发表机构
Institute of Mathematical of the Czech Academy of Sciences; L.N. Gumilyov Eurasian National University（捷克科学院数学研究所; L.N.古米廖夫欧亚国立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文完整刻画了离散分数阶极大算子在加权Lorentz序列空间间的有界性，通过尖锐重排估计将其归结为上确界算子的有界性，并利用近期结果得到最终判定。
AI 中文摘要
本文给出了定义在$\mathbb{Z}^n$（$n\in\mathbb{N}$，$\gamma\in [0,n)$）上的离散分数阶极大算子$M_\gamma $在经典Lorentz序列空间$\lambda_p(v)$与$\lambda_q(w)$之间有界性的完整刻画，其中$0<p,q<\infty$，$\{v(m)\}$和$\{w(m)\}$是定义在$\mathbb{N}$上的非负序列。我们首先获得了离散分数阶极大算子$M_\gamma x$的非递增重排的尖锐上界估计，该估计涉及一个包含离散Hardy算子的上确界算子以及序列$\{x(k)\}$的非递增重排，并通过针对特殊序列的下界估计证明了该上界估计的尖锐性。利用这一结果，我们将离散分数阶极大算子$M_\gamma$在离散加权Lorentz序列空间之间的有界性刻画归结为离散上确界算子在加权Lebesgue序列空间上、限制于非递增序列的有界性刻画。这一问题在作者近期的论文中已得到研究，从而使我们能够获得最终的刻画。
英文摘要
In this paper, we give the complete characterization of the boundedness of the discrete fractional maximal operator $M_γ$, defined on $\mathbb{Z}^n$, $n\in\mathbb{N}$, $γ\in [0,n)$, between the classical Lorentz sequence spaces $λ_p(v)$ and $λ_q(w)$, where $0<p,q<\infty$, $\{v(m)\}$ and $\{w(m)\}$ are non-negative sequences defined on $\mathbb{N}$. We first obtain a sharp upper estimate for the nonincreasing rearrangement of the discrete fractional maximal operator $M_γx$, with a supremal operator involving a discrete Hardy operator and the nonincreasing rearrangement of the sequence $\{x(k)\}$, and we show that this estimate is sharp by showing a lower estimate for special sequences. Using this result, we can reduce the characterization of the boundedness of the discrete fractional maximal operator $ M_γ$ between the discrete weighted Lorentz sequence spaces to the boundedness of the discrete supremum operator between weighted Lebesgue sequence spaces, restricted to nonincreasing sequences. This problem was investigated in the authors' recent paper, which enables us to obtain the final characterization.
Comments9 pages
