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In this study, we first define the local potential associated to a weaklypositive closed supercurrent in analogy to the one investigated by Ben Messaoudand El Mir in the complex setting. Next, we study the definition and thecontinuity of the $m$-superHessian operator for unbounded $m$-convex functions.As an application, we generalize our previous work on Demailly-Lelong numbersand several related results in the superformalism setting. Furthermore,strongly inspired by the complex Hessian theory, we introduce the Cegrell-typeclasses as well as a generalization of some $m$-potential results in the classof $m$-convex functions.

$m$-potential theory and $m$-generalized Lelong numbers associated with $m$-positive supercurrents