Banach lattice-valued $q$-variation and convexity
Abstract
In this paper, we show that the $q$-variation for differential operator is not bounded in $L^p(\mathbb{R};L^{\infty}(\mathbb{R}))$ for any $1<p<\infty$. As a consequence, the $q$-variation operator can not be used to characterize the Hardy-Littlewood property of the underlying Banach lattice. Moreover, for Köthe function spaces $X$ with $X^*$ norming such that $X$ is $r$-convex for some large $r$, and $X$ is not $s$-convex for any $s$, $r<s<\infty$, we obtain lower bounds of the $(L^p(\mathbb{R};X),L^p(\mathbb{R};X)$-bounds of the $q$-variation operator, which tends to $\infty$, as $r$ tends to $\infty$.
- Publication:
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arXiv e-prints
- Pub Date:
- October 2014
- DOI:
- 10.48550/arXiv.1410.1575
- arXiv:
- arXiv:1410.1575
- Bibcode:
- 2014arXiv1410.1575H
- Keywords:
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- Mathematics - Functional Analysis;
- Mathematics - Classical Analysis and ODEs;
- Variational inequalities;
- Hardy-Littlewood property;
- Behaviour in $L^\infty(\R)$