Semivariation in $L^p$-spaces

Abstract

summary: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