Hessian measure estimates for Perron envelopes in Cegrell classes

Let Ω⋐Cn be a bounded m-hyperconvex domain. For f∈Em(Ω)∩C(Ω),g∈Ema(Ω), we consider the Perron envelope under the obstacle f with a lower Hessian measure constraint, Φ:=(sup{φ∈Em(Ω):φ≤f q.e., NPm⁡(Hm(φ))≥Hm(g)})∗. Since Hm(g) does not charge m-polar sets, this constraint is equivalent to Hm(φ)≥Hm(g). We prove that Φ∈Em(Ω) and Hm(g)≤Hm(Φ)≤1{Φ=f}Hm(f)+1{−∞<Φ<f}Hm(g). Consequently, Hm(Φ)=Hm(g) on the non-contact set {−∞<Φ<f}, while the mass on the contact set is controlled by Hm(f).