An Orlicz scale of integrals (Q1119762)

Let \(\Phi\) : \({\mathbb{R}}\to {\mathbb{R}}\) be a convex, even function with \(\Phi (0)=0\), \(\Phi (x)>0\) for \(x>0\), \(x^{-1}\Phi (x)\to 0\) as \(x\to 0\), \(x^{-1}\Phi (x)\to \infty\) as \(x\to \infty\), satisfying the condition \((\Delta_ 2)\) for small \(| x|\). Such functions are called in the paper Orlicz functions. \(\Phi\) generates spaces \(W_{\Phi}[a,b]\) of functions of bounded \(\Phi\)-variation \(V_{\Phi}(f,[a,b])\) and \(AC_{\Phi}[a,b]\) of \(\Phi\)-absolutely continuous functions. A function \(f:\quad [a,b]\to {\mathbb{R}}\) is called \(D_{\Phi}\)-integrable on \([a,b]\), \(f\in D_{\Phi}[a,b]\), if it is Perron integrable on \([a,b]\) and has a major function belonging to \(W_{\Phi}[a,b].\) Among others, there are given necessary and sufficient conditions in order that \(f\in D_{\Phi}[a,b]\), formulated by means of its primitive F. There is established the general form of linear continuous functionals on \(D_{\Phi}[a,b]\) provided with norm generated by \(\Phi\)-variation.