ARTICLE DOWNLOAD

Critical Hardy–Littlewood inequality for multilinear forms

10$
ARTICLE DOWNLOAD

Critical Hardy–Littlewood inequality for multilinear forms

10$

Djair Paulino 

Abstract

The Hardy–Littlewood inequalities for m-linear forms on ℓpℓp spaces are known just for p>mp>m. The critical case p=mp=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≥2m≥2 we have

supj1⎛⎝⎜⎜∑j2=1n⎛⎝⎜…(∑jm=1n∣∣T(ej1,…,ejm)∣∣sm)1sm⋅sm−1…⎞⎠⎟1s3s2⎞⎠⎟⎟1s2≤2m−22∥T∥supj1(∑j2=1n(…(∑jm=1n|T(ej1,…,ejm)|sm)1sm⋅sm−1…)1s3s2)1s2≤2m−22‖T‖

for all m-linear forms T:ℓnm×⋯×ℓnm→K=RT:ℓmn×⋯×ℓmn→K=R or CC with sk=2m(m−1)mk−2k+2sk=2m(m−1)mk−2k+2 for all k=2,…,mk=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).

Only units of this product remain
Year 2020
Language English
Format PDF
DOI 10.1007/s12215-019-00409-0