Disk Envelopes of Functions, II

Given a Suslin functionφon a domainYin a Banach spaceXone can define two envelopes ofφrelated to the pluripotential theory. The first one, denoted byPφ, is the supremum of all plurisubharmonic functions less thanφ. The value of the other one, denoted byDφ, at some pointzis equal to the infimum of integrals over the boundaries of all closed analytic disks centered atz. In this paper we study the relationship between these envelopes. We show thatDφ⩽P*φfor a large class of functions on domains in Banach spaces, whereP*φis the upper regularization ofPφ. IfX=Cn, then we prove thatD*φis always equal toP*φ.