Banach lattice-valued $q$-variation and convexity

TL;DR

This paper investigates the boundedness of the $q$-variation operator in Banach lattice-valued $L^p$ spaces, revealing limitations in its use for characterizing certain properties and establishing bounds that grow unboundedly under specific conditions.

Contribution

It demonstrates the unboundedness of the $q$-variation operator in certain Banach lattice-valued $L^p$ spaces and provides lower bounds that tend to infinity under specific convexity conditions.

Findings

The $q$-variation operator is not bounded in $L^p( eal;L^{inite}( eal))$ for any $1<p<inite$.

The $q$-variation operator cannot characterize the Hardy-Littlewood property of Banach lattices.

Lower bounds of the operator's bounds tend to infinity as the convexity parameter $r$ increases.

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 K\"othe 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 $\infty$.