β∑k=0sfn+k,Mβ−fn=supr∈ℕ1r+11−β∑k=−r0fn+k.
The second aim of this paper is to study the regularity of the discrete analogues ofb,Mβ+ and Mb,β+. Let us introduce some definitions.Definition 10.
Letb:ℤ⟶ℝ be a discrete function and β∈0,1. The commutators of the discrete one-sided fractional maximal function b,Mβ+ and b,Mβ− are defined by
(37)b,Mβ+fn=bnMβ+fn−Mβ+bfn,n∈ℤ,b,Mβ−fn=bnMβ−fn−Mβ−bfn,n∈ℤ.Definition 11.
Letb:ℤ⟶ℝ be a discrete function. For β∈0,1, the maximal commutators of Mb,β+ and Mb,β− with b are defined, respectively, by
(38)Mb,β+fn=supr∈ℕ1r+11−β∑k=0rbn−bn+kfn+k,n∈ℤ,Mb,β−fn=supr∈ℕ1r+11−β∑k=0rbn−bn−kfn−k,n∈ℤ.
We now formulate the rest of main results as follows:Theorem 12.
Letb∈BVℤ. Then, the map b,M+:BVℤ⟶BVℤ is bounded. Moreover,
(39)Varb,M+f≤6bBVℤfBVℤ,forallf∈BVℤ.
The same conclusions also hold for the operatorb,M−.Theorem 13.
Letb∈BVℤ and β∈0,1. Then, the map b,Mβ+:ℓ1ℤ⟶BVℤ is bounded and continuous. Moreover, for any β∈0,1, it holds that
(40)Varb,Mβ+f≤5bBVℤfℓ1ℤ,forallf∈ℓ1ℤ.
The same conclusions also hold for the operatorb,Mβ−.Theorem 14.
Letb:ℤ⟶ℝ be a discrete function such that Varb<∞ and β∈0,1. Then, the map f↦Mb,β+f′ is bounded and continuous from ℓ1ℤ to ℓ1ℤ. Moreover,
(41)Mb,β+f′ℓ1ℤ≤2Varbfℓ1ℤ,forallf∈ℓ1ℤ.
The same conclusions also hold for the operatorMb,β−.This paper will be organized as follows. Section2 is devoted to proving Theorems 6 and 7. The proofs of Theorems 12–14 will be given in Section 3. We remark that the proofs of Theorems 6 and 7 are motivated by [21, 28]. The main ideas in the proofs of Theorems 12–14 are motivated by [6, 26], but some techniques are needed. In particular, in the proof of the continuity part of Theorem 14, we give a useful application of the Brezis-Lieb lemma in [35].Throughout this paper, the letterC will stand for positive constants, not necessarily the same one at each occurrence but independent of the essential variables. In particular, the letter Cα,β denotes the positive constants that depend on the parameters α,β.


## 2. Proofs of Theorems6 and 7

In this section, we shall prove Theorems6 and 7. Before giving our proofs, let us give some notations and lemmas. Let f∈Lpℝ with p≥1. For all h∈ℝ with h≠0, we define
(42)fhx=fτh−fxhandfτhx=fx−h.For convenience, we set(43)Gf;p=limsuph→0fτh−fLpℝb.According to Section 7.11 in [36], one has that for 1<p<∞,
(44)f∈W1,pℝ⇔f∈LpℝandGf;p<∞.Moreover, for functionsf∈W1,pℝ for 1<p≤∞, we have (see [36], Section 7.11) that fh⟶f′ in Lpℝ when h⟶0.In order to prove Theorems6 and 7, we need the following lemma, which follows from [23].Lemma 15. (see [23]).
Let1<p1,p2,p<∞ and 1/p=1/p1+1/p2. If f∈W1,p1ℝ and g∈W1,p2ℝ, then fg∈W1,pℝ. Moreover,
(45)fg′x=f′xgx+fxg′x,for almost every x∈ℝ. In particular, it holds that
(46)fg1,p≤f1,p1g1,p2.
Now we are in a position to prove Theorems6 and 7.Proof of Theorem 6.
