  Let $\mathcal{F}$ be a holomorphic foliation by curves defined in a
neighborhood of $0$ in $\mathbb{C}^n$ ($n\geq 2$) having $0$ as a weakly
hyperbolic singularity. Let $T$ be a positive harmonic current directed by
$\mathcal{F}$ which does not give mass to any of the $n$ coordinate invariant
hyperplanes $\{z_j=0\}$ for $1\leq j\leq n.$ Then we show that the Lelong
number of $T$ at $0$ vanishes. Moreover, an application of this local result in
the global context is given. We discuss also the relation between several basic
notions such as directed positive harmonic currents, directed positive
ddc-closed currents, Lelong numbers etc. in the framework of singular
holomorphic foliations.
