Brian Jefferies, Susumu Okada
Semivariation in $L^p$-spaces

Comment.Math.Univ.Carolinae 46,3 (2005) 425-436.

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. \par 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.

Keywords: absolutely $p$-summing, bilinear integration, semivariation, tensor product
AMS Subject Classification: Primary 28B05, 46G10; Secondary 46B42, 47B65