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Endpoint regularity of discrete multilinear fractional nontangential maximal functions
Advances in Difference Equations volume 2019, Article number: 403 (2019)
Abstract
Given \(m\geq 1\), \(0\leq \lambda \leq 1\), and a discrete vector-valued function \(\vec{f}=(f_{1},\ldots,f_{m})\) with each \(f_{j}:\mathbb{Z} ^{d}\rightarrow \mathbb{R}\), we consider the discrete multilinear fractional nontangential maximal operator
where \(\mathcal{B}\) is the collection of all open balls \(B\subset \mathbb{R}^{d}\), \(B_{r}(\vec{x})\) is the open ball in \(\mathbb{R}^{d}\) centered at \(\vec{x}\in \mathbb{R}^{d}\) with radius r, and \(N(B_{r}(\vec{x}))\) is the number of lattice points in the set \(B_{r}(\vec{x})\). We show that the operator \(\vec{f}\mapsto |\nabla \mathrm{M}_{\alpha, \mathcal{B}}^{\lambda }(\vec{f})|\) is bounded and continuous from \(\ell ^{1}(\mathbb{Z}^{d})\times \ell ^{1}(\mathbb{Z} ^{d})\times \cdots \times \ell ^{1}(\mathbb{Z}^{d})\) to \(\ell ^{q}(\mathbb{Z} ^{d})\) if \(0\leq \alpha < md\) and \(q\geq 1\) such that \(q>\frac{d}{md- \alpha +1}\). We also prove that the same result also holds for the discrete multilinear fractional nontangential maximal operators associated with cubes. These results we obtained represent significant and natural extensions of what was known previously.
1 Introduction
The regularity theory of maximal operators has been the subject of many recent articles in harmonic analysis. The first work was due to Kinnunen [12] who observed that the centered Hardy–Littlewood maximal operator \(\mathcal{M}\) is bounded on the first order Sobolev space \(W^{1,p}(\mathbb{R}^{d})\) for \(1< p\leq \infty \). Since then, the regularity properties of the various kinds of maximal operators have been studied by many authors. See [8, 11, 13,14,15,16, 19, 20, 24, 25, 28, 30, 31] for example. Since \(\mathcal{M}:L^{1}(\mathbb{R}^{d}) \rightarrow L^{1}(\mathbb{R} ^{d})\) is not bounded, the endpoint regularity of maximal operators seems to be a deeper issue. A crucial question was posed by Hajłasz and Onninen in [11].
Problem 1.1
([11])
Is the operator \(f\mapsto |\nabla \mathcal{M}f|\) bounded from \(W^{1,1}(\mathbb{R}^{d})\) to \(L^{1}(\mathbb{R}^{d})\)?
In 2002, Tanaka [35] first proved that the uncentered Hardy–Littlewood maximal function \(\widetilde{\mathcal{M}}f\) is weakly differentiable and satisfies
if \(f\in W^{1,1}(\mathbb{R})\). Later on, Tanaka’s result was refined by Aldaz and Pérez Lázaro [1] who showed that if f is of bounded variation on \(\mathbb{R}\), then \(\widetilde{\mathcal{M}}f\) is absolutely continuous and
where \(\operatorname{Var}(f)\) denotes the total variation of f. The above result directly implies that
if \(f\in W^{1,1}(\mathbb{R})\) (see also [23] for a new proof). Notice that inequalities (2) and (3) are sharp. In the centered case, Kurka [17] showed that if f is of bounded variation on \(\mathbb{R}\), then
It was also shown in [17] that if \(f\in W^{1,1} (\mathbb{R})\), then \(\mathcal{M}f\) is weakly differentiable and inequality (3) also holds for \(\mathcal{M}\) with constant \(C=240{,}004\). It is currently unknown whether inequalities (2) and (3) also hold for \(\mathcal{M}\). Recently, Carneiro and Madrid [7] extended inequalities (2) and (3) to a fractional setting. Very recently, we [26] extended the result of [7] to a multisublinear setting. Other interesting works related to this theory are [4, 9, 10, 22, 33].