We only prove Theorem6 for b,Mβ+ since another case can be obtained similarly. Let 1<p1,p2,q<∞, 0≤β<1/p1, 1/q=1/p1+1/p2−β, 1/p1+1/p2<1, b∈W1,p2ℝ, and f∈W1,p1ℝ. Let r,p be such that 1/p=1/p1+1/p2 and 1/r=1/p1−β. It is clear that 1/q=1/p2+1/r=1/p−β, p∈1,∞, and 0≤β<1/p.
(i)
We first prove the bounds forb,Mβ+. By Theorem 2, we have Mβ+f∈W1,rℝ. By Lemma 15, we have that bMβ+f∈W1,qℝ and bf∈W1,pℝ. By Theorem 2 again, we see that Mβ+bf∈W1,qℝ. Hence, b,Mβ+f∈W1,qℝ(ii)
We now prove (16). Applying Theorem 2, one has
(47)Mβ+f1,r≤Cp1,βf1,p1,which together with Lemma 15 yields that
(48)bMβ+f1,q≤2b1,p2Mβ+f1,r≤Cp1,βb1,p2f1,p1.
By Lemma15 again, it holds that
(49)bf1,p≤2b1,p2f1,p1,which together with Theorem 2 implies that
(50)Mβ+bf1,q≤Cp,βbf1,p≤Cp1,p2,βb1,p2f1,p1.
Combining (50) with (48) leads to (16).
(iii)
We now prove the continuity part. Letfj⟶f in W1,p1ℝ. It suffices to show that(51)b,Mβ+fj−b,Mβ+f1,q⟶0asj⟶∞.
By Lemma15 and applying the continuity result in Theorem 2, one can get
(52)bMβ+fj−bMβ+f1,q=bMβ+fj−Mβ+f1,q≤2b1,p2Mβ+fj−Mβ+f1,r⟶0asj⟶∞,(53)bfj−bf1,p=bfj−f1,p≤2b1,p2fj−f1,p1.
Combining (53) with the continuity result in Theorem 2 implies that
(54)Mβ+bfj−Mβ+bf1,q⟶0asj⟶∞.
Then, (51) follows from (52) and (54).
(iv)
It remains to prove (15). By Lemma 15, we have(55)bf′x=b′xfx+bxf′x,(56)b,Mβ+f′x=bMβ+f′x−Mβ+bf′x=b′xMβ+fx+bxMβ+f′x−Mβ+bf′x,for almost every x∈ℝ. By Theorem 2, one can get
(57)Mβ+f′x≤Mβ+f′x,(58)Mβ+bf′x≤Mβ+bf′x,for almost every x∈ℝ. It follows from (55)–(58) that
(59)b,Mβ+f′x≤b′xMβ+fx+bxMβ+f′x+Mβ+bf′x≤b′xMβ+fx+bxMβ+f′x+Mβ+b′fx+Mβ+bf′x,for almost every x∈ℝ. This proves (15) and completes the Proof of Theorem 6.Proof of Theorem 7.
We only prove Theorem7 for Mb,β+ since another case can be obtained similarly. Let 1<p1,p2,q<∞, 0≤β<1/p1, 1/q=1/p1+1/p2−β, 1/p1+1/p2<1, b∈W1,p2ℝ, and f∈W1,p1ℝ. Let r,p be such that 1/p=1/p1+1/p2 and 1/r=1/p1−β. It is clear that 1/q=1/p2+1/r=1/p−β, p∈1,∞ and 0≤β<1/p.
(i)
We first prove thatMb,β+f∈W1,qℝ. Fix x,h∈ℝ, it is easy to see that(60)Mb,β+fτhx=supr>01r1−β∫xx+rbτhx−bτhyfτhydy,which gives that
(61)Mb,β+fτhx−Mb,β+fx≤supr>01r1−β∫xx+rbτhx−bτhyfτhy−bx−byfydy≤Mbτh,β+fτh−fx+bτhx−bxMβ+fx+Mβ+bτh−bfx.
By (61), Hölder’s inequality, and the bounds for Mb,β+ and Mβ+, one can get
(62)Mb,β+fτh−Mb,β+fLqℝ≤Mbτh,β+fτh−fLqℝ+bτh−bMβ+fLqℝ+Mβ+bτh−bfLqℝ≤Cp1,p2,βbτhLp2ℝfτh−fLp1ℝ+bτh−bLp2ℝMβ+fLrℝ+bτh−bfLpℝ≤Cp1,p2,βbLp2ℝfτh−fLp1ℝ+bτh−bLp2ℝfLp1ℝ.
