# A Note on the Regularity of the Two-Dimensional One-Sided Hardy-Littlewood Maximal Function

Authors: Feng Liu; Lei Xu
Journal: Journal of Function Spaces
 (2018)

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## Abstract

We investigate the regularity properties of the two-dimensional one-sided Hardy-Littlewood maximal operator. We point out that the above operator is bounded and continuous on the Sobolev spacesWs,p(R2) for 0≤s≤1 and 1<p<∞. More importantly, we establish the sharp boundedness and continuity for the discrete two-dimensional one-sided Hardy-Littlewood maximal operator from l1(Z2) to BV(Z2). Here BV(Z2) denotes the set of all functions of bounded variation on Z2.


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## Body

## 1. Introduction

The purpose of this paper is to present new results related to the regularity properties of the two-dimensional one-sided Hardy-Littlewood maximal operator, in both the continuous and discrete setting. We start with a recall of some recent developments on the general regularity theory of maximal operators. In 1997, Kinnunen [1] first studied the Sobolev regularity of the usual centered Hardy-Littlewood maximal function M and showed that M is bounded on the first-order Sobolev spaces W1,p(Rd) for all 1<p≤∞ (the same conclusion also holds for the uncentered version M~ by a simple modification of Kinnunen’s arguments or [2, Theorem 1]). Later on, Kinnunen’s result was extended to a local version in [3], to a fractional version in [4] and to a multisublinear version in [5, 6] and to a one-sided version in [7]. Due to the lack of the sublinearity for the derivative of the maximal function, the continuity of M:W1,p(Rd)→W1,p(Rd) for 1<p<∞ is certainly a nontrivial issue. This problem was addressed by Luiro [8] in the affirmative and was later extended to the local version in [9] and the multisublinear version in [5, 10].Since the maximal operatorM is not of type (1,1), the W1,1-regularity for maximal operator is more delicate. A crucial question was posed by Hajłasz and Onninen in [2]: Is the operator f↦|∇Mf| bounded from W1,1(Rd) to L1(Rd)? A complete solution was achieved only in dimension d=1 in [11–14] and partial progress on the general dimension d≥2 was given by Hajłasz and Malý [15] and Luiro [16]. Tanaka [14] first observed that if f∈W1,1(R), then M~f is weakly differentiable and (1)M~f′L1R≤2f′L1R.Inequality (1) with the sharp constant C=1 was later proved by Liu et al. in [13]. An important improvement of Tanaka’s result was given by Aldaz and Pérez Lázaro [11] who proved that if f is of bounded variation on R, then M~f is absolutely continuous and (2)var⁡M~f≤var⁡f.Here var⁡f denotes the total variation of f. Notice that the constant C=1 in inequality (2) is sharp. Recently, inequality (2) was extended to a fractional setting in [17, Theorem 1] and to a multisublinear fractional setting in [18, Theorems 1.3-1.4]. In the centered setting, Kurka [12] showed that if f is of bounded variation on R, then inequality (2) holds for M (with constant C=240,004). It was also shown in [12] that if f∈W1,1(R), then Mf is weakly differentiable and (1) holds for M with constant C=240,004. Very recently, Carneiro et al. [19] proved that the operator f↦M~f′ is continuous from W1,1(R) to L1(R). It is currently unknown whether inequality (1) with C=1 also holds for M and the map f↦Mf′ is also continuous from W1,1(R) to L1(R). For other interesting works related to this theory, we refer the reader to consult [20, 21], among others.In order to address theW1,1-regularity of the one-dimensional Hardy-Littlewood maximal function, Tanaka [14] first studied the W1,1-regularity of the one-sided Hardy-Littlewood maximal function (3)M+fx=sups>0⁡1s∫xx+sfydyand proved that if f∈W1,1(R), then the distributional derivative of M+f is integrable, and (4)M+f′L1R≤f′L1R.It is observed that M+f is also absolutely continuous on R by a combination of arguments in [13, 14]. We remark that M+ maps W1,∞(R) into W1,∞(R) boundedly by similar arguments to those in [1, Remark (iii)]. Recently, Liu and Mao [7] proved the following result.Theorem A (see [7]).
