On the Hardy--Littlewood theorem for functions of bounded variation (Q2782977)

A function \(f\in L^1(a,b)\) is called to be of bounded essential variation, if there exists a function \(g\in \text{BV}(a,b)\) such that \(f= g\) a.e. and \(g\) is normal, i.e., the value \(g(x)\) is between the two one-side limits of \(g\) at \(x\) for \(x\in (a,b)\) and \(g\) is right-continuous at \(a\) and left-continuous at \(b\). The essential variation of \(f\) is defined as \(\text{TV}_{\text{ess}}[f; a,b]= \inf\{\text{TV}[u; a,b]: u\in \text{BV}(a,b)\), \(u= f\) a.e.\}. Extending a well-known Hardy-Littlewood result it is shown that if \(f\in L^1(a,b)\) and \(\limsup_{h\to 0+}{1\over h} \int^{b-h}_a|f(x+ h)- f(x)|dx= C<\infty\), then \(f\) is of essential bounded variation on \([a,b]\) and \(\text{TV}_{\text{ess}}[f; a,b]= C\).