Because off∈W1,p1ℝ and b∈W1,p2ℝ, then by (44), we can get
(63)Gf;p1<∞,Gb;p2<∞.
It follows from (62) and (63) that
(64)GMb,β+f;q<∞.
Combining (64) with (44) and the bounds for Mb,β+ yields Mb,β+f∈W1,qℝ.
(ii)
We now prove (17). Since Mb,β+f∈W1,qℝ, f∈W1,p1ℝ and b∈W1,p2ℝ, then fh⟶f′ in Lp1ℝ as h⟶0, bτh⟶b and bh⟶b′ in Lp2ℝ as h⟶0, bτhfh⟶bf′ in Lpℝ as h⟶0, Mβ+fh−f′⟶0 in Lrℝ as h⟶0, Mβ+bhf−b′f⟶0 in Lqℝ as h⟶0, Mβ+bτhfh−bf′⟶0 in Lqℝ as h⟶0, Mb,β+fh⟶Mb,β+f′ in Lqℝ when h⟶0. Therefore, there exists a sequence of real numbers hk satisfying limk→∞hk=0 and a measurable set E satisfying ℝ\E=0 such that bτhkx⟶bx, bhkx⟶b′x, Mβ+fhk−f′x⟶0, Mβ+bτhkfhk−bf′x⟶0, and Mb,β+fhkx⟶Mb,β+f′x as k⟶∞ for all x∈E. From (61) and (13) we have that for all h∈ℝ(65)Mb,β+fτhx−Mb,β+fx≤bτhxMβ+fτh−fx+Mβ+bτhfτh−fx+bτhx−bxMβ+fx+Mβ+bτh−bfx.
It follows that(66)Mb,β+f′x=limk→∞Mb,β+fhkx≤limk→∞bτhkxMβ+fhkx+bhkxMβ+fx+Mβ+bτhkfhkx+Mβ+bhkfx=bxMβ+f′x+b′xMβ+fx+Mβ+bf′x+Mβ+b′fx,for any x∈E, which gives (17).
(iii)
By (17), the bounds for Mβ+, and Hölder’s inequality, one can get(67)Mb,β+f′Lqℝ≤bMβ+f′Lqℝ+b′Mβ+fLqℝ+Mβ+bf′Lqℝ+Mβ+b′fLqℝ≤bLp2ℝMβ+f′Lrℝ+b′Lp2ℝMβ+fLrℝ+Cp,βbf′Lpℝ+b′fLpℝ≤Cp1,p2,βb1,p2f1,p1,which together with (14) yields (18).


## 3. Proofs of Theorems12–14

This section is devoted to presenting the proofs of Theorems12–14.Proof of Theorem 12.
It is clear that(68)bf′n=bn+1fn+1−bnfn=b′nfn+1+bnf′n,
for alln∈ℤ. By (68) one has
(69)bf′ℓ1ℤ≤Varbfℓ∞ℤ+bℓ∞ℤVarf.
By (69) and Theorem 8, it holds that
(70)Varb,M+f=b,M+f′ℓ1ℤ≤bM+f′ℓ1ℤ+M+bf′ℓ1ℤ≤VarbM+fℓ∞ℤ+bℓ∞ℤVarM+f0+Varbfℓ∞ℤ+bℓ∞ℤVarf≤2Varbfℓ∞ℤ+bℓ∞ℤVarf.
On the other hand, one can easily check that(71)Varbfℓ∞ℤ+bℓ∞ℤVarf≤bnM+fn+M+bfn≤2bℓ∞ℤfℓ∞ℤ,which together with (70) and (25) yields that
(72)b,M+fBVℤ≤6bBVℤfBVℤ.Proof of Theorem 13.
(i) It is clear that(73)bfℓ1ℤ≤bℓ∞ℤfℓ1ℤ.
One can easily check that(74)Mβ+fn≤fℓ1ℤ,for all n∈ℤ. In light of (73) and (74) we would have
(75)b,Mβ+fℓ∞ℤ≤bℓ∞ℤMβ+fℓ∞ℤ+Mβ+bfℓ∞ℤ≤bℓ∞ℤfℓ1ℤ+bfℓ1ℤ≤2bℓ∞ℤfℓ1ℤ.