On the other hand, the investigation of the regularity of discrete maximal operators has also attracted the attention of many authors (cf. [2, 5, 7, 18, 21, 24, 27, 29, 32, 36, 37]). Let us recall some definitions and background. For \(d\geq 1\) and \(\vec{n}=(n_{1},n_{2}, \ldots, n_{d})\in \mathbb{Z}^{d}\), we set \(|\vec{n}|=(\sum_{i=1}^{d} |n_{i}|^{2})^{1/2}\) and \(\|\vec{n}\|_{\infty }= \sup_{1\leq i\leq d}|n _{i}|\). For a discrete function \(f:\mathbb{Z}^{d}\rightarrow \mathbb{R}\) and \(1\leq p\leq \infty \), we define its \(\ell ^{p}( \mathbb{Z}^{d})\)-norm by \(\|f\|_{\ell ^{p}(\mathbb{Z}^{d})}=( \sum_{\vec{n}\in \mathbb{Z}^{d}}|f(\vec{n})|^{p})^{1/p}\) if \(1\leq p<\infty \) and \(\|f\|_{\ell ^{\infty }(\mathbb{Z}^{d})}= \sup_{\vec{n} \in \mathbb{Z}^{d}}|f(\vec{n})|\). Formally, define the discrete analogue of the Sobolev spaces by
where ∇f is the gradient of a discrete function f and is defined by
and \(D_{l} f(\vec{n})\) is the partial derivative of f denoted by
and \(\vec{e}_{l}=(0,\ldots,0,1,0,\ldots,0)\) is the canonical lth base vector, \(l=1,2,\ldots,d\). Observe that
It follows that
We denote by \(\mathrm{BV}(\mathbb{Z}^{d})\) the set of all functions of bounded variation defined on \(\mathbb{Z}^{d}\), where the total variation of \(f:\mathbb{Z}^{d}\rightarrow \mathbb{R}\) is defined by
It is clear that
Let \(m\geq 1\) and \(0\leq \alpha < md\). For a vector-valued function \(\vec{f}=(f_{1},\ldots,f_{m})\) with each \(f_{j}: \mathbb{Z}^{d} \rightarrow \mathbb{R}\) being a discrete function, we define the discrete centered m-sublinear fractional maximal operator \(\mathrm{M}_{\alpha }\) by
where \(B_{r}(\vec{n})\) is the open ball in \(\mathbb{R}^{d}\) centered at n⃗ with radius r and \(N(B_{r}(\vec{n}))\) is the number of lattice points in the set \(B_{r}(\vec{n})\). The uncentered version of \(\mathrm{M}_{\alpha }\) is given by
where the supremum is taken over all open balls \(B_{r}\) in \(\mathbb{R}^{d}\) containing the point n⃗ with radius r. When \(m=1\), the operator \(\mathrm{M}_{\alpha }\) (resp., \(\widetilde{\mathrm{M}}_{\alpha }\)) reduces to the discrete centered (resp., uncentered) fractional maximal operator \(M_{\alpha }\) (resp., \(\widetilde{M}_{\alpha }\)). Particularly, when \(\alpha =0\), the operator \(M_{\alpha }\) (resp., \(\widetilde{M}_{\alpha }\)) is just the discrete centered (resp., uncentered) Hardy–Littlewood maximal operator M (resp., M̃).
