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A new multilinear insight on Littlewood’s 4/3-inequality. (English) Zbl 1171.46034
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 [J. E. Littlewood, Q. J. Math., Oxf. Ser. 1, 164–174 (1930; JFM 56.0335.01)], its multilinear extension due to H. F. Bohnenblust and 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 G. Bennett [J. Funct. Anal. 13, 20–27 (1973; Zbl 0255.47033)] and B. Carl [Math. Nachr. 63, 353–360 (1974; Zbl 0292.47019)], and a recent result of F. Bombal, D. Pérez–García and 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.

##### MSC:
 46G25 (Spaces of) multilinear mappings, polynomials 46B07 Local theory of Banach spaces 47B10 Linear operators belonging to operator ideals (nuclear, $$p$$-summing, in the Schatten-von Neumann classes, etc.)
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