On the other hand, by Theorem9 and (73), (69), and (25), we have
(76)Varb,Mβ+f=b,Mβ+f′ℓ1ℤ≤bMβ+f′ℓ1ℤ+Mβ+bf′ℓ1ℤ≤VarbMβ+fℓ∞ℤ+bℓ∞ℤVarMβ+f+2bfℓ1ℤ≤5bBVℤfℓ1ℤ.
Combining (76) with (75) yields that the map b,Mβ+:ℓ1ℤ⟶BVℤ is bounded.
(iii)
We now prove the continuity result. Letf∈ℓ1ℤ, fjj=1∞⊂ℓ1ℤ, and fj⟶f in ℓ1ℤ as j⟶∞. We want to show that(77)b,Mβ+fj−b,Mβ+fBVℤ⟶0asj⟶∞.
By the sublinearity ofMβ+ and (25), (73), and (74), it holds that
(78)b,Mβ+fjn−b,Mβ+fn≤bnMβ+fjn−Mβ+fn+Mβ+bfjn−Mβ+bfn≤bnMβ+fj−fn+Mβ+bfj−fn≤bℓ∞ℤfj−fℓ1ℤ+bfj−fℓ1ℤ≤2bBVℤfj−fℓ1ℤ,for all n∈ℤ. From (78), we see that b,Mβ+fjn⟶b,Mβ+fn uniformly for n∈ℤ. Therefore, to prove (77), it suffices to show that
(79)Varb,Mβ+fj−b,Mβ+f⟶0asj⟶∞.
By (73) and (25), we get
(80)bfj−bfℓ1ℤ=bfj−fℓ1ℤ≤2bℓ∞ℤfj−fℓ1ℤ≤2bBVℤfj−fℓ1ℤ,which yields that bfj⟶bf in ℓ1ℤ as j⟶∞. This together with Theorem 9 implies that
(81)VarMβ+bfj−Mβ+bf⟶0asj⟶∞.
On the other hand, by Theorem9 again, we get
(82)VarMβ+fj−Mβ+f⟶0asj⟶∞.
By (69), (74), and (25) and the sublinearity for Mβ+, we have
(83)VarbMβ+fj−bMβ+f=bMβ+fj−bMβ+f′ℓ1ℤ=bMβ+fj−Mβ+f′ℓ1ℤ≤VarbMβ+fj−Mβ+fℓ∞ℤ+bℓ∞ℤVarMβ+fj−Mβ+f≤bBVℤfj−fℓ1ℤ+VarMβ+fj−Mβ+f.
This together with (82) yields that
(84)VarbMβ+fj−bMβ+f⟶0asj⟶∞.
Combining (84) with (81) implies that
(85)Varb,Mβ+fj−b,Mβ+f=b,Mβ+fj−b,Mβ+f′ℓ1ℤ≤bMβ+fj−bMβ+f′ℓ1ℤ+Mβ+bfj−Mβ+bf′ℓ1ℤ=VarbMβ+fj−bMβ+f+VarMβ+bfj−Mβ+bf⟶0asj⟶∞,which proves (79) and finishes the Proof of Theorem 13.Proof of Theorem 14.
We only prove Theorem14 for Mb,β+ since another one is analogous. The proof will be divided into two steps:Step 1.
Proof of the boundedness part. Letf∈ℓ1ℤ. Without loss of generality we may assume f≥0 since Mb,β+f=Mb,β+f. For convenience, we define the function Γ:0,∞⟶ℝ by Γx=x+1β−1−x+2β−1 for any x≥0. It is clear that Γx is decreasing on 0,∞ and ∑n∈ℕΓn=1. Fix n∈ℤ and r∈ℕ, it holds that
(86)1r+11−β∑k=0rbn−bn+kfn+k≤Varbfℓ1ℤr+1β−1.
This yields that for any fixedn∈ℤ, there exists rn∈ℕ such that
(87)Mb,β+fn=Arnfn≔rn+1β−1∑k=0rnbn−bn+kfn+k.
Let(88)X+=n∈ℤ:Mb,β+fn+1>Mb,β+fn,X−=n∈ℤ:Mb,β+fn>Mb,β+fn+1.