The study of the regularity properties of discrete maximal operators was initiated by Bober et al. [2] in 2012 when they observed that
and
It was noticed that inequality (8) is sharp and (9) is not sharp. Inequality (9) with the sharp constant \(C=2\) was proved by Madrid in [32] (see [32, Theorem 1]). It was known that inequality (8) for M was established by Temur in [36] (with constant \(C=294\text{,}912\text{,}004\)). It is unknown whether inequality (8) also holds for M. Recently, Carneiro and Madrid [7] and Liu [18] extended (8) and [32, Theorem 1] to the fractional setting, respectively. More recently, in the remarkable paper [6], Carneiro et al. established the \(\mathrm{BV}(\mathbb{Z})\)-continuity of the discrete centered and uncentered Hardy–Littlewood maximal operator. For general dimension \(d\geq 1\), Carneiro and Hughes [5] showed that both M and M̃ map \(\ell ^{1}(\mathbb{Z}^{d})\) into \(\mathrm{BV}(\mathbb{Z}^{d})\) boundedly and continuously. Recently, Carneiro and Madrid [7] extended the result of [5] to a fractional setting. Very recently, we [27] extended the result of [5] to a multisublinear fractional setting.
Let us recall the main result of [27], which can be stated as follows.
Theorem 1.2
([27])
Let \(0\leq \alpha <(m-1)d+1\). Then both \(\mathrm{M}_{\alpha }\) and \(\widetilde{\mathrm{M}}_{\alpha }\) map \(\ell ^{1}(\mathbb{Z}^{d})\times \ell ^{1}(\mathbb{Z}^{d}) \times \cdots \times \ell ^{1}(\mathbb{Z}^{d})\) into \(\mathrm{BV} (\mathbb{Z} ^{d})\) boundedly and continuously.
The aim of this paper is to investigate the endpoint regularity of the discrete multilinear fractional nontangential maximal operator associated with balls or cubes.
Definition 1.3
Let \(0\leq \alpha < md\) and \(0\leq \lambda \leq 1\). For a vector-valued function \(\vec{f}=(f_{1},\ldots,f_{m})\) with each \(f_{j}:\mathbb{Z} ^{d}\rightarrow \mathbb{R}\) being a discrete function, we define the discrete multilinear fractional nontangential maximal operator associated with balls \(\mathrm{M}_{\alpha, \mathcal{B}}^{\gamma }\) by
where \(\mathcal{B}\) is the collection of all open balls \(B\subset \mathbb{R}^{d}\). The discrete multilinear fractional nontangential maximal operator associated with cubes \(\mathrm{M}_{\alpha, \mathcal{R}}^{\lambda }\) is defined by
where \(\mathcal{R}\) is the collection of all open cubes \(R\subset \mathbb{R}^{d}\) with sides parallel to the coordinate axes and \(R_{r}(\vec{x})\) is the open cube in \(\mathbb{R}^{d}\) centered at x⃗ with length of side 2r.
One can easily check that
By relationships (10)–(11) and the bounds for \(\widetilde{\mathrm{M}}_{\alpha }\), we obtain
if \(1< p_{i}\leq \infty \) \((i=1,\ldots,d)\), \(1\leq q\leq \infty \) for \(\alpha =0\), and \(1< p_{i}<\infty \) \((i=1, \ldots,d)\), \(1\leq q< \infty \) for \(0<\alpha <md\) and \(\frac{1}{q}\leq \frac{1}{p_{1}}+ \cdots +\frac{1}{p_{m}}- \frac{\alpha }{d}\). One can easily check that
where \(\vec{f}=(f_{1},\ldots,f_{m})\), \(\vec{g}=(g_{1}, \ldots,g_{m})\) and \(\vec{F}_{j}=(f_{1},\ldots,f_{j-1}, f_{j}-g_{j},g_{j+1},\ldots,g _{m})\). It follows from (4) and (12)–(13) that both \(\mathrm{M}_{\alpha,\mathcal{B}}^{\lambda }\) and \(\mathrm{M} _{\alpha,\mathcal{R}}^{\lambda }\) are bounded and continuous from \(W^{1,p_{1}}(\mathbb{Z}^{d}) \times W^{1,p_{2}}(\mathbb{Z}^{d})\times \cdots \times W^{1,p_{m}}(\mathbb{Z}^{d})\rightarrow W^{1,q}( \mathbb{Z}^{d})\) provided that \(1< p_{i}\leq \infty \) \((i=1,\ldots,m)\), \(1\leq q\leq \infty \) for \(\alpha =0\), and \(1< p_{i}\leq \infty\ (i=1, \ldots,m)\), \(1\leq q<\infty \) for \(0<\alpha <md\), and \(\frac{1}{q} \leq \frac{1}{p_{1}} +\cdots +\frac{1}{p_{m}}-\frac{\alpha }{d}\). In addition, it is clear that both \(\mathrm{M}_{\alpha, \mathcal{B}} ^{\lambda }\) and \(\mathrm{M}_{\alpha, \mathcal{R}}^{\lambda }\) are not bounded from \(\ell ^{1} (\mathbb{Z}^{d})\times \ell ^{1}(\mathbb{Z}^{d}) \times \cdots \times \ell ^{1}(\mathbb{Z}^{d})\) to \(\ell ^{1}( \mathbb{Z}^{d})\).
