  Let $(M,\omega)$ be a Kahler manifold. An integrable function on M is called
$\omega^q$-plurisubharmonic if it is subharmonic on all q-dimensional complex
subvarieties. We prove that a smooth $\omega^q$-plurisubharmonic function is
q-convex. A continuous $\omega^q$-plurisubharmonic function admits a local
approximation by smooth, $\omega^q$-plurisubharmonic functions. For any closed
subvariety $Z\subset M$, $\dim Z < q$, there exists a strictly
$\omega^q$-plurisubharmonic function in a neighbourhood of $Z$ (this result is
known for q-convex functions). This theorem is used to give a new proof of
Sibony's lemma on integrability of positive closed (p,p)-forms which are
integrable outside of a complex subvariety of codimension >p.
