A Calderón couple of down spaces
The down space construction is a variant of the Köthe dual, restricted to the cone of non-negative, non-increasing functions. The down space corresponding to \(L^{1}\) is shown to be \(L^{1}\) itself. An explicit formula for the norm of the down space \(D^{\infty }\) corresponding to \(L^{\infty }\) is given in terms of the Hardy averaging operator. A formula for the Peetre \(K\)-functional follows and is used to show that \((L^{1},D^{\infty })\) is a uniform Calderón couple with constant of \(K\)-divisibility equal to one. These results are first proved for the Lebesgue measure on \((0,\infty)\) and then, using this important special case and some transfer arguments for the \(K\)-functional, for any measure \(\lambda\) on the Borel subsets of \(\mathbb R\) satisfying \(\lambda(-\infty,x]<\infty\) for all \(x \in \mathbb R\). As a consequence, a complete description of all exact interpolation spaces between \(L^{1}\) and \(D^{\infty }\) is obtained, similar to the classical result for the couple \((L^1,L^\infty)\). Whereas in the classical case, the main characterization is stated in terms of integral inequalities for the decreasing rearrangement of functions, here for \((L^1,D^\infty)\) integral inequalities for so-called level functions are involved. These interpolation spaces are shown to be closely related to the rearrangement invariant spaces via the down space construction.
scientific article; zbMATH DE number 5083447
scientific article