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Multiple summing maps: coordinatewise summability, inclusion theorems and \(p\)-Sidon sets. (English) Zbl 1391.46057
Let \(T: X_1 \times \cdots \times X_n \rightarrow Y\) be an \(n\)-linear operator and consider the following question: if the restriction of \(T\) to each \(X_j\), \(j=1,\dots, n\), is \((r,p)\)-summing (that is, \(T\) is separately \((r,p)\)-summing), what can we say about the multiple \((s,t)\)-summability of the mapping \(T\)?
The case \(p=t=1\) was settled by A. Defant et al. [J. Funct. Anal. 259, No. 1, 220–242 (2010; Zbl 1205.46026)]: if \(Y\) has cotype \(q\), \(r \in [1,q]\) and \(T\) is separately \((r,1)\)-summing, then \(T\) is multiple \((s,1)\)-summing, with \[ \frac{1}{s} = \frac{n-1}{nq} + \frac{1}{nr}. \] Note the constraint imposed in this result on the parameter \(r\) by the cotype of the target space. This kind of result has been improved, using an interpolative technique, in some recent papers as in the work of N. Albuquerque et al. [J. Funct. Anal. 269, No. 6, 1636–1651 (2015; Zbl 1331.46036)].
In this well written paper, the author settles the above question for a large range of possible values of \(t \geq p \geq 1\), including the case when \(r\) exceeds the cotype of the target space, recovering the known cases as particular instances. Interesting consequences of the developed apparatus are presented as best inclusion results for multiple summing operators and applications to harmonic analysis, for the product of \(p\)-Sidon sets. The last part of the paper is devoted to the discussion of the optimality (sharpness) of the obtained parameters of all results.

MSC:
46G25 (Spaces of) multilinear mappings, polynomials
47H60 Multilinear and polynomial operators
47L22 Ideals of polynomials and of multilinear mappings in operator theory
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