\(q\)-concavity and \(q\)-Orlicz property on symmetric sequence spaces (Q5944182)

Let \(1\leq q<\infty\). A Banach lattice \(X\) is said to be \(q\)-concave if there exists a constant \(C\geq 0\) such that \((\sum_{k=1}^n \|x_k\|_X^q)^{1/q}\leq C\|(\sum_{k=1}^n |x_k|^q)^{1/q}\|_X\) for every choice of elements \(x_1,\dots, x_n\) in \(X\). A Banach lattice \(X\) is said to satisfy a lower \(q\)-estimate if there exists a constant \(C\geq 0\) so that, for every choice of elements \(x_1,\dots, x_n\) in \(X\) we have \((\sum_{k=1}^n \|x_k\|_X^q)^{1/q}\leq C\|(\sum_{k=1}^n |x_k|)\|_X\). Two related concepts from the theory of Banach spaces are the following concepts. A Banach space \(X\) is said to have cotype \(q, 2\leq q<\infty\), if there exists a constant \(C\geq 0\) so that \((\sum_{k=1}^n \|x_k\|_X^q)^{1/q}\leq C\int_o^1 \|\sum_{k=1}^n r_k(t) x_k\|_X dt\) for every choice of elements \(x_1,\dots, x_n\) in \(X\), where \(r_k\) stands for the Radamacher functions. \(X\) is said to have the \(q\)-Orlicz property if the identity operator \(id:X\to X\) is \((q,1)\)-summing. There are many connections between all these notions [see \textit{J. Lindenstrauss} and \textit{L. Tzafriri}, ``Classical Banach Spaces, II. Function spaces'', Berlin-Heidelberg-New York (1979; Zbl 0403.46022)]. NEWLINENEWLINENEWLINEFor \(2<q<\infty\), we have that: NEWLINE\[NEWLINE q\text{-concavity \(\Rightarrow\) cotype \(q\) \(\Leftrightarrow\) \(q\)-Orlicz property \(\Leftrightarrow\) lower \(q\)-estimate}.NEWLINE\]NEWLINE For \(q=2\), we have that: NEWLINE\[NEWLINE2\text{-concavity \(\Leftrightarrow\) cotype 2 \(\Rightarrow\) 2-Orlicz property \(\Rightarrow\) lower 2-estimate}.NEWLINE\]NEWLINE \textit{M. Talagrand} constructed an example with the 2-Orlicz property but without cotype 2 [Invent. Math. 107, No. 1, 1-40 (1992; Zbl 0788.47022)]. Moreover, he was even able to construct a counterexample in the setting of symmetric sequence space [Isr. J. Math. 87, No. 1-3, 181-192 (1994; Zbl 0835.46015)]. NEWLINENEWLINENEWLINEIn the paper the study of the relationship between all these notions is continued and a general method, inspired by Talagrand's technique, is given for the constructing symmetric sequence spaces that satisfy a lower \(q\) estimate but fail to be \(q\) concave, \(1<q<2\), and that have the \(q\)-Orlicz property but fail to be \(q\)-concave for \(q\geq 2\).