Critical Hardy–Littlewood inequality for multilinear forms

  • Djair PaulinoEmail author


The Hardy–Littlewood inequalities for m-linear forms on \(\ell _{p}\) spaces are known just for \(p>m\). The critical case \(p=m\) was overlooked for obvious technical reasons and, up to now, the only known estimate is the trivial one. In this paper we deal with this critical case of the Hardy–Littlewood inequality. More precisely, for all positive integers \(m\ge 2\) we have
$$\begin{aligned} \sup _{j_{1}}\left( \sum _{j_{2}=1}^{n}\left( \ldots \left( \sum _{j_{m}=1} ^{n}\left| T\left( e_{j_{1}},\ldots ,e_{j_{m}}\right) \right| ^{s_{m} }\right) ^{\frac{1}{s_{m}}\cdot s_{m-1}}\ldots \right) ^{\frac{1}{s_{3}}s_{2} }\right) ^{\frac{1}{s_{2}}}\le 2^{\frac{m-2}{2}}\left\| T\right\| \end{aligned}$$
for all m-linear forms \(T{:}\,\ell _{m}^{n}\times \cdots \times \ell _{m} ^{n}\rightarrow \mathbb {K}=\mathbb {R}\) or \(\mathbb {C}\) with \(s_{k} =\frac{2m(m-1)}{mk-2k+2}\) for all \(k=2,\ldots ,m\) and for all positive integers n. As a corollary, for the classical case of bilinear forms investigated by Hardy and Littlewood in 1934 our result is sharp in a strong sense (both exponents and constants are optimal for real and complex scalars).


Hardy–Littlewood inequality Multilinear forms Operator multiple summing 

Mathematics Subject Classification

47A63 47H60 



The author is very grateful to the referee(s) for his/her very important suggestions and remarks that helped to improve and clarify the final version of this paper.


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© Springer-Verlag Italia S.r.l., part of Springer Nature 2019

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade Federal da ParaíbaJoão PessoaBrazil

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