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 $C_{p_{1},...,p_{m}}^{Y}geq 1$ such that begin{equation*} left( sum_{j_{1}=1}^{infty }left( sum_{j_{2}=1}^{infty }cdots left( sum_{j_{m}=1}^{infty }leftVert A(e_{j_{1}},...,e_{j_{m}})rightVert ^{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}leftVert ArightVert end{equation*} for all continuous $m-$linear operators $A:ell _{p_{1}}times cdots times ell _{p_{m}}rightarrow Y.$ (b) The exponents $q_{1},...,q_{m}$ satisfy begin{equation*} q_{1}geq lambda _{m,cot Y}^{p_{1},...,p_{m}},q_{2}geq lambda _{m-1,cot Y}^{p_{2},...,p_{m}},...,q_{m}geq lambda _{1,cot Y}^{p_{m}}, end{equation*} where, for $k=1,...,m,$ begin{equation*} lambda _{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.
تحميل البحث