Coordinatewise multiple summing operators in Banach spaces
The present paper is a continuation and improvement of [\textit{A.\,Defant} and \textit{P.\,Sevilla-Peris}, J.~Funct.\ Anal.\ 256, No.\,5, 1642--1664 (2009; Zbl 1171.46034)]. Let \(1\leq p\leq q<\infty\) and \(E_{1},\dots ,E_{m}\) and \(F\) be Banach spaces. A continuous \(m\)-linear mapping \(T:E_{1}\times\cdots\times E_{m}\rightarrow F\) is multiple \((q;p)\)-summing if there exists a constant \(L_{m}\geq0\) such that \[ \left( \sum_{j_{1},\dots ,j_{m}=1}^{N}\| T(x_{j_{1}}^{(1)},\dots ,x_{j_{m} }^{(m)})\|^{q}\right) ^{1/q}\leq L_{m}\prod\limits_{k=1}^{m}\left( \sup_{\varphi\in B_{E_{k}^{\ast}}}\sum\limits_{j=1}^{N}\left| \varphi(x_{j}^{(k)})\right| ^{p}\right) ^{1/p}\text{ } \] for every positive integer \(N\) and \(x_{j}^{(k)}\in E_{k}\), with \(k\in \{1,\dots ,m\}\) and \(j\in\{1,\dots ,N\}\). This class of multilinear mappings was introduced, independently, by \textit{M.\,C.\thinspace Matos} [Collect.\ Math.\ 54, No.\,2, 111--136 (2003; Zbl 1078.46031)] and by \textit{F.\,Bombal, D.\,Pérez-García} and \textit{I.\,Villanueva} [Q.\ J.\ Math.\ 55, No.\,4, 441--450 (2004; Zbl 1078.46030)]. However, this idea of summability of multilinear maps appears, for particular bilinear maps, in the famous Littlewood \(4/3\) theorem and, for multilinear maps, in the generalization of Littlewood's \(4/3\) theorem to multilinear mappings due to \textit{H.\,F.\thinspace Bohnenblust} and \textit{E.\,Hille} [Ann.\ Math.\ (2) 32, No.\,3, 600--622 (1931; Zbl 0001.26901 and JFM 57.0266.05)]. More precisely, a reformulation of the Bohnenblust-Hille inequality asserts that every continuous \(m\)-linear form \(T:E_{1}\times\dots\times E_{m}\rightarrow \mathbb{C}\) is multiple \((\frac{2m}{m+1};1)\)-summing and \(L_{m}\leq 2^{\frac{m-1}{2}}\). The result is also true for Banach spaces over the real scalar field. The main result of the paper is a beautiful vector-valued extension of Bohnenblust-Hille inequality, involving the notion of cotype. Also, the results of the paper re-prove a recent result by Marcela Luciano Vilela de Souza (from her PhD thesis, 2003) and by \textit{F.\,Bombal, D.\,Pérez-García} and \textit{I.\,Villanueva} [loc.\,cit.] which asserts that every bounded \(m\)-linear mapping with values in a cotype \(q\) space is multiple \((q;1)\)-summing. The nice proof of the main result combines the notion of coordinatewise multiple summing operators, introduced by the authors, and some lemmas (Khinchin's inequality, a variant of an inequality of Blei, and an inequality involving cotype). Moreover, the results of the paper are used, for example, for the generalization of a theorem due to S. Kwapien and in the convergence theory of products of vector valued Dirichlet series.
scientific article; zbMATH DE number 5731607
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