Then, we have(89)Mb,β+f′ℓ1ℤ=∑n∈X+Mb,β+fn+1−Mb,β+fn+∑n∈X−Mb,β+fn−Mb,β+fn+1.
We can write(90)∑n∈X+Mb,β+fn+1−Mb,β+fn≤∑n∈X+Arn+1fn+1−Arn+1+1fn=∑n∈X+rn+1+1β−1∑k∈ℤbn+1−bkfkχn+1,n+rn+1+1k−rn+1+2β−1∑k∈ℤbn−bkfkχn,n+rn+1+1k≤∑n∈X+bn+1−bnrn+1+1β−1∑k∈ℤfkχn+1,n+rn+1+1k+Varb∑n∈X+Γrn+1∑k∈ℤfkχn+1,n+rn+1+1k,(91)∑n∈X−Mb,β+fn−Mb,β+fn+1≤∑n∈X−Arnfn−Arn+1fn+1=∑n∈X−rn+1β−1∑k∈ℤbn−bkfkχn,n+rnk−rn+2β−1∑k∈ℤbn+1−bkfkχn+1,n+rn+2k≤∑n∈X−rn+1β−1∑k∈ℤbn−bkfkχn+1,n+rn+1k−rn+2β−1∑k∈ℤbn+1−bkfkχn+1,n+rn+2k≤∑n∈X−bn+1−bnrn+1β−1∑k∈ℤfkχn+1,n+rn+1k+∑n∈X−Γrn∑k∈ℤbn+1−bkfkχn+1,n+rn+1k.
By Hölder’s inequality with exponentsp=1/β and p′=1/1−β and the fact that ℓqℤ⊆ℓpℤ for all 1≤q≤p≤∞, it holds that
(92)rn+1+1β−1∑k∈ℤfkχn+1,n+rn+1+1k≤fℓ1/βℤ≤fℓ1ℤ,(93)rn+1β−1∑k∈ℤfkχn+1,n+rn+1k≤fℓ1/βℤ≤fℓ1ℤ.
In light of (90) and (92), we would have
(94)∑n∈X+Mb,β+fn+1−Mb,β+fn≤fℓ1ℤ∑n∈X+bn+1−bn+Varb∑n∈X+Γrn+1∑k∈ℤfkχn+1,n+rn+1+1k.
By (91) and (93), we have
(95)∑n∈X−Mb,β+fn−Mb,β+fn+1≤fℓ1ℤ∑n∈X−bn+1−bn+Varb∑n∈X−Γrn∑k∈ℤfkχn+1,n+rn+1k.
It follows from (89), (94), and (95) that
(96)Mb,β+f′ℓ1ℤ≤∑n∈ℤbn+1−bnfℓ1ℤ+Varb∑n∈ℤ∑k∈ℤfkΓk−n−1χn+1,∞k≤Varbfℓ1ℤ+Varb∑k∈ℤfk∑n∈ℤΓk−n−1χn+1,∞k.
Fixk∈ℤ, one can easily check that
(97)∑n∈ℤΓk−n−1χn+1,∞k=∑n=−∞k−1Γk−n−1=1.
This together with (96) yields that
(98)Mb,β+f′ℓ1ℤ≤2Varbfℓ1ℤ.
This proves (41).Step 2.
Proof of the continuity part. Letf∈ℓ1ℤ, fjj=1∞⊂ℓ1ℤ, and fj⟶f in ℓ1ℤ when j⟶∞. Without loss of generality, we may assume that all fj≥0 and f≥0 since fj−f≤fj−f. We want to show that
(99)Mb,β+fj′−Mb,β+f′ℓ1ℤ⟶0asj⟶∞.
By the sublinearity ofMb,β+, we can get
(100)Mb,β+fjn−Mb,β+fn≤Mb,β+fj−fn≤Varbfj−fℓ1ℤ,which yields that Mb,β+fj⟶Mb,β+f uniformly in ℤ. Fix ε∈0,1, there exists N1>0 depending only on ε,f such that
(101)fj−fℓ1ℤ<ε,forallj≥N1;(102)fjℓ1ℤ≤fj−fℓ1ℤ+fℓ1ℤ≤fℓ1ℤ+1,forallj≥N1.