Let1<p<∞. Then M+ is bounded and continuous on W1,p(R).In this paper we focus on the regularity of two-dimensional one-sided Hardy-Littlewood maximal operator. In 2011, Forzani et al. [22] first introduced and studied the weighted weak estimates for the two-dimensional one-sided Hardy-Littlewood maximal operator M2+, which is defined by (5)M2+fx=sups>0⁡1s2∫x1x1+s∫x2x2+sfydy,where x=(x1,x2)∈R2. It is not difficult to see that (6)M2+fx≤M~fx≤CMfx∀x∈R2.This together with the Lp bounds for M implies that M2+ is of type (p,p) for 1<p≤∞ and of weak type (1,1). One can easily check that M2+ is a sublinear operator and commutes with translations. These facts together with Theorem 1 in [2] imply that M2+ maps W1,p(R2)→W1,p(R2) boundedly with 1<p<∞. On the other hand, by the arguments similar to those used in deriving [8, Theorem 4.1] we can obtain that M2+ maps W1,p(R2)→W1,p(R2) continuously with 1<p<∞. It was shown in [23, Lemma 2.2] that (7)∑j∈ZMfj,ζLrR2q1/qLpR2≤C∑j∈Zfj,ζLrR2q1/qLpR2for all 1<p,q,r<∞. Here R2={ζ∈R2;1/2<|ζ|≤1}. Combining (7) with (6) yields (8)∑j∈ZM2+fj,ζLrR2q1/qLpR2≤C∑j∈Zfj,ζLrR2q1/qLpR2for all 1<p,q,r<∞. On the other hand, one can easily check that (9)ΔζM2+fx≤M2+Δζfx∀x,ζ∈R2and (10)M2+f-M2+g≤M2+f-gfor arbitrary functions f,g defined on R2. Here Δζf(x)=f(x+ζ)-f(x). By using (8)-(10) and the arguments similar to those used to derive [21, Theorems 1 and 3] we can obtain that M2+ is bounded and continuous on the inhomogeneous Triebel-Lizorkin spaces Fsp,q(R2) and the inhomogeneous Besov spaces Bsp,q(R2) for 0<s<1 and 1<p,q<∞. For other interesting works on this topic we refer the readers to consult [24–28]. We denote by Ws,p(Rd) the fractional Sobolev spaces defined by the Bessel potentials. Since Ws,p(Rd)=Fsp,2(Rd) for all s>0 and 1<p<∞ and W0,p(Rd)=Lp(Rd), the above analyses lead to the following result.Theorem 1.
M 2 + is bounded and continuous from Ws,p(R2) to Ws,p(R2) for all 0≤s≤1 and 1<p<∞.Our main motivation of this paper is to investigate the regularity of the discrete version ofM2+, which is defined by (11)M2+fn1,n2=supN∈N⁡1N+12∑i=0N∑j=0Nfn1+i,n2+j,where (n1,n2)∈Z2 and N={0,1,…}. Let us recall some pertinent definitions, notations, and background. For a discrete function f:Z2→R and 1≤p<∞, we define its lp(Z2)-norm by flpZ2≔(∑n∈Z2fnp)1/p and l∞(Z2)-norm by fl∞Z2≔supn∈Z2⁡fn. We also define the total variation of f on Z2 by (12)var⁡f=D1fl1Z2+D2fl1Z2,where D1f(n1,n2)=f(n1+1,n2)-f(n1,n2) and D2f(n1,n2)=f(n1,n2+1)-f(n1,n2). We denote by BV(Z2) the set of functions of bounded variation defined on Z2. We also denote by BV(Z) the set of functions of bounded variation f defined on Z satisfying var⁡(f)=‍∑n∈Zfn+1-fn<+∞.Recently, the investigation of the regularity properties of the discrete maximal operators has also attracted the attention of many authors. A good start was due to Bober et al. [29] in 2012 when they proved that (13)var⁡M~f≤var⁡fif f∈BV(Z), and (14)var⁡Mf≤2+146315fl1Zif f∈l1(Z). Here the operators M and M~ are the one-dimensional discrete centered and uncentered Hardy-Littlewood maximal operators, respectively, which are defined by (15)Mfn=supr∈N⁡1r+1∑k=0rfn+k,M~fn=supr,s∈N⁡1r+s+1∑k=-rsfn+k. We notice that inequality (13) is sharp. Subsequently, Temur [30] proved (13) for M (with constant C=294,912,004) following Kurka’s breakthrough [12]. Inequality (14) is not optimal, and it was asked in [29] whether the sharp constant for inequality (14) is in fact C=2; this question was resolved in the affirmative by Madrid in [31]. Later on, the above results were extended to a fractional case in [17, 32, 33], to a one-sided case in [7], and to a high dimensional case in [34]. For other interesting works we can consult [19, 20, 35–37]. In particular, Liu and Mao [7] investigated the regularity of the one-dimensional discrete one-sided Hardy-Littlewood maximal operator (16)M+fn=supN∈N⁡1N+1∑k=0Nfn+kand proved the following result.Theorem B (see [7]).
