Endpoint regularity of discrete multilinear fractional nontangential maximal functions

• Main Author: Daiqing Zhang
• Format: Article
• Language: English
• Published: SpringerOpen 2019-09-01
• Series: Advances in Difference Equations
• Subjects: Discrete multilinear fractional nontangential maximal operator; Discrete multilinear fractional maximal operator; Bounded variation; Continuity
• Online Access: http://link.springer.com/article/10.1186/s13662-019-2257-3

Abstract Given m≥1 $m\geq 1$, 0≤λ≤1 $0\leq \lambda \leq 1$, and a discrete vector-valued function f→=(f1,…,fm) $\vec{f}=(f_{1},\ldots,f_{m})$ with each fj:Zd→R $f_{j}:\mathbb{Z} ^{d}\rightarrow \mathbb{R}$, we consider the discrete multilinear fractional nontangential maximal operator Mα,Bλ(f→)(n→)=supr>0,x→∈Rd|n→−x→|≤λr1N(Br(x→))m−αd∏j=1m∑k→∈Br(x→)∩Zd|fj(k→)|, $$ \mathrm{M}_{\alpha,\mathcal{B}}^{\lambda }(\vec{f}) (\vec{n})=\mathop{\sup_{r>0, \vec{x}\in \mathbb{R}^{d}}}_{ \vert \vec{n}-\vec{x} \vert \leq \lambda r}\frac{1}{N(B _{r}(\vec{x}))^{m-\frac{\alpha }{d}}} \prod _{j=1}^{m}\sum_{\vec{k}\in B_{r}(\vec{x})\cap \mathbb{Z}^{d}} \bigl\vert f_{j}(\vec{k}) \bigr\vert , $$ where B $\mathcal{B}$ is the collection of all open balls B⊂Rd $B\subset \mathbb{R}^{d}$, Br(x→) $B_{r}(\vec{x})$ is the open ball in Rd $\mathbb{R}^{d}$ centered at x→∈Rd $\vec{x}\in \mathbb{R}^{d}$ with radius r, and N(Br(x→)) $N(B_{r}(\vec{x}))$ is the number of lattice points in the set Br(x→) $B_{r}(\vec{x})$. We show that the operator f→↦|∇Mα,Bλ(f→)| $\vec{f}\mapsto |\nabla \mathrm{M}_{\alpha, \mathcal{B}}^{\lambda }(\vec{f})|$ is bounded and continuous from ℓ1(Zd)×ℓ1(Zd)×⋯×ℓ1(Zd) $\ell ^{1}(\mathbb{Z}^{d})\times \ell ^{1}(\mathbb{Z} ^{d})\times \cdots \times \ell ^{1}(\mathbb{Z}^{d})$ to ℓq(Zd) $\ell ^{q}(\mathbb{Z} ^{d})$ if 0≤α<md $0\leq \alpha < md$ and q≥1 $q\geq 1$ such that q>dmd−α+1 $q>\frac{d}{md- \alpha +1}$. 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.