The exact correction theorem for spaces of functions of generalized bounded variation (Q1905307)

A sequence space \(X\) is called symmetric if for any sequence \(\{\lambda_i:|\lambda_i|\leq 1\}\) and for any \(\{a_i\}\in X\), we have \(\{\lambda_ia_i\}\in X\) and \(\|\{\lambda_i a_i\}\|_X\leq \|\{a_i\}\|_X\) and \(\|\{a_i\}\|_X= \|\{a^*_i\}\|_X\), where \(\{a^*_i\}\) is a permutation of the sequence \(\{| a_i|\}\) in nonascending order. Let \(X\) be a symmetric space of sequences. The space \(\text{BV}(X)\) of functions of generalized bounded variation is the set of all functions \(f: [0,1]\to \mathbb{R}\) such that the norm \[ \| f\|_{\text{BV}(X)}= \sup_{\{J_i\}} \|\{f(b_i)- f(a_i)\}\|_X+ | f(0)| \] is finite, here \(\{J_i\}\) denotes the set of pairwise nonoverlapping intervals \(J_i= (a_i, b_i)\) of \([0,1]\). We say that it is corrected by the Lipschitz space \(\text{Lip }w\) if for an arbitrary \(\varepsilon> 0\) and for a function \(f\in \text{BV}(X)\) there exists a function \(f_\varepsilon\in \text{Lip }w\) such that \(\mu(\{t: f(t)\neq f_\varepsilon(t)\})< \varepsilon\). Theorem 1. Let \(X\) be a symmetric space. Then the space \(\text{BV}(X)\) is corrected by a Lipschitz space \(\text{Lip }h^{-1}\) if and only if the following continuous embedding holds: \(X\subset l_h\). Theorem 5. A symmetric space \(X\) of sequences coincides with some Orlicz space if and only if the space \(\text{BV}(X)\) contains the same Lipschitz space by which it is corrected.