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Critical Hardy--Littlewood inequality for multilinear forms | | Djair Paulino
; | | Date: |
21 Oct 2017 | | Abstract: | The Hardy--Littlewood inequalities for $m$-linear forms on $ell_{p}$ 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 $mgeq2$ we have [
sup_{j_{1}}left( sum_{j_{2}=1}^{n}left( .....left( sum_{j_{m}=1}
^{n}leftvert Tleft( e_{j_{1}},dots,e_{j_{m}}
ight)
ightvert ^{s_{m}
}
ight) ^{frac{1}{s_{m}}cdot s_{m-1}}.....
ight) ^{frac{1}{s_{3}}s_{2}
}
ight) ^{frac{1}{s_{2}}}leq2^{frac{m-2}{2}}leftVert T
ightVert ] for
all $m$--linear forms $T:ell_{m}^{n} imescdots imesell_{m}
^{n}
ightarrowmathbb{K}=mathbb{R}$ or $mathbb{C}$ with $s_{k}
=frac{2m(m-1)}{m+mk-2k}$ 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). | | Source: | arXiv, 1710.7835 | | Services: | Forum | Review | PDF | Favorites | |
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