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Multiple Rademacher means and their applications. (English) Zbl 1233.47016
J. Math. Anal. Appl. 386, No. 2, 699-708 (2012); corrigendum ibid. 403, No. 1, 331-332 (2013).
In the theory of bounded linear operators, the role played by Rademacher means is evinced in concepts as important as cotype. The multilinear version of Rademacher means leads to a multiple cotype inequality, which is the starting point of the paper. The paper establishes several inclusion theorems among different classes of multilinear operators that extend absolutely summing linear operators. Among other results, it is proved that for any Banach spaces \(X_1,\dots, X_n\) and any Banach space \(Y\) that has cotype \(q\), the class of \((t;p_1,\dots,p_n)\)-dominated (\(1\leq p_1,\dots, p_n<\infty\), \(\frac1t=\frac 1{p_1}+\cdots +\frac1{p_n}\)) \(n\)-linear maps from \(X_1\times \cdots \times X_n\) into \(Y\) and the class of \(p\)-semi-integral (\(1\leq p<\infty\)) \(n\)-linear maps from \(X_1\times \cdots \times X_n\) into \(Y\) are contained in the class of all multiple \((q;2,\dots,2)\)-summing \(n\)-linear maps.
The dominated case extends a result in [D. Pérez-García and I. Villanueva, J. Math. Anal. Appl. 285, No. 1, 86–96 (2003; Zbl 1044.46037)]. It is also proved a multilinear extension of Q.-Y. Bu’s theorem [Contemp. Math. 328, 145–149 (2003; Zbl 1066.47019)]. A former multilinear extension of Bu’s theorem can be found in [D. Achour and L. Mezrag, J. Math. Anal. Appl. 327, No. 1, 550–563 (2007; Zbl 1121.47013)] and a polynomial version in [D. Achour and K. Saadi, Collect. Math. 61, No. 3, 291–301 (2010; Zbl 1226.46042)].

MSC:
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
47H60 Multilinear and polynomial operators
46G25 (Spaces of) multilinear mappings, polynomials
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