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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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