Disjointly homogeneous Banach lattices and applications

This is a survey on disjointly homogeneous Banach lattices and their applicactions. Several structural properties of this class are analyzed. In addition we show how these spaces provide a natural framework for studying the compactness of powers of operators allowing for a unified treatment of well-known results.Comment: 20 pages, to appear in Proceedings Positivity VII Conference (2013

The least doubling constant of a metric measure space

We study the least doubling constant $C_{(X,d)}$, among all doubling measures $\mu$ supported on a metric space $(X,d)$. In particular, we prove that for every metric space with more than one point, $C_{(X,d)}\ge 2$. We also describe some further properties of $C_{(X,d)}$ and compute its value for several important examples.Comment: 16 pages. Theorem 3.1 has been included improving previous estimates for general metric spaces, and therefore some partial results from the older version have been remove