Abstract
The Hardy–Littlewood inequalities for multilinear forms on sequence spaces state that for all positive integers \(m,n\ge 2\) and all m-linear forms \(T:\ell _{p_{1}}^{n}\times \cdots \times \ell _{p_{m}}^{n}\rightarrow {\mathbb {K}}\) (\({\mathbb {K}}={\mathbb {R}}\) or \({\mathbb {C}}\)) there are constants \(C_{m}\ge 1\) (not depending on n) such that
where \(\rho =\frac{2m}{m+1-2\left( \frac{1}{p_{1}}+\cdots +\frac{1}{p_{m}}\right) }\) if \(0\le \frac{1}{p_{1}}+\cdots +\frac{1}{p_{m}}\le \frac{1}{2}\) or \(\rho =\frac{1}{1-\left( \frac{1}{p_{1}}+\cdots +\frac{1}{p_{m}}\right) }\) if \(\frac{1}{2}\le \frac{1}{p_{1}}+\cdots +\frac{1}{p_{m}}<1\). Good estimates for the Hardy–Littlewood constants are, in general, associated to applications in Mathematics and even in Physics, but the exact behavior of these constants is still unknown. In this note we give some new contributions to the behavior of the constants in the case \(\frac{1}{2}\le \frac{1}{p_{1}}+\cdots +\frac{1}{p_{m}}<1\). As a consequence of our main result, we present a generalization and a simplified proof of a result due to Aron et al. on certain Hardy–Littlewood type inequalities.
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We thank the anonymous referees for the careful reading and the valuable comments that helped to improve the presentation of the paper.
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Araújo, G.S., Câmara, K.S. Universal Bounds for the Hardy–Littlewood Inequalities on Multilinear Forms. Results Math 73, 124 (2018). https://doi.org/10.1007/s00025-018-0886-6
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DOI: https://doi.org/10.1007/s00025-018-0886-6