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Some applications of the regularity principle in sequence spaces. (English) Zbl 06861658
Summary: The Hardy-Littlewood inequalities for $$m$$-linear forms have their origin with the seminal paper of G. H. Hardy and J. E. Littlewood [Q. J. Math., Oxf. Ser. 5, 241–254 (1934; Zbl 0010.36101)]. Nowadays it has been extensively investigated and many authors are looking for the optimal estimates of the constants involved. For $$m<p\leq 2m$$ it asserts that there is a constant $$D_{m,p}^{\mathbb K}\geq 1$$ such that $\left(\sum_{j_1,\dots,j_m=1}^{n}| T(e_{j_1},\dots,e_{j_m})| ^{\frac{p}{p-m}}\right) ^{\frac{p-m}{p}}\leq D_{m,p} ^{\mathbb K}\| T\| ,$ for all $$m$$-linear forms $$T:\ell _p^n\times \cdots \times \ell_p ^n\rightarrow \mathbb K=\mathbb R$$ or $$\mathbb C$$ and all positive integers $$n$$. Using a regularity principle recently proved by Pellegrino, Santos, Serrano and Teixeira, we present a straightforward proof of the Hardy-Littlewood inequality and show that: cm
(1)
If $$m<p_1\leq p_2\leq 2m$$ then $$D_{m,p_1}^{\mathbb K}\leq D_{m,p_2}^{\mathbb K}$$;
(2)
$$D_{m,p}^{\mathbb K}\leq D_{m-1,p}^{\mathbb K}$$ whenever $$m<p\leq 2(m-1)$$ for all $$m\geq 3$$.

MSC:
 47H60 Multilinear and polynomial operators 11Y60 Evaluation of number-theoretic constants 47A63 Linear operator inequalities 46G25 (Spaces of) multilinear mappings, polynomials
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References:
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