Lagrangian traces for the Johnson filtration of the handlebody group

We define trace-like operators on a subspace of the space of derivations of the free Lie algebra generated by the first homology group $H$ of a surface $\mathrm{\Sigma }$. This definition depends on the choice of a Lagrangian of $H$, and we call these operators the \emph{Lagrangian traces}. We suppose that $\mathrm{\Sigma }$ is the boundary of a handlebody with first homology group $H^{\mathrm{\prime }}$, and we show that the Lagrangian traces corresponding to the Lagrangian $\mathrm{Ker}(H\to H^{\mathrm{\prime }})$ vanish on the image by the Johnson homomorphisms of the elements of the Johnson filtration that extend to the handlebody.