Based on the above bounds for \(\mathrm{M}_{\alpha, \mathcal{B}}^{ \lambda }\) and \(\mathrm{M}_{\alpha, \mathcal{R}}^{\lambda }\), Theorem 1.2, (7), and (10)–(11), a question that arises naturally is the following.
Problem 1.4
Are both the operators \(\mathrm{M}_{\alpha, \mathcal{B}}^{\lambda }\) and \(\mathrm{M}_{\alpha, \mathcal{R}}^{\lambda }\) bounded and continuous from \(\ell ^{1}(\mathbb{Z}^{d})\times \ell ^{1}(\mathbb{Z} ^{d}) \times \cdots \times \ell ^{1}(\mathbb{Z}^{d})\) to \(\mathrm{BV}( \mathbb{Z}^{d})\)?
We would like to point out that Problem 1.4 seems to be affirmative and expected. However, we will present a positive answer to Problem 1.4 by the following more general conclusion.
Theorem 1.5
Let \(0\leq \alpha < md\), \(0\leq \lambda \leq 1\), and \(0\leq \beta \leq 1\). Let \(q\geq 1\) such that \(q>\frac{d}{md-\alpha +\beta }\). Then the operator \(\vec{f}\mapsto |\nabla \mathrm{M}_{\alpha, \mathcal{B}} ^{\lambda }(\vec{f})|\) is bounded and continuous from \(\ell ^{1}( \mathbb{Z}^{d})\times \ell ^{1} (\mathbb{Z}^{d})\times \cdots \times \ell ^{1}(\mathbb{Z}^{d})\) to \(\ell ^{q}(\mathbb{Z}^{d})\). Moreover,
for all \(\vec{f}=(f_{1},\ldots,f_{m})\) with each \(f_{j}\in \ell ^{1}( \mathbb{Z}^{d})\). The same results hold for the operator \(\mathrm{M} _{\alpha,\mathcal{R}}^{\lambda }\).
Remark 1.6
It should be pointed out that inequality (14) holds only if \(q\geq \frac{d}{md-\alpha +\beta }\). To see this, let l be an integer such that \(l\geq 8\sqrt{d}\). Taking \(f_{j}(\vec{n})= \chi _{\{|\vec{n}|\leq l\}}(\vec{n})\) for all \(1\leq j\leq m\), one can verify that \(\|f_{j}\|_{\ell ^{1}(\mathbb{Z}^{d})}\sim _{d} l^{d}\), \(\|\nabla f_{j}\|_{\ell ^{1}(\mathbb{Z}^{d})}\sim _{d} l^{d-1}\) and \(\|\nabla \mathrm{M}_{\alpha,\mathcal{B}}^{\lambda }(\vec{f}) \|_{ \ell ^{q}(\mathbb{Z}^{d})}\gtrsim _{d}l^{\frac{d}{q}+\alpha -1}\). It follows that
This yields our claim by letting \(l\rightarrow \infty \).
