Regularity of one-sided multilinear fractional maximal functions

Theorem A

([27]). Let 1 < p < ∞, 0 ≤ β < 1/p and q = p/(1 − pβ). Then bothMβ+andMβ−map W^1,p(ℝ) into W^1,q(ℝ) boundedly and continuously. Moreover, if f ϵ W^1,p(ℝ), then

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

Let1<p1,…,pm<∞,0≤β<∑i=1m1/pi,1/q=∑i=1m1/pi−βand1≤q<∞.ThenMβ+maps W^1,p₁ (ℝ) ×· ··× W^1,pm (ℝ) into W^1,q(ℝ) boundedly and continuously. Especially, if f ➝= (f₁, … , f_m) with each f_i ∈ W^1,pi (ℝ), then the weak derivative(Mβ+(f→))′exists almost everywhere. More precisely,

for almost every x ϵ ℝ, wheref→j=(f1,…,fj−1,fj′,fj+1,…,fm).Moreover,

The same results hold forMβ−.

Theorem 1.2

Letf→=(f1,…,fm)with eachfi∈Lpi(R)for1<p1,…,pm<∞and1≤β<∑i=1m1/pi.

1. Then the weak derivative(Mβ+(f→))′exists almost everywhere. Precisely,

for almost every x ϵ ℝ.

1. Let1/q=∑i=1m1/pi−β+1.Then

The same results hold forMβ−.

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 ℓ^p-norm and the ℓ^∞-norm of f by ∥f∥_ℓp(ℤ) = (Σ_nϵZ |f(n)|^p)^1/p and ∥f∥ℓ∞(Z)=supn∈Z|f(n)|.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

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

and

Here M(resp.,M~)denotes the discrete centered (resp., uncentered) Hardy-Littlewood maximal operator, which are defined by

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 MβandM~βare bounded and continuous from ℓ¹(ℤ) to BV(ℤ) (also see [29, 34]). Here M _β and M~βare the discrete centered and uncentered fractional maximal operators, which are defined by

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

where 0≤β<mandf→=(f1,…,fm)with each fi∈Lloc1(Z).When β = 0, the operators Mβ+andMβ−reduce to the discrete one-sided multilinear Hardy-Littlewood maximal operators M⁺ and M⁻, respectively. When m = 1, the operators Mβ+andMβ−reduce to the discrete one-sided fractional maximal operators Mβ+andMβ−respectively. Particularly, the discrete one-sided Hardy-Littlewood maximal operators M⁺ and M⁻ correspond to the special case of Mβ+andMβ−when β = 0, respectively. Recently, Liu and Mao [6] proved that both ⁺⁻M⁺ and M⁻ are bounded and continuous from ℓ¹(ℤ) to BV(ℤ). Moreover, if f ∈ BV(ℤ), then

We notice that the constant C = 1 in inequality (1.7) is sharp. Very recently, Liu [27] pointed out that Mβ+and Mβ−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. ThenMβ+is bounded and continuous from ℓ¹(ℤ) to BV(ℤ).Moreover, if f ϵ ℓ¹(ℤ), then

and the constant C = 2 is the best possible. The same results hold forMβ−.

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

Theorem 1.3

Let 0 ≤ β < m. ThenMβ+is bounded and continuousfrom ℓ¹(ℤ) ×· ··× ℓ¹(ℤ) to BV(ℤ). Moreover, iff→=(f1,…,fm)with each f_i ϵ ℓ¹(ℤ), then

The same results hold forMβ−.

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.

