Embedding \(\ell^ n_ p\) into \(r\)-Banach spaces, \(0<r\leq p<2\)
If \((X_ 1, \|\;\|_ 1)\) and \((X_ 2, \|\;\|_ 2)\) are quasi-Banach spaces, we say that for \(0< \varepsilon< 1\) the space \(X_ 1\) \((1+ \varepsilon)\)-embeds in \(X_ 2\) if there is an operator \(T: X_ 1\to X_ 2\) so that \((1- \varepsilon)\| x\|_ 1\leq \| Tx\|_ 2\leq (1+ \varepsilon)\| x\|_ 1\), for all \(x\) in \(X\). In an earlier paper the authors showed that \(\ell^ K_ p\) \((1+ \varepsilon)\)-embeds in \(\ell^ n_ r\) whenever \(0< r< p< 2\) and \(n\geq C\cdot K\) (where \(C\) is a constant depending only on \(\varepsilon\), \(p\), and \(r\)). In this paper they prove an analogue of a result of \textit{Q. Pisier} [Trans. Am. Math. Soc. 276, 201-211 (1983; Zbl 0509.46016)] concerning \((1+ \varepsilon)\)-embeddings of \(\ell^ n_ p\) into a Banach space for the case of such embeddings of \(\ell^ n_ p\) into an \(r\)-Banach space, where \(0< r< p< 2\). The results are technical in nature and will not be detailed here.
scientific article; zbMATH DE number 484023