As several direct corollaries of Theorem 1.5, we obtain the following.
Corollary 1.7
Let \(m\geq 1\), \(0\leq \lambda \leq 1\), and \(0\leq \alpha < (m-1)d+1\). Then both \(\mathrm{M}_{\alpha, \mathcal{B}}^{\lambda }\) and \(\mathrm{M}_{\alpha,\mathcal{R}}^{\lambda }\) are bounded and continuous from \(\ell ^{1}(\mathbb{Z}^{d}) \times \ell ^{1}(\mathbb{Z}^{d})\times \cdots \times \ell ^{1} (\mathbb{Z}^{d})\) to \(\mathrm{BV}(\mathbb{Z} ^{d})\).
Corollary 1.8
Let \(m>1\), \(0\leq \lambda \leq 1\), and \(0\leq \alpha <(m-1)d\). Then both \(\mathrm{M}_{\alpha,\mathcal{B}}^{\lambda }\) and \(\mathrm{M}_{\alpha ,\mathcal{R}}^{\lambda }\) are bounded from \(\mathrm{BV}(\mathbb{Z} ^{d})\times \ell ^{1}(\mathbb{Z}^{d})\times \cdots \times \ell ^{1} ( \mathbb{Z}^{d})\) to \(\mathrm{BV}(\mathbb{Z}^{d})\).
Remark 1.9
When \(\beta =1\) (resp., \(\beta =0\)) and \(0\leq \alpha < (m-1)d+1\) (resp., \(0\leq \alpha <(m-1)d\)), then \(\frac{d}{md-\alpha +\beta }<1\). Thus Theorem 1.5 yields Corollaries 1.7 and 1.8. On the other hand, Corollary 1.7 extends Theorem B, which corresponds to the case \(\lambda =0\) and \(\lambda =1\).
Remark 1.10
Our main results are new even in the special case \(m=1\) and \(\alpha =0\).
The paper is organized as follows. Section 2 contains some preliminary notation and a useful subtle summability lemma. The proof of Theorem 1.5 will be given in Sect. 3. We would like to remark that our arguments are motivated by [7], but our methods and techniques are more simple and direct than those of [7]. It is worth mentioning that there are three virtues: (i) In the previous papers [5, 7, 27], the authors established the endpoint regularities of the discrete maximal operator and its fractional version and multilinear fractional version by dealing with their centered case and uncentered case individually. Here, we give a uniform handling method of proving the regularity properties of discrete centered and uncentered maximal operators. (ii) In the precise papers [5, 7, 27], the proofs of the corresponding continuity results are highly dependent on the Brezis–Lieb lemma [3]. Moreover, the discrete versions of Luiro’s lemma (see [5, Lemmas 3–4] and [7, Lemmas 4–5] played key roles in the proofs of the corresponding continuity results in [5, 7]. However, these useful lemmas are unnecessary in our proofs. (iii) Although our main result greatly improves the main result of [27], our methods and techniques are more simple than those of [27].
Throughout this paper, if there exists a constant \(c>0\) depending only on Ï‘ such that \(A\leq cB\), we then write \(A \lesssim _{\vartheta }B\) or \(B\gtrsim _{\vartheta }A\); and if \(A\lesssim _{\vartheta }B \lesssim _{\vartheta }A\), we then write \(A\sim _{\vartheta }B\). In what follows, for a set \(E\subset \mathbb{R}^{d}\), we set \(E^{c}=\{x\in \mathbb{R}^{d};x\notin E\}\). We shall use the conventions \(\prod_{i\in \emptyset }a_{i}=1\) and \(\sum_{i\in \emptyset } a_{i}=0\).
