Endpoint regularity of the discrete multisublinear fractional maximal operators
The main results of this article are about the discrete centered and uncentered \(m\)-sublinear fractional maximal operators, \( \mathfrak{M}_\alpha \) and \( \widetilde{ \mathfrak{M} }_\alpha \) (respectively). First, a theorem on endpoint regularity is obtained for \( \widetilde{ \mathfrak{M} }_\alpha \) from the \(m\)-fold Cartesian product of \( \text{BV}(\mathbb{Z}) \) (\(0 \leq \alpha < 1 \)) or \( \ell^1 (\mathbb{Z}) \) (\( m-1 \leq \alpha < m \)) into the space of functions with bounded \(q\)-variation, where \(q\) depends on the subscript \(\alpha\). Here \( \text{BV}(\mathbb{Z}) \) denotes the space of functions of bounded variation defined on \(\mathbb{Z}\). After that, a theorem showing the boundedness of \(\mathfrak{M}_\alpha \) and \( \widetilde{ \mathfrak{M} }_\alpha \), \( 0 \leq \alpha < m \), from the \(m\)-fold Cartesian product of \( \ell^1 (\mathbb{Z}) \) into \( \text{BV}(\mathbb{Z}) \) is obtained. Both theorems can be considered as the discrete version of the corresponding results contained in [\textit{F. Liu} and \textit{H. Wu}, Can. Math. Bull. 60, No. 3, 586--603 (2017; Zbl 1372.42015)] and also extend the corresponding already known results of \textit{E. Carneiro} and \textit{J. Madrid} [Trans. Am. Math. Soc. 369, No. 6, 4063--4092 (2017; Zbl 1370.26022)] and \textit{F. Liu} [Bull. Aust. Math. Soc. 95, No. 1, 108--120 (2017; Zbl 1364.42020)].
discrete multisublinear fractional maximal operator
scientific article; zbMATH DE number 7369089
scientific article