2017/10/21 by Paulino, Djair
#FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.1710.07835
The Hardy--Littlewood inequalities for m-linear forms on ℓp 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≥2 we have sup_j1( ∑_j2=1n( .....( ∑_jm=1 n\vert T( e_j1,…,e_jm) \vert ^sm ) ^\frac1sm⋅ sm-1.....) ^\frac1s3s2 ) ^\frac1s2≤2(m-2)/(2)\Vert T\Vert for all m--linear forms T:ℓmn×⋯×ℓm n→\mathbbK=ℝ or ℂ with sk =(2m(m-1))/(m+mk-2k) for all k=2,....,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).