Endpoint regularity of discrete multisublinear fractional maximal operators associated with ℓ1 $\ell^{1}$-balls

Abstract

Abstract In this paper we investigate the endpoint regularity of the discrete m-sublinear fractional maximal operator associated with ℓ1 $\ell^{1}$-balls, both in the centered and uncentered versions. We show that these operators map ℓ1(Zd)×⋯×ℓ1(Zd) $\ell^{1}(\mathbb{Z}^{d})\times\cdots\times \ell^{1}(\mathbb{Z}^{d})$ into BV(Zd) $\operatorname{BV}(\mathbb{Z}^{d})$ boundedly and continuously. Here BV(Zd) $\operatorname{BV}(\mathbb{Z}^{d})$ represents the set of functions of bounded variation defined on Zd $\mathbb{Z}^{d}$.

Keywords

• Discrete multisublinear fractional maximal function
• ℓ 1 $\ell^{1}$ -balls
• Sobolev space
• Bounded variation
• Continuity