A new multilinear insight on Littlewood's 4/3-inequality
For a positive integer \(m\) and \(1\leq p\leq q\), the authors let \(\rho={2m\over m+2({1\over p}-\max\{{{1\over q},{1\over 2}\}})}\) if \(p\leq 2\) and \(\rho = p\) if \(p\geq 2\). They show that there is \(C>0\) such that for every \(m\)-linear mapping \(A: c_0\times\dots \times c_0\to\ell_p\), \[ \left(\sum_{i_1,\dots, i_m}^\infty\|A(e_{i_1},\dots, e_{i_m})\|_q^\rho\right)^{1/ \rho}\leq C\|A\|, \] where \(\|A\|=\sup_{\|x_j\|\leq 1}\|A(x_1,\dots, x_m)\|\). Moreover, this value of \(\rho\) is the best possible. This result unifies Littlewood's 4/3-inequality [\textit{J.\,E.\thinspace Littlewood}, Q.\ J.\ Math., Oxf.\ Ser.\ 1, 164--174 (1930; JFM 56.0335.01)], its multilinear extension due to \textit{H.\,F.\thinspace Bohnenblust} and \textit{E.\,Hille} [Ann.\ Math.\ (2) 32, 600--622 (1931; Zbl 0001.26901 and JFM 57.0266.05)], a vector-valued inequality for operators from \(c_0\) to \(\ell_p\) discovered independently by \textit{G.\,Bennett} [J.~Funct.\ Anal.\ 13, 20--27 (1973; Zbl 0255.47033)] and \textit{B.\,Carl} [Math.\ Nachr.\ 63, 353--360 (1974; Zbl 0292.47019)], and a recent result of \textit{F.\,Bombal, D.\,Pérez--García} and \textit{I.\,Villanueva} [Q.\ J.\ Math.\ 55, No.\,4, 441--450 (2004; Zbl 1078.46030)]. As an application, it is shown that, if \(\sum_{\alpha\in {\mathbb N}^{(\mathbb N)}}c_\alpha(P) z^\alpha\) is the monomial expansion of an \(m\) homogeneous polynomial \(P: c_0\to\ell_p\), then \[ \left(\sum_{\alpha\in {\mathbb N}^{({\mathbb N})}}\|c_\alpha (P)\|_q^\rho\right)^{1/\rho}\leq C\|P\| \] with \(\rho\) the best possible constant. The authors say that a linear operator between Banach spaces, \(v: X\to Y\), is \((r,1)\)-summing of order \(m\) is there is \(C>0\) such that for every \(m\)-linear mapping \(A: c_0\times\dots \times c_0\to X\) \[ \left(\sum_{i_1,\dots, i_m}^\infty\|v\circ A(e_{i_1},\dots, e_{i_m})\|^r\right) ^{1/r}\leq C\|A\|. \] They examine the indices \(\inf\{r:v\text{ is }(r,a)\text{-summing of order m}\}\) for certain identity and inclusion mappings and relate \((r,1)\)-summing operators of a given order to summing operators.
scientific article; zbMATH DE number 5541681
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