2 Preliminaries
We start by presenting some preliminary notation. It was shown in [34] that
where \(c_{d}=\frac{2\pi ^{d/2}}{\varGamma (d/2)d}\). It is clear that
Here \([x]=\{k\in \mathbb{Z};k\leq x\}\). Fix \(\vec{x} \in \mathbb{R} ^{d}\backslash \mathbb{Z}^{d}\), there exist two lattice points \(\vec{n_{1}}\in \mathbb{Z}^{d}\) and \(\vec{n_{2}}\in \mathbb{Z}^{d}\) such that \(|\vec{n_{1}}-\vec{x}| \leq \sqrt{d}/2\) and \(\|\vec{n_{2}}- \vec{x}\|_{\infty }\leq 1/2\) and
Consequently,
It follows from (18) that
Define the functions \(F(r)\) and \(G(r)\) on \((0,\infty )\) by
Observe from (17) and (19) that
We can claim that
To see this, fix \(\vec{x}\in \mathbb{R}^{d}\), when \(r\geq 4\sqrt{d}\), by (17) and the differential mean value theorem,
When \(0< r<4\sqrt{d}\), we get from (15) that
This together with (22) yields (21).
Fix \(r>0\) and \(\vec{x}\in \mathbb{R}^{d}\), if there exists \(\vec{n} \in \mathbb{Z}^{d}\) such that \(\vec{n}\in R_{r}(\vec{x})\). It follows easily from (16) and (19) that
The following lemma is two refined summability estimates, which play key roles in our proofs.
Lemma 2.1
Let \(R>\sqrt{d}\) and \(\gamma >d\). Then
Proof
For any \(\vec{n}\in \mathbb{Z}^{d}\), let \(Q(\vec{n})=\{x\in \mathbb{R}^{d}:-1/2< x_{i}-n_{i}\leq 1/2, 1\leq i\leq d\}\). Clearly, \(Q(\vec{n})\cap Q(\vec{l})= \emptyset \) for \(\vec{n}\neq \vec{l}\) and \(\bigcup_{|\vec{n}| \geq R, \vec{n}\in \mathbb{Z}^{d}}Q(\vec{n}) \subset \{x\in \mathbb{R}^{d}:|x|\geq R/2\}\). When \(\vec{x}\in Q( \vec{n})\) and \(|\vec{n}|\geq R\), we have
It follows that
Note that \(\frac{|\vec{n}|}{\sqrt{d}}\leq \|\vec{n}\|_{\infty } \leq |\vec{n}|\). Then (24) leads to
 □
3 Proof of Theorem 1.5
3.1 Proof of Theorem 1.5 for \(\mathrm{M}_{\alpha, \mathcal{B}}^{\lambda }\)
The proof will be divided into two parts.
Step 1. Proof of the boundedness part. Let \(0\leq \beta \leq 1\) and \(q\geq 1\) such that \(q>\frac{d}{md-\alpha +\beta }\). Let \(\vec{f}=(f_{1}, \ldots,f_{m})\) with each \(f_{j}\in \ell ^{1}( \mathbb{Z}^{d})\). Without loss of generality we may assume that all \(f_{j}\geq 0\). We want to show that
for all \(1\leq l\leq d\). We shall prove (25) for \(l=d\), and other cases are analogous. In what follows, we set \(\vec{n}=(n',n_{d})\in \mathbb{Z}^{d}\) with \(n'=(n_{1},\ldots,n_{d-1})\in \mathbb{Z}^{d-1}\). Then
For each \(n'\in \mathbb{Z}^{d-1}\), let
Hence,
Thus, to prove (25), it suffices to prove that
We only prove (26) since (27) is analogous. For \(r>0\), we define the function \(\mathrm{A}_{r}(\vec{f}): \mathbb{R}^{d}\rightarrow \mathbb{R}\) by
