Interpolation theorems for rearrangement invariant -spaces of functions, , and some applications
We prove that the basic facts of the real interpolation method remain true for couples of cones obtained by intersection of the cone of concave functions with rearrangement invariant spaces.
We give conditions on Gromov-Hausdorff convergent inverse systems of metric measure graphs which imply that the measured Gromov-Hausdorff limit (equivalently, the inverse limit) is a PI space i.e., it satisfies a doubling condition and a Poincaré inequality in the sense of Heinonen-Koskela [12]. The Poincaré inequality is actually of type (1, 1). We also give a systematic construction of examples for which our conditions are satisfied. Included are known examples of PI spaces, such as Laakso spaces,...