Capacities and Hessians in the class of \(m\)-subharmonic functions
An upper semicontionuous function \(u\) on a domain \(D\subset{\mathbb C}^n\) is called \(m\)-subharmonic if the current \(dd^cu\wedge dd^cv_1\ldots\wedge dd^c v_{m-1}\wedge\beta^{n-m}\) is positive for any smooth functions \(v_j\) satisfying \((dd^c v_j)^k\wedge\beta^{n-k}\) for all \(k\leq m\); here \(\beta=dd^c|z|^2\). The authors develop the basics of an \(m\)-pluripotential theory similar to the classical one (where \(m=n\)), including: a definition of the Hessian \((dd^c u)^m\wedge\beta^{n-m}\) for continuous \(m\)-subharmonic \(u\) as the limit of the Hessians for smooth \(u_j\) decreasing to \(u\); equivalence of local and global \(m\)-polarity; relative extremal \(m\)-subharmonic functions (called here \({\mathcal P}_m\)-measures); the notion of \(m\)-capacity; quasicontinuity of \(m\)-subharmonic functions; continuity of the Hessians for decreasing, locally bounded sequences; \(m\)-polarity of the sets \(\{\limsup u_j<(\limsup u_j)^*\}\). Most of the proofs are skipped.
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