Special Banach lattices and their applications. (Q2760178)

The article provides a self-contained survey of certain Banach lattices central to analysis. The Lorentz spaces \(L_{p,q}\) and their properties including relations to interpolation theory are introduced. An analysis of questions of embedding \(l_p\) or \(l_q\) in \(L_{p,q}\) is presented. Orlicz spaces are introduced including Orlicz and Luxemburg norms, questions of duality, and relations to interpolation theory. The Nikishin factorization theorem stating that a continuous linear operator from a \(p\)-Banach space (triangle inequality for the \(p\)th power of the norm) to \(L_0\) can be factorized through \(L_{p,\infty}\) is proved. Then abstractions of Aldous's Theorem which states that every closed subspace of \(L_1(0,1)\) contains a subspace isomorphic to \(l_p\) for some \(p\) between \(1\) and \(2\) is considered. An interplay of Lorentz and Orlicz spaces with probability is also included. The final section of the article discusses the embedding of \(L_{p,q}\) into \(L_q\).NEWLINENEWLINEThis survey is well-written and contains extensive references to pertinent literature.NEWLINENEWLINEFor the entire collection see [Zbl 0970.46001].