Spaces of Lorentz type and complex interpolation
For a nonnegative measurable function \(w\) on \((0,\infty)\) (a ``weight''), one can define the space of Lorentz type \[ \Lambda^ q(w)=\left\{ f: \int^ \infty_ 0\bigl[ w(s) f^*(s)\bigr]^ q ds<\infty\right\}. \] The authors study complex interpolation of the spaces \(\Lambda^ q(w)\). Two approaches are used, which lead to different conditions on weights. The first approach is based on a certain reiteration formula. The second one is based on the characterization of the pairs of weights \((u,v)\) such that the operator \(S: f\mapsto t^{-1}\int^ t_ 0 f(s)ds\) maps \(L^ q(u)\) to \(L^ q(v)\), and on a similar fact for \(S^*\). This characterization is due to \textit{B. Muckenhoupt} [Stud. Math. 44, 31-38 (1972; Zbl 0236.26015)].
scientific article; zbMATH DE number 60548