The operatorM+ is bounded and continuous from l1(Z) to BV(Z). Moreover, if f∈BV(Z), then (17)var⁡M+f≤2fl1Z,and the constant C=2 is the best possible.Based on the above, it is natural to ask whetherM2+:l1(Z2)→BV(Z2) is bounded and continuous, which can be addressed by the following.Theorem 2.
The operatorM2+ is bounded and continuous from l1(Z2) to BV(Z2). In particular, if f∈l1(Z2), then (18)var⁡M2+f≤2π23fl1Z2and the constant C=2π2/3 is the best possible one.Remarks. (i) It should be pointed out that (19)M2+l1Z2→BVZ2≔supfl1Z2>0⁡var⁡M2+ffl1Z2=2π23.To see this, let us consider the characteristic function f(n1,n2)=χ{(0,0)}(n1,n2). It is clear that fl1Z2=1 and M2+f(n1,n2)=1/max⁡n1,n2+12χ{n1≤0,n2≤0}(n1,n2). One can easily check that var⁡(M2+f)=2π2/3. This together with Theorem 2 yields the claim.(ii) The main results of this paper can be extended to the general dimensiond≥2 by using similar arguments.The rest of this paper is organized as follows. In Section2 we shall prove Theorem 2. We remark that the proof of the boundedness part in Theorem 2 is motivated by the method in [31], but our proof is simpler and more direct than that of [31]. The proof of the continuity part in Theorem 2 relies on the previous boundedness result and an useful application of the Brezis-Lieb lemma in [38]. 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. Proof of Theorem2

In this section we shall prove Theorem2. For convenience, for any N∈N and a discrete function f:Z2→R, we define the average function AN:Z2→R by (20)ANfn1,n2=1N+12∑i=0N∑j=0Nfn1+i,n2+j.We shall divide the proof of Theorem2 into two steps.Step 1 (the boundedness part).
Letf∈l1(Z2). Without loss of generality we may assume that f≥0 since M2+f=M2+f and fl1Z2≤fl1(Z2). To prove (18), it suffices to show that (21)∑k1∈Z∑k2∈ZM2+fk1+1,k2-M2+fk1,k2≤π23fl1Z2and (22)∑k1∈Z∑k2∈ZM2+fk1,k2+1-M2+fk1,k2≤π23fl1Z2.
We only prove (21) since (22) is analogous. Fix i∈Z, and let (23)Ri=k∈Z:M2+fi+1,k<M2+fi,k,Ri′=k∈Z:M2+fi+1,k>M2+fi,k.Then (21) reduces to the following inequalities: (24)∑i∈Z∑k∈RiM2+fi,k-M2+fi+1,k≤π26fl1Z2;(25)∑i∈Z∑k∈Ri′M2+fi+1,k-M2+fi,k≤π26fl1Z2.We now prove (24). Since f∈l1(Z2), then for any i∈Z and k∈Ri, there exists N(i,k)∈N such that M2+f(i,k)=AN(i,k)f(i,k). It follows that (26)∑i∈Z∑k∈RiM2+fi,k-M2+fi+1,k≤∑i∈Z∑k∈RiANi,kfi,k-ANi,kfi+1,k.Note that, for any i∈Z and k∈Ri, the following holds:(27)ANi,kfi,k-ANi,kfi+1,k≤1Ni,k+12∑k1=0Ni,k∑k2=0Ni,kfi+k1,k+k2-1Ni,k+12∑k1=0Ni,k-1∑k2=0Ni,kfi+1+k1,k+k2χNi,k≥1+fi,kχNi,k=0≤1Ni,k+12∑k2=0Ni,kfi,k+k2χNi,k≥1+fi,kχNi,k=0≤∑j=0+∞11+j2fi,k+j.This together with (26) yields (28)∑i∈Z∑k∈RiM2+fi,k-M2+fi+1,k≤∑j=0+∞∑i∈Z∑k∈Ri11+j2fi,k+j≤∑j=0+∞11+j2fl1Z2≤π26fl1Z2,which gives (24).
