  Due to its nonlocal nature, the $r$-variation norm Carleson operator $C_r$
does not yield to the sparse domination techniques of Lerner, Di Plinio and
Lerner, Lacey. We overcome this difficulty and prove that the dual form to
$C_r$ can be dominated by a positive sparse form involving $L^p$ averages. Our
result strengthens the $L^p$ estimates by Oberlin et. al. As a corollary, we
obtain quantitative weighted norm inequalities improving on previous results by
Do and Lacey. Our proof relies on the localized outer $L^p$-embeddings of Di
Plinio-Ou and Uraltsev.