Since all \(f_{j}\in \ell ^{1}(\mathbb{Z}^{d})\), then, for any \(\vec{x}\in \mathbb{R}^{d}\), \(\lim_{r\rightarrow \infty }\mathrm{A} _{r}(\vec{f})(\vec{x})=0\). It follows that, for any \((n',n_{d})\in \mathbb{Z}^{d}\) with \(X_{n'}^{+}\), there exist \(\vec{x}\in \mathbb{R} ^{d}\) and \(r(n',n_{d})>0\) such that \(|(n',n_{d})-\vec{x}|\leq \lambda r(n',n_{d})\) and \(\mathrm{M}_{\alpha, \mathcal{B}}^{\lambda }( \vec{f})(n',n_{d})= \mathrm{A}_{r(n',n_{d})}(\vec{f})(\vec{x})\). Note that \(|(n',n_{d}+1)-(\vec{x}+\vec{e}_{d})|\leq \lambda r(n',n_{d})\). Consequently,
This together with (20) and (21) implies that
On the other hand, (20) yields that
Note that \(B_{\vec{r}(n',n_{d})}(\vec{x})\subset B_{(\lambda +1) \vec{r}(n',n_{d})}(n',n_{d})\). It follows from (28) and (29) that
For convenience, fix \(1\leq \mu \leq m\), we set
Then (30) leads to
By (31) and Hölder’s inequality with exponents \(p=\frac{1}{1- \beta }\) and \(p'=\frac{1}{\beta }\),
Note that
Fix \(\vec{k}\in \mathbb{Z}^{d}\). By (15) we have
Since \(q(md-\alpha +\beta )>d\). Invoking Lemma 2.1, we have
Combining (35) with (34) yields that
(36) together with (33) yields that
Similarly,
Then (26) follows from (32) and (37)–(38).
Step 2. Proof of the continuity part. Let \(\vec{f} =(f_{1}, \ldots,f_{m})\) with each \(f_{j}\in \ell ^{1} (\mathbb{Z}^{d})\) and \(g_{i,j}\rightarrow f_{j}\) in \(\ell ^{1}(\mathbb{Z}^{d})\) for any \(1\leq j\leq m\) as \(i\rightarrow \infty \). Let \(\vec{g_{i}}=(g_{i,1}, \ldots,g_{i,m})\) for \(i\in \mathbb{Z}\). Since \(||g_{i,j}|-|f_{j}|| \leq |g_{i,j}-f_{j}|\) for all \(1\leq j\leq d\), we may assume without loss of generality that all \(g_{i,j}\geq 0\) and \(f_{j}\geq 0\). It suffices to show that
for each \(l=1,2,\ldots,d\).
We only prove (39) for \(l=d\) (since other cases are analogous). By the boundedness part, we see that \(D_{d}\mathrm{M}_{\alpha,\mathcal{B}} ^{\lambda }(\vec{f}) \in \ell ^{q}(\mathbb{Z}^{d})\). Then, fix \(\epsilon \in (0,1)\), there exist \(N_{1}=N_{1}(\epsilon,\vec{f})>0\) and \(\varLambda >8\sqrt{d}(\lambda +1)\) such that
Combining (41) with (40) yields that
For any \(\vec{n}\in \mathbb{Z}^{d}\) and \(i\geq N_{1}\), we can write
which together with (40) implies that \(\mathrm{M}_{\alpha, \mathcal{B}} ^{\lambda }(\vec{g_{i}})(\vec{n})\rightarrow \mathrm{M} _{\alpha, \mathcal{B}}^{\lambda }(\vec{f})(\vec{n})\) as \(i\rightarrow \infty \) for any \(\vec{n}\in \mathbb{Z}^{d}\). Consequently,
By (43), there exists \(N_{2}=N_{2}(\epsilon,\varLambda )>0\) such that
It follows from (44) that
Fix \(i\geq \max \{N_{1},N_{2}\}\). We can write
We first estimate \(A_{1}\). For each \(n'\in \mathbb{Z}^{d-1}\) with \(|n'|\geq 2\varLambda \), let
Then
We now prove that
We only prove (48), and (49) is analogous. Since all \(g_{i,j}\in \ell ^{1}(\mathbb{Z}^{d})\), then for any \((n',n_{d})\in \mathbb{Z}^{d}\) with \(n_{d}\in Y_{n'}^{+}\), there exist \(\vec{x}\in \mathbb{R}^{d}\) and \(r(n', n_{d})>0\) such that \(|(n',n_{d})-\vec{x}|\leq \lambda r(n',n _{d})\) and \(\mathrm{M}_{\alpha, \mathcal{B}}^{\lambda }(\vec{g}_{i})(n',n _{d})= \mathrm{A}_{r(n',n_{d})}(\vec{g}_{i})(\vec{x})\). By the argument similar to those used in deriving (32),
