Semivariation in $L^p$-spaces

Jefferies, Brian, and Okada, Susumu. "Semivariation in $L^p$-spaces." Commentationes Mathematicae Universitatis Carolinae 46.3 (2005): 425-436. <http://eudml.org/doc/249537>.

@article{Jefferies2005,
abstract = {Suppose that $X$ and $Y$ are Banach spaces and that the Banach space $X\hat\{\otimes \}_\tau Y$ is their complete tensor product with respect to some tensor product topology $\tau $. A uniformly bounded $X$-valued function need not be integrable in $X\hat\{\otimes \}_\tau Y$ with respect to a $Y$-valued measure, unless, say, $X$ and $Y$ are Hilbert spaces and $\tau $ is the Hilbert space tensor product topology, in which case Grothendieck’s theorem may be applied. In this paper, we take an index $1 \le p < \infty $ and suppose that $X$ and $Y$ are $L^p$-spaces with $\tau _p$ the associated $L^p$-tensor product topology. An application of Orlicz’s lemma shows that not all uniformly bounded $X$-valued functions are integrable in $X\hat\{\otimes \}_\{\tau _p\} Y$ with respect to a $Y$-valued measure in the case $1\le p < 2$. For $2 < p <\infty $, the negative result is equivalent to the fact that not all continuous linear maps from $\ell ^1$ to $\ell ^p$ are $p$-summing, which follows from a result of S. Kwapien.},
author = {Jefferies, Brian, Okada, Susumu},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {absolutely $p$-summing; bilinear integration; semivariation; tensor product; absolutely -summing; bilinear integration; semivariation; tensor product},
language = {eng},
number = {3},
pages = {425-436},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Semivariation in $L^p$-spaces},
url = {http://eudml.org/doc/249537},
volume = {46},
year = {2005},
}

TY - JOUR
AU - Jefferies, Brian
AU - Okada, Susumu
TI - Semivariation in $L^p$-spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2005
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 46
IS - 3
SP - 425
EP - 436
AB - Suppose that $X$ and $Y$ are Banach spaces and that the Banach space $X\hat{\otimes }_\tau Y$ is their complete tensor product with respect to some tensor product topology $\tau $. A uniformly bounded $X$-valued function need not be integrable in $X\hat{\otimes }_\tau Y$ with respect to a $Y$-valued measure, unless, say, $X$ and $Y$ are Hilbert spaces and $\tau $ is the Hilbert space tensor product topology, in which case Grothendieck’s theorem may be applied. In this paper, we take an index $1 \le p < \infty $ and suppose that $X$ and $Y$ are $L^p$-spaces with $\tau _p$ the associated $L^p$-tensor product topology. An application of Orlicz’s lemma shows that not all uniformly bounded $X$-valued functions are integrable in $X\hat{\otimes }_{\tau _p} Y$ with respect to a $Y$-valued measure in the case $1\le p < 2$. For $2 < p <\infty $, the negative result is equivalent to the fact that not all continuous linear maps from $\ell ^1$ to $\ell ^p$ are $p$-summing, which follows from a result of S. Kwapien.
LA - eng
KW - absolutely $p$-summing; bilinear integration; semivariation; tensor product; absolutely -summing; bilinear integration; semivariation; tensor product
UR - http://eudml.org/doc/249537
ER -