Endpoint regularity of discrete multisublinear fractional maximal operators associated with ℓ 1 $\ell^{1}$ -balls

In this paper we investigate the endpoint regularity of the discrete m-sublinear fractional maximal operator associated with $\ell^{1}$ -balls, both in the centered and uncentered versions. We show that these operators map $\ell^{1}(\mathbb{Z}^{d})\times\cdots\times \ell^{1}(\mathbb{Z}^{d})$ into $\operatorname{BV}(\mathbb{Z}^{d})$ boundedly and continuously. Here $\operatorname{BV}(\mathbb{Z}^{d})$ represents the set of functions of bounded variation defined on $\mathbb{Z}^{d}$ .