Interpolation of operators in quasinormed groups of measurable functions (Q1920837)

An Abelian group with a quasinorm \(|\cdot|\) is called a quasinormed group. The author considers complete quasinormed groups of Lebesgue measurable real functions on \((0,\infty)\) with the usual addition. An important example of such a group is the Lebesgue space \(L_p(0,\infty)\), \(0<p\leq\infty\). For \(0<p\leq 1\) an equivalent norm on \(L_p(0,\infty)\) is given by \[ |x|_p= \int^\infty_0|x(s)|^pds.\tag{\(*\)} \] Following \textit{J. Peetre} and \textit{G. Sparr} [Ann. Mat. Pura Appl., IV. Ser., 92, 217-262 (1972; Zbl 0237.46039)] one can pass in \((*)\) to the limit as \(p\to 0\) and define \(L_0\) as the set of all measurable function on \((0,\infty)\) which are finite a.e. and satisfy \[ |x|_0:=\text{meas }(\text{supp }f)<\infty. \] Then \(L_0\) is a rearrangement-invariant quasinormed group. The author considers the pair \((L_0,L_\infty)\) and his aim is to characterize the orbit of \(\text{Orb}(a;L_0,L_\infty)\) in \(L_0+L_\infty\) for any \(a\in L_0+L_\infty\) (Theorem 1). Furthermore, he shows that there is a function \(a\in L_0+L_\infty\) such that \[ \text{Orb }(a;L_0,L_\infty)\neq \text{KO}(a;L_0,L_\infty), \] where \(\text{KO}(a;L_0,L_\infty)\) stands for the \(K\)-orbit of \(a\). This represents the essential distinction between the pairs \((L_0,L_\infty)\) and \((L_1,L_\infty)\).