Lemma 2.1

Let 1 < p₁, … , p_m < ∞ and 1 ≤ q < ∞ with1/q=∑i=1m1/pi−β.Letf→j=(f1,j,…,fm,j)and f̄ = (f₁, … , f_m) such that f_i,j ➝ f_i in L_pⁱ (ℝ) when j ➝ ∞. Then, for all ℝ > 0 and λ > 0, it holds that

Proof. Without loss of generality, we may assume that all f_i,j ≥ 0 and f_i ≥ 0. By the similar argument as in the proof of Lemma 2.2 in [7], we can conclude that the set {x∈R:Rβ+(f→j)(x)⊈Rβ+(f→)(x)(λ)}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

Otherwise, for almost every x ϵ (−R, R), there exists a bounded sequence of radii {sk}k=1∞such that

We can choose a subsequence {rk}k=1∞of{sk}k=1∞such that r_k ➝ s as k ➝ ∞. Then we have s∈Rβ+(f→)(x)and d(s,Rβ+(f→)(x))≥λ,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

Notice that

It yields that

where

Hence,

Let Ā be the set of all points x such that x is a Lebesgue point of all f_j. Note that |R \ Ā| = 0 and A_3,j∩ Ā⊂{x∈R:Rβ+(f→j)(x)⊂Rβ+(f→)(x)(λ)}.This together with (2.3) implies

It follows that

We can write

for any x ϵ ℝ, where f→jl=(f1,…,fl−1,fl,j−fl,fl+1,j,…,fm,j).From (2.5) we have

Since f_i,j ➝ f_i in L^pi (ℝ) as j ➝ ∞, then there exists N₀ = N₀(ϵ, γ ) ∈ ℕ \ {0} such that

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

On the other hand, one can easily check that

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

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

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

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

Letf→=(f1,…,fm)with each f_i ∈ L^pi (ℝ) for 1 < p₁, … , p_m < ∞. Let 1 ≤ q < ∞ and 1/q = ∑i=1m1/pi−β.Then, for all ℝ > 0 and λ > 0, we have

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, … , f_m) with each f_i ϵ W^1,pi (ℝ) for 1 < p_i < ∞. Let 1 ≤ q < ∞ and1/q=∑i=1m1/pi−β.

Then, for almost every x ∈ ℝ, we have

Proof. We may assume that all f_i ≥ 0 since |f| ϵ W^1,p(ℝ) if f ∈ W^1,p(ℝ) with 1 < p < ∞. By the boundedness part in Theorem 1.1 we see that Mβ+(f→)∈W1,q(R).Invoking Lemma 2.2, we can choose a sequence {sk}k=1∞sk>0such that lim_k➝∞s_k = 0 and limk→∞π(Rβ+(f→)(x),Rβ+(f→)(x+sk))=0for almost every x ϵ (−R, R). For 1 ≤ i ≤ m and h ϵ ℝ, we set

It was known that

Here (Mβ+(f→))sk(x)=1sk(Mβ+(f→)(x+sk)−Mβ+(f→)(x)).Furthermore, there exists a subsequence {hk}k=1∞of{sk}k=1∞and a measurable set A₁ (−R, R) with |(−R, R)\A₁| = 0 such that

1. fτ(hk)i(x)→fi(x),fhki(x)→fi′(x),M+(fτ(hk)i−fi)(x)→0,M+(fhki−fi′)(x)→0and(Mβ+(f→))hk(x)→(Mβ+(f→))′(x)when k ➝ ∞ for any x ϵ A₁ and 1 ≤ i ≤ m;

2. limk→∞π(Rβ+(f→)(x),Rβ+(f→)(x+hk))=0for any x ∈ A₁.

Let

It is obvious that |(−R, R)\A_j| = 0 for j = 2, 3, 4. Let x ϵ A₁∩ A₂∩ A₃∩ A₄ be a Lebesgue point of all fiandfi′.Fix s∈Rβ+(f→)(x), there exists radii rk∈Rβ+(f→)(x+hk)such that lim_k➝∞r_k = s. We consider the following two cases:

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

Since fτ(hk)νχ(x,x+rk)→fνχ(x,x+s)and fhklχ(x,x+rk)→fl′χ(x,x+s)in L¹(ℝ) as k ➝ ∞. Then (2.12) yields that

On the other hand,

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