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

(b) The exponents q1,...,qm satisfyq1≥λm,cot⁡Yp1,...,pm,q2≥λm−1,cot⁡Yp2,...,pm,...,qm≥λ1,cot⁡Ypm, where, for k=1,...,m,λm−k+1,cot⁡Ypk,...,pm:=cot⁡Y1−(1pk+...+1pm)cot⁡Y. As an application of the above result we generalize one of the classical Hardy-Littlewood inequalities for bilinear forms to the m-linear setting. Our result is sharp in a very strong sense: the constants and exponents are optimal, even if we consider mixed sums.