On the endpoint regularity of discrete maximal operators (Q390211)

Let \(M\) denote the Hardy-Littlewood maximal operator. In 2004 \textit{P. Hajłasz} and \textit{J. Onninen} [Ann. Acad. Sci. Fenn., Math. 29, No. 1, 167--176 (2004; Zbl 1059.46020)] posed the question: is the operator \(\displaystyle f\rightarrow \nabla Mf\) bounded from \(\displaystyle W^{1,1}(\mathbb{R}^d)\) to \(\displaystyle L^1(\mathbb{R}^d)\)? In this paper the authors address this problem in the discrete setting. Let \(\displaystyle \Omega\subset \mathbb{R}^d\) be a bounded open subset that is convex with Lipschitz boundary and contains the origin and let \(f\) be a given function, \(\displaystyle f :\mathbb{Z}^d\rightarrow\mathbb{R} \). The centered discrete maximal operator associated to \(\Omega\), \(\displaystyle Mf(\vec{n})=\sup_{r\geq 0}\frac{1}{N(r)}\sum_{\vec{m}\in \bar{\Omega}_r}|f(\vec{n}+\vec{m})|\), where \(\{\bar{\Omega}_r\}_{r\geq 0}\) are dilations of \(\Omega\) and \( N(r)\) is the number of lattice points in \(\bar{\Omega}_r\), is being considered. The authors prove that the operator \( f\rightarrow \nabla Mf\) is bounded and continuous form \(\displaystyle l^1(\mathbb{Z}^d)\) to \(\displaystyle l^1(\mathbb{Z}^d)\). The same result is also shown for the non-centered version of the aforementioned maximal operator. The boundedness result provides a positive answer to the question by Hajłasz and Onninen in the discrete setting, in all dimensions and for this general family of centered or non-centered maximal operators with convex \(\Omega-\)balls. The continuity result is a novelty in the endpoint regularity theory.