  Let $\Gamma$ be a closed Jordan curve, and $f$ the conformal mapping that
sends the unit disc $\mathbb{D}$ onto the interior domain of $\Gamma$. If $\log
f'$ belongs to the Dirichlet space $\mathcal{D}$, we call $\Gamma$ a
Weil-Petersson curve. The purpose of this note is to extend recent results,
obtained by G. Cui and Ch. Bishop in the case of Weil-Petersson curves, to the
case when $\log f'$ belongs to either some $Q_{p,0},$ space, for $0<p\leq 1$,
or to some weighted-Dirichlet space contained in $\mathcal{D}$. More precisely,
we will characterize the quasiconformal extensions of $f$, and describe some of
the geometric properties of $\Gamma$, that arise in this context.