It follows from (100) that
(103)Mb,β+fj′n⟶Mb,β+f′nasj⟶∞,for all n∈ℤ. By the boundedness part, we have that Mb,β+f′∈ℓ1ℤ. This together with the classical Brezis-Lieb lemma in [3] implies that (99) reduces to the following:
(104)limj→∞Mb,β+fj′ℓ1ℤ=Mb,β+f′ℓ1ℤ.
By (103) and Fatou’s lemma, one can get
(105)Mb,β+f′ℓ1ℤ≤liminfj→∞Mb,β+fj′ℓ1ℤ.
Therefore, to prove (104), it suffices to show that
(106)limsupj→∞Mb,β+fj′ℓ1ℤ≤Mb,β+f′ℓ1ℤ.
We now prove (106). Since f∈ℓ1ℤ, then there exists a sufficiently large positive integer Λ1>0 depending only on ε,f such that
(107)∑n≥Λ1fn<ε.
There exists an integerΛ2>0 such that sβ−1<ε if s≥Λ2. Since Varb<∞, then there exists a large positive integer Λ3 depending only on b,ε such that
(108)∑n≥Λ3bn+1−bn<ε.
LetΛ=maxΛ1,Λ2,Λ3. By (103), there exists N2>0 depending only on ε,Λ such that
(109)Mb,β+fj′n−Mb,β+f′n≤ε4Λ+2,foralln≤2Λ,j≥N2.
It follows from (109) that
(110)Mb,β+fj′ℓ1ℤ≤∑n≤2ΛMb,β+fj′n−Mb,β+f′n+Mb,β+f′ℓ1ℤ+∑n≥2ΛMb,β+fj′n≤ε+Mb,β+f′ℓ1ℤ+∑n≥2ΛMb,β+fj′n,for all j≥N2. Fix j≥N2, we set
(111)Xj+=n≥2Λ:Mb,β+fjn+1>Mb,β+fjn,Xj−=n≥2Λ:Mb,β+fjn≥Mb,β+fjn+1.
Sincefj∈ℓ1ℤ, then for n∈ℤ, there exists rn∈ℕ such that Mb,β+fjn=Arnfjn. Then, we can write
(112)∑n≥2ΛMb,β+fj′n=∑n∈Xj+Mb,β+fjn+1−Mb,β+fjn+∑n∈Xj−Mb,β+fjn−Mb,β+fjn+1.
Similar arguments to those used in deriving (94) and (95) may yield that
(113)∑n∈Xj+Mb,β+fjn+1−Mb,β+fjn≤fjℓ1ℤ∑n∈Xj+bn+1−bn+Varb∑n∈Xj+Γrn+1∑k∈ℤfjkχn+1,n+rn+1+1k,(114)∑n∈Xj−Mb,β+fjn−Mb,β+fjn+1≤fjℓ1ℤ∑n∈Xj−bn+1−bn+Varb∑n∈Xj−Γrn∑k∈ℤfjkχn+1,n+rn+1k.
It follows from (102), (108), and (112)–(114) that
(115)∑n≥2ΛMb,β+fj′n≤∑n≥2Λbn+1−bnfjℓ1ℤ+∑n≥2ΛVarb∑k∈ℤfjkΓk−n−1χn+1,∞k≤fℓ1ℤ+1ε+Varb∑k∈ℤfjk∑n≥2ΛΓk−n−1χn+1,∞k.
By (101) and (107), we can get
(116)∑k∈ℤfjk∑n≥2ΛΓk−n−1χn+1,∞k≤∑k≥Λfjk∑n=−∞k−1Γk−n−1+∑∣k∣<Λfjk∑n≥2Λn<kΓk−n−1≤∑k≥Λfk+fj−fℓ1ℤ+∑k<Λfjk∑n≤−2ΛΓk−n−1≤2ε+∑∣k∣<ΛfjkΛβ−1≤3+fℓ1ℤε.
It follows from (115) and (116) that
(117)∑n≥2ΛMb,β+fj′n≤3+fℓ1ℤVarb+fℓ1ℤ+1ε.
Inequality (117) together with (110) implies that
(118)Mb,β+fj′ℓ1ℤ≤Cε+Mb,β+f′ℓ1ℤ,
for allj≥maxN1,N2. Here, C>0 depends only on b,f. This leads to (106) and completes the Proof of Theorem 14.



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*Source: 1369454-2020-07-13.xml*