The Minimal Function of a BV Function

In this paper we introduce a class of one dimensional non-centered minimal operator \({\widetilde{m}}_\Phi \) ${\tilde{m}}_{\mathrm{\Phi }}$ associated to a function \(\Phi \) $\mathrm{\Phi }$ , which covers the usual minimal operator. We establish the boundedness of \({\widetilde{m}}_\Phi :\textrm{BV}(\mathbb {R})\rightarrow \textrm{BV}(\mathbb {R})\) ${\tilde{m}}_{\mathrm{\Phi }}:\text{BV}(\mathbb{R})\to \text{BV}(\mathbb{R})$ under a more restrictive condition on \(\Phi \) $\mathrm{\Phi }$ . Here \(\textrm{BV}(\mathbb {R})\) $\text{BV}(\mathbb{R})$ is the set of functions of bounded variation defined on \(\mathbb {R}\) $\mathbb{R}$ . In the discrete setting, we prove the discrete minimal operator is bounded and continuous from \(\textrm{BV}(\mathbb {Z})\) $\text{BV}(\mathbb{Z})$ to itself under a more restrictive condition on \(\Phi \) $\mathrm{\Phi }$ .