  In this paper we establish the persistence property for solutions of the
quartic generalized Korteweg-de Vries equation with initial data in weighted
Sobolev spaces $H^{s}(\mathbb{R})\cap L^2(|x|^{2r}dx)$ for $s =1/12 +
\varepsilon$ and any $r\in (0,R)$, for some $0 < \varepsilon < 1/4$ and
$0<R<s/2$.
