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A regularity principle in sequence spaces and applications. (English) Zbl 1404.46041
The authors prove a rather general regularity principle, from which they deduce several results in different directions: inclusion theorems for multiple summing operators, Grothendieck-type theorems and Hardy-Littlewood inequalities. We emphasise two of them.
On the one hand it is shown that every \(m\)-linear mapping \(T: \ell_{1} \times \cdots \times \ell_{1} \to \ell_{2}\) is multiple \((q;p)\)-summing if and only if \(p \leq 2\) or \(q>p>2\).
The second result that we emphasise is a version of the Hardy-Littlewood inequality for bilinear forms on \(\ell_{p} \times \ell_{q}\). Consider \(2 \leq p,q \leq \infty\) with \(\frac{1}{p} + \frac{1}{q} <1\) and \(s,t>0\). There exists a constant \(C>0\) such that \[ \bigg( \sum_{i=1}^{n} \bigg( \sum_{j=1}^{n} | T(e_{i}, e_{j}) |^{s} \bigg)^{\frac{t}{s}} \bigg)^{\frac{1}{t}} \leq C \| T \| \] for every \(T : \ell_{p}^{n} \times \ell_{q}^{n} \to \mathbb{K}\) (note that \(C\) depends on \(p,q,s,t\) but not on \(n\)) if and only if the following three conditions are satisfied: \(\frac{q}{q-1} \leq s < \infty\), \(\frac{pq}{pq-p-q} \leq t < \infty\) and \(\frac{1}{s} + \frac{1}{t} \leq \frac{3}{2} - \big(\frac{1}{p} + \frac{1}{q} \big)\).
If these conditions are not satisfied, then the constant \(C\) in the inequality depends on \(n\). A careful study is performed, giving the precise asymptotic growth on \(n\) of the constant for fixed \(p,q,s,t\).

MSC:
46G25 (Spaces of) multilinear mappings, polynomials
46B45 Banach sequence spaces
47L22 Ideals of polynomials and of multilinear mappings in operator theory
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