  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 $\infty$.

Functional Analysis;Classical Analysis and ODEs