It remains to prove (25). We want to show that (29)∑i∈Z∑k∈Ri′M2+fi+1,k-M2+fi,k=∑i∈Z∑k∈RiM2+fi,k-M2+fi+1,k.This can be seen by the following: (30)∑i∈Z∑k∈Ri′M2+fi+1,k-M2+fi,k-∑i∈Z∑k∈RiM2+fi,k-M2+fi+1,k=∑i∈Z∑k∈Ri′M2+fi+1,k-M2+fi,k+∑i∈Z∑k∈RiM2+fi+1,k-M2+fi,k=∑i∈Z∑k∈ZM2+fi+1,k-M2+fi,k=∑k∈Z∑i∈ZM2+fi+1,k-M2+fi,k=0since, for fixed k∈Z, the following holds: (31)∑i∈ZM2+fi+1,k-M2+fi,k=limN1→+∞⁡M2+fN1,k-limN2→-∞⁡M2+fN2,k=0.Combining (29) with (24) yields (25).Step 2 (the continuity part).
Letfj→f in l1(Z2) as j→∞. Without loss of generality we may assume that fj≥0, for all j, and that f≥0 since fj-f≤fj-f. It suffices to show that (32)limj→∞⁡D1M2+fj-D1M2+fl1Z2=0since limj→∞⁡D2M2+fj-D2M2+fl1(Z2)=0 can be proved analogously.
We now prove (32). By the sublinearity of M2+, we have (33)M2+fjn1,n2-M2+fn1,n2≤M2+fj-fn1,n2≤fj-fl1Z2,which implies that M2+fj→M2+f pointwise as j→∞ and (34)limj→∞⁡D1M2+fjn1,n2=D1M2+fn1,n2for all (n1,n2)∈Z2. Applying (34) and the classical Brezis-Lieb lemma in [38], to get (32), it suffices to show that (35)limj→∞⁡D1M2+fjl1Z2=D2M2+fl1Z2.By (34) and Fatou’s lemma we get (36)D1M2+fl1Z2≤liminfj→∞⁡D1M2+fjl1Z2.Thus, (35) reduces to the following: (37)limsupj→∞⁡D1M2+fjl1Z2≤D1M2+fl1Z2.By Step 1 we have D1M2+f∈l1(Z2). Therefore, given ϵ∈(0,1), there exists an integer Λ>0 such that (38)max⁡D1M2+fχn1,n2≥Λl1Z2,fχn1,n2≥Λl1Z2,Λ-1/2<ϵ.For the above ϵ>0, since fj→f in l1(Z2) as j→∞, then there exists an integer N1>0 such that (39)fj-fl1Z2<ϵ,fjl1Z2<fl1Z2+1for all j≥N1. By (34) again, there exists a positive integer N2=N2(ϵ,Λ) such that (40)D1M2+fjn1,n2-D1M2+fn1,n2≤164Λ2ϵ∀j≥N2,n1,n2≤4Λ. And (40) and (38) yield the following: (41)D1M2+fjl1Z2≤D1M2+fj-D1M2+fl1Z2+D1M2+fl1Z2≤D1M2+fj-D1M2+fχn1,n2≥4Λl1Z2+D1M2+fj-D1M2+fχn1,n2<4Λl1Z2+D1M2+fl1Z2≤D1M2+fjχn1,n2≥4Λl1Z2+D1M2+fχn1,n2≥4Λl1Z2+ϵ+D1M2+fl1Z2≤D1M2+fjχn1,n2≥4Λl1Z2+2ϵ+D1M2+fl1Z2for all j≥N2. Therefore, (37) reduces to the following: (42)limj→∞⁡D1M2+fjχn1,n2≥4Λl1Z2=0.
Below we shall prove (42). Fix j≥max⁡{N1,N2}. We can write (43)D1M2+fjχn1,n2≥4Λl1Z2≤∑n1≥2Λ∑n2∈ZM2+fjn1+1,n2-M2+fjn1,n2+∑n2≥2Λ∑n1∈ZM2+fjn1+1,n2-M2+fjn1,n2≕A1+A2.
