  Let $(M,J,g,\omega)$ be a complete Hermitian manifold of complex dimension
$n\ge2$. Let $1\le p\le n-1$ and assume that $\omega^{n-p}$ is
$(\partial+\overline{\partial})$-bounded. We prove that, if $\psi$ is an $L^2$
and $d$-closed $(p,0)$-form on $M$, then $\psi=0$. In particular, if $M$ is
compact, we derive that if the Aeppli class of $\omega^{n-p}$ vanishes, then
$H^{p,0}_{BC}(M)=0$. As a special case, if $M$ admits a Gauduchon metric
$\omega$ such that the Aeppli class of $\omega^{n-1}$ vanishes, then
$H^{1,0}_{BC}(M)=0$.
