$m$-Positivity and Regularisation

Starting from the notion of $m$-plurisubharmonic function introduced recently by Dieu and studied, in particular, by Harvey and Lawson, we consider $m$-(semi-)positive $(1,\,1)$-currents and Hermitian holomorphic line bundles on complex Hermitian manifolds and prove two kinds of results: vanishing theorems and $L^2$-estimates for the $\bar\partial$-equation in the context of $C^\infty$ $m$-positive Hermitian fibre metrics; global and local regularisation theorems for $m$-semi-positive $(1,\,1)$-currents whose proofs involve the use of viscosity subsolutions for a certain Monge-Amp\`ere-type equation and the associated Dirichlet problem.