We only estimateA1 since A2 is analogous. Fix n1∈Z with n1≥2Λ. We denote (44)Xn1+=n2∈Z;M2+fjn1+1,n2>M2+fn1,n2;Xn1-=n2∈Z;M2+fjn1+1,n2<M2+fn1,n2.Then we can write (45)A1=∑n1≥2Λ∑n2∈Xn1+M2+fjn1+1,n2-M2+fjn1,n2+∑n1≥2Λ∑n2∈Xn1-M2+fjn1+1,n2-M2+fjn1,n2≕A11+A12.Since fj∈l1(Z2), then for any n1∈Z with n1≥2Λ and n2∈Xn1+, there exists Nn1+1,n2∈N such that M2+fj(n1+1,n2)=AN(n1+1,n2)fj(n1+1,n2). Then we can write (46)A11=∑n1≥2Λ∑n2∈Xn1+M2+fjn1+1,n2-M2+fjn1,n2≤∑n1≥2Λ∑n2∈Xn1+ANn1+1,n2fjn1+1,n2-ANn1+1,n2+1fjn1,n2=∑n1≥2Λ∑n2∈Xn1+1Nn1+1,n2+12∑i=0Nn1+1,n2∑l=0Nn1+1,n2fjn1+1+i,n2+l-1Nn1+1,n2+22∑i=0Nn1+1,n2+1∑l=0Nn1+1,n2+1fjn1+i,n2+l≤∑n1≥2Λ∑n2∈Xn1+2Nn1+1,n2+3Nn1+1,n2+14∑i=1Nn1+1,n2+1∑l=0Nn1+1,n2fjn1+i,n2+l.Observe that (47)2Nn1+1,n2+3Nn1+1,n2+14∑i=1Nn1+1,n2+1∑l=0Nn1+1,n2fjn1+i,n2+l≤4∑k1∈Z∑k2∈Zfjk1,k21Nn1+1,n2+13χn1+1,n1+1+Nn1+1,n2k1χn2,n2+Nn1+1,n2k2≤∑k1∈Z∑k2∈Zfjk1,k21k1-n13/2k2-n2+13/2χn1+1,+∞k1χn2,+∞k2,which together with (46) and (38)-(39) yields the following: (48)A11≤4∑k1∈Z∑k2∈Zfjk1,k2∑n1≥2Λ1k1-n13/2χn1+1,+∞k1∑n2∈Xn1+1k2-n2+13/2χn2,+∞k2≤C∑k1∈Z∑k2∈Zfjk1,k2∑n1<k1n1≥2Λ1k1-n13/2≤C∑k1>Λ∑k2∈Zfjk1,k2∑n1<k1n1≥2Λ1k1-n13/2+C∑k1<Λ∑k2∈Zfjk1,k2∑n1<k1n1≤-2Λ1k1-n13/2≤Cfjχn1,n2≥Λl1Z2+Cfjl1Z2∑k1=Λ+∞1k13/2≤Cfχn1,n2≥Λl1Z2+Cfj-fl1Z2+Cfjl1ZΛ-1/2≤Cϵ.
We now estimateA12. Since fj∈l1(Z2), then, for any n1∈Z with n1≥2Λ and n2∈Xn1-, there exists N(n1,n2)∈N such that M2+fj(n1,n2)=AN(n1,n2)fj(n1,n2). Then we have (49)
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.By similar arguments to those in deriving (48) we can obtain (50)∑n1≥2Λ∑n2∈Xn1-2Nn1,n2+3Nn1,n2+14∑i=1Nn1,n2+1∑l=0Nn1,n2fjn1+i,n2+l≤Cϵ.On the other hand, we get from (38) and (39) that (51)∑n1≥2Λ∑n2∈Xn1-1Nn1,n2+12∑l=0Nn1,n2fjn1,n2+l≤∑n1≥2Λ∑n2∈Xn1-∑k2∈Zfjn1,k21Nn1,n2+12χn2,n2+Nn1,n2k2≤∑n1≥2Λ∑n2∈Xn1-∑k2∈Zfjn1,k21k2-n2+12χn2,+∞k2≤∑n1≥2Λfjn1,k2∑k2∈Z∑n2≤k2n2∈Xn1-1k2-n2+12≤Cfjχn1,n2≥2Λl1Z2≤Cfχn1,n2≥2Λl1Z2+Cfj-fl1Z2≤Cϵ.It follows from (49)-(51) that (52)A12≤Cϵ.Combining (52) with (45) and (48) implies that (53)A1≤Cϵ.Similarly, we can get (54)A2≤Cϵ.Also (43) together with (53)-(54) yields (42) and completes the proof of Theorem 2.



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*Source: 6759632-2018-08-05.xml*