On the dual of Orlicz-Lorentz space (Q2781333)

A description of the Köthe duals is given for symmetric Orlicz-Lorentz spaces defined on either nonatomic or purely atomic measure space. Since the proofs work mutatis mutandis only the case of Orlicz-Lorentz spaces defined on \((I,m)\), where either \(I = (0,1)\) or \(I = (0,\infty)\) and \(m\) Lebesgue measure, is considered. Following the ideas from \textit{S. Reisner} [Indiana Univ. Math. J. 31, 65-72 (1982; Zbl 0494.46032)] it is proved, under the assumption that \(\varphi\) is an \(N\)-function satisfying the \(\Delta_2\)-condition, that the regularity of the weight function \(w\) is a necessary and sufficient condition for the dual of Orlicz-Lorentz space on \((I,m)\) to consist exactly of those functions \(f\) for which \(f^*/w\) belongs to the Orlicz function space \(L_{\varphi_*}\) on \((I, wdm)\), where \({\varphi_*}\) is the Young conjugate of \(\varphi\). Some partial results are obtained for the case when \(\varphi\) is an arbitrary Orlicz function.