Archiv der Mathematik

, Volume 105, Issue 3, pp 285–295 | Cite as

Optimal Hardy–Littlewood type inequalities for m-linear forms on \({\ell_{p}}\) spaces with \({1\leq p\leq m}\)



The Hardy–Littlewood inequalities for m-linear forms on \({\ell _{p}}\) spaces are stated for \({p > m}\). In this paper, among other results, we investigate similar results for \({1\leq p\leq m.}\) Let \({\mathbb{K}}\) be \({ \mathbb{R}}\) or \({\mathbb{C}}\) and \({m\geq 2}\) be a positive integer. Our main results are the following sharp inequalities:
  1. (i)
    If \({\left( r,p\right) \in \left( \lbrack 1,2]\times \lbrack 2,2m)\right) \cup \left( \lbrack 1,\infty )\times \lbrack 2m,\infty \right] ) }\), then there is a positive constant \({D_{m,r,p}^{\mathbb{K}}}\) (not depending on n) such that
    $$\left(\sum\limits_{j_{1}, \ldots, j_{m} = 1}^{n}\left\vert T(e_{j_{1}}, \ldots, e_{j_{m}})\right\vert^{r} \right)^{\frac{1}{r}} \leq D_{m,r,p}^{\mathbb{K}}n^{\max\left\{\frac{2mr + 2mp - mpr - pr}{2pr}, 0\right\}} \left\Vert T\right\Vert$$
    for all m-linear forms \({T:\ell _{p}^{n} \times \cdots \times \ell _{p}^{n} \rightarrow \mathbb{K}}\) and all positive integers n.
  2. (ii)
    If \({\left( r,p\right) \in \lbrack 2,\infty )\times (m,2m]}\), then
    $$\left( \sum\limits_{j_{1},\ldots,j_{m}=1}^{n}\left\vert T(e_{j_{1}},\ldots,e_{j_{m}})\right\vert ^{r}\right)^{\frac{1}{r}}\leq \left( \sqrt{2}\right) ^{m-1}n^{\max \left\{\frac{p+mr-rp}{pr},0\right\}}\left\Vert T\right\Vert$$
    for all m-linear forms \({T:\ell _{p}^{n} \times \cdots \times \ell _{p}^{n} \rightarrow \mathbb{K}}\) and all positive integers n.
Moreover, the exponents \({\max \{(2mr+2mp-mpr-pr)/2pr,0\}}\) in (i) and \({\max \{(p+mr-rp)/pr,0\}}\) in (ii) are optimal. The cases \({\left( r,p\right) =\left( 2m/\left( m+1\right), \infty \right) }\) and \({\left( r,p\right) =\left( 2mp/\left( mp+p-2m\right), p\right) }\) for \({p\geq 2m}\) and \({\left( r,p\right) =\left( p/\left( p-m\right), p\right) }\) for \({m < p < 2m}\) recover the classical Bohnenblust–Hille and Hardy–Littlewood inequalities.


Bohnenblust–Hille inequality Hardy–Littlewood inequality Absolutely summing operators 

Mathematics Subject Classification

32A22 47H60 


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Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.CAPES FoundationMinistry of Education of BrazilBrasilia–DFBrazil
  2. 2.Departamento de MatemáticaUniversidade Federal da ParaíbaJoão PessoaBrazil

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