A Calderón couple of down spaces

Abstract

The down space construction is a variant of the Köthe dual, restricted to the cone of non-negative, nonincreasing 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 ∞ 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 (L 1 , 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 L 1 and D ∞ is obtained. These interpolation spaces are shown to be closely related to the rearrangement invariant spaces via the down space construction.

Structure of Cesàro function spaces: a survey

“…Note that the proof of the second equality in (14) for the spaces on I = [0, ∞) in [11] is essentially based on some results from the paper [71]. Moreover, another proof of this equality in the case I = [0, ∞) was also given by Sinnamon [ …”

Isomorphic Structure of Cesàro and Tandori Spaces

“…Most commonly the classical Cesàro spaces appear as optimal domains of the Cesàro (Hardy) operator or some its versions (see [DS07], [NP10], [LM15b]). Moreover, they can coincide with the so-called down spaces introduced and investigated by Sinnamon (see [KMS07], [MS06], [Si94], [Si01], [Si07]), but having their roots in the papers of Halperin and Lorentz. Comparing to the function case, there is much more rich literature devoted to Cesàro sequence spaces and their duals (see the classical paper of Bennett [Be96] and also [CH01], [CMP00], [Ja74], [KK12], [MPS07]).…”

A new algorithm for approximating the least concave majorant

“…To begin we must ad to the partition points 8,9,10 the critical point 8.5 and the inflection points 9.2 and 9.4. It is then found that the intervals on which s is strictly concave and increasing are The unique component interval in [8,10]. See Figure 6.…”

“…We now seek component intervals contained in [8,10]. To begin we must ad to the partition points 8,9,10 the critical point 8.5 and the inflection points 9.2 and 9.4.…”