\(q\)-concavity and related properties on symmetric sequence spaces (Q1863733)

The authors define a new property of a Banach lattice \(X\), namely being \(q\)-concave with respect to \(X_1(l^1)\), where \(X_1\) is a Banach lattice such that \(X\subset X_1\) with continuous inclusion, and \(1\leq q< \infty\). \(X\) has this property if for some constant \(c\) \[ \Biggl( \sum_{k=1}^n \| x_k\| _X^q\Biggr)^{1/q} \leq c \max\Biggl\{ \Biggl\| \sum_{k=1}^n | x_k| \Biggr\| _{X_1}, \Biggl\| \Biggl( \sum_{k=1}^n | x_k| ^q\Biggr)^{1/q} \Biggr\| _X \Biggr\} \] for all elements \(x_1,\dots,x_n\in X\). The nice thing about this property is that it is implied by \(X\) being \(q\)-concave and it implies a lower \(q\)-estimate in \(X\). The authors then go on to construct for every \(1<q<\infty\) a maximal symmetric sequence space that is \(q\)-concave with respect to \(l^\infty(l^1)\) but is not \(q\)-concave. This provides a unified approach to all of the following examples: \(\bullet\) \(1<q<2\): A Banach lattice that satisfies a lower \(q\)-estimate but is not \(q\)-concave. Previously, it was known that the Lorentz spaces \(L_{q,1}\) work here, see \textit{J. Creekmore} [Indag. Math. 43, 145--152 (1981; Zbl 0483.46014)]. \(\bullet\) \(q=2\): A Banach space that has the Orlicz property but is not of cotype \(2\). This is originally due to \textit{M. Talagrand} [Isr. J. Math. 87, 181--192 (1994; Zbl 0835.46015)]. \(\bullet\) \(2<q<\infty\): A Banach lattice that is of cotype \(q\) but is not \(q\)-concave. The first examples of this type are due to G. Pisier, see \textit{J. Lindenstrauss} and \textit{L. Tzafriri} [Classical Banach spaces. II: Function spaces. (Ergebnisse der Mathematik und ihrer Grenzgebiete 97, Springer-Verlag, Berlin-Heidelberg-New York) (1979, Zbl 0403.46022)]. The method employed in the present paper is a clever refinement of Talagrand's original example for the case \(q=2\).