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 L1 is shown to be L1 itself. An explicit formula for the norm of the down space D∞ corresponding to L∞ is given in terms of the Hardy averaging operator. A formula for the Peetre K-functional follows and is used to show that (L1,D∞) is a uniform Calderón couple with constant of K-divisibility equal to one. As a consequence a complete description of all exact interpolation spaces between L1 and D∞ is obtained. These interpolation spaces are shown to be closely related to the rearrangement invariant spaces via the down space construction.