Note that
Fix \(\vec{k}=(k',k_{d})\). When \(|k'|\geq \varLambda \), we get from (36) that
When \(|k'|<\varLambda \), note that \(|n'-k'|>\varLambda > 8\sqrt{d}(\lambda +1)\) and \(q(md-\alpha +\beta )>d\). Then by Lemma 2.1 and (41) we have
It follows from (40), (42), and (51)–(53) that
Similarly,
Then (48) follows from (40), (50), and (54)–(55). It follows from (47)–(49) that
It remains to estimate \(A_{2}\). For each \(n'\in \mathbb{Z}^{d-1}\), let
Then we have
We want to show that
We only prove (58), and (59) is analogous. Since all \(g_{i,j}\in \ell ^{1}(\mathbb{Z}^{d})\), then for any \((n',n_{d}) \in \mathbb{Z}^{d}\) with \(n_{d}\in Z_{n'}^{+}\), there exist \(\vec{x}\in \mathbb{R}^{d}\) and \(r(n',n_{d})>0\) such that \(|(n',n_{d})-\vec{x}|\leq \lambda r(n',n _{d})\) and \(\mathrm{M}_{\alpha,\mathcal{B}}^{\lambda }(\vec{g}_{i}) (n',n _{d})=\mathrm{A}_{r(n',n_{d})}(\vec{g}_{i})(\vec{x})\). By the arguments similar to those used to derived (32),
Note that
Fix \(\vec{k}=(k',k_{d})\). When \(|k_{d}|<\varLambda \), note that \(|n_{d}-k_{d}|>\varLambda > 8\sqrt{d}(\lambda +1)\) and \(q(md-\alpha + \beta )>d\). Invoking Lemma 2.1, we have
When \(|k_{d}|\geq \varLambda \), we get easily from (36) that
It follows from (40) and (61)–(63) that
Similarly,
Combining (60) with (40) and (64)–(65) yields (58). We get from (57)–(59) that
It follows from (45)–(46), (56), and (66) that
This yields (39) for \(l=d\).
3.2 Proof of Theorem 1.5 for \(\mathrm{M}_{\alpha, \mathcal{R}}^{\lambda }\)
The proof of Theorem 1.5 for \(\mathrm{M}_{\alpha, \mathcal{R}} ^{\lambda }\) is similar as for \(\mathrm{M}_{\alpha,\mathcal{B}}^{ \lambda }\). We only replace the norm \(|\cdot |\) with \(\|\cdot \|_{ \infty }\). The details are left to the interested reader.
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The author thanks anonymous referees for their valuable suggestions.
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This work was supported by the Natural Science Foundation of University Union of Science and Technology Department of Fujian Province (No. 2019J01784), the Youth Foundation of Fujian Province (Grant No. JAT170398), and the Natural Science Foundation of Fujian University of Technology (Nos. GY-Z15124, GY-Z160129).
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Zhang, D. Endpoint regularity of discrete multilinear fractional nontangential maximal functions. Adv Differ Equ 2019, 403 (2019). https://doi.org/10.1186/s13662-019-2257-3
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DOI: https://doi.org/10.1186/s13662-019-2257-3