Optimal exponents for Hardy--Littlewood inequalities for $m$-linear operators

Abstract

The Hardy--Littlewood inequalities on ${\ell }_p$ spaces provide optimal exponents for some classes of inequalities for bilinear forms on ${\ell }_p$ spaces. In this paper we investigate in detail the exponents involved in Hardy--Littlewood type inequalities and provide several optimal results that were not achieved by the previous approaches. Our first main result asserts that for $q_1,...,q_m>0$ and an infinite-dimensional Banach space $Y$ attaining its cotype $\cot Y$, if \begin{equation*} \frac{1}{p_{1}}+...+\frac{1}{p_{m}}<\frac{1}{\cot Y}, \end{equation*} then the following assertions are equivalent: (a) There is a constant $C_{p_1,...,p_m}^Y\ge 1$ such that \begin{equation*} \left( \sum_{j_{1}=1}^{\infty }\left( \sum_{j_{2}=1}^{\infty }\cdots \left( \sum_{j_{m}=1}^{\infty }\left\Vert A(e_{j_{1}},...,e_{j_{m}})\right\Vert ^{q_{m}}\right) ^{\frac{q_{m-1}}{q_{m}}}\cdots \right) ^{\frac{q_{1}}{q_{2}} }\right) ^{\frac{1}{q_{1}}}\leq C_{p_{1},...,p_{m}}^{Y}\left\Vert A\right\Vert \end{equation*} for all continuous $m-$linear operators $A:{\ell }_{p_1}\times \cdots \times {\ell }_{p_m}\to Y.$ (b) The exponents $q_1,...,q_m$ satisfy \begin{equation*} q_{1}\geq λ_{m,\cot Y}^{p_{1},...,p_{m}},q_{2}\geq λ_{m-1,\cot Y}^{p_{2},...,p_{m}},...,q_{m}\geq λ_{1,\cot Y}^{p_{m}}, \end{equation*} where, for $k=1,...,m,$ \begin{equation*} λ_{m-k+1,\cot Y}^{p_{k},...,p_{m}}:=\frac{\cot Y}{1-\left( \frac{1}{ p_{k}}+...+\frac{1}{p_{m}}\right) \cot Y}. \end{equation*} As an application of the above result we generalize to the $m$-linear setting one of the classical Hardy--Littlewood inequalities for bilinear forms. Our result is sharp in a very strong sense: the constants and exponents are optimal, even if we consider mixed sums.

20 pages

References in corpus (2)

• Sharp generalizations of the multilinear Bohnenblust--Hille inequality
• Remarks on the Hardy--Littlewood inequality for $m$-homogeneous polynomials and $m$-linear forms

Cited by in corpus (6)

• Anisotropic Regularity Principle in sequence spaces and applications
• Polynomial and multilinear Hardy--Littlewood inequalities: analytical and numerical approaches
• Optimal Hardy--Littlewood inequalities uniformly bounded by a universal constant
• Sharp anisotropic Hardy--Littlewood inequality for positive multilinear forms
• Optimal constants of classical inequalities in complex sequence spaces
• The Orlicz inequality for multilinear forms