Disk envelopes of functions. II
Given a Suslin function \(\varphi\) on a domain \(Y\) in a Banach space \(X\) one can define two envelopes of \(\varphi\) related to the pluripotential theory. The first one, denoted by \(P\varphi\), is the supremum of all plurisubharmonic functions less than \(\varphi\). The value of the other one, denoted by \(D\varphi\), at some point is equal to the infimum of integrals over the boundaries of all closed analytic disks centered at \(z\). In this paper, we study the relationship between these envelopes. We show that \(D\varphi \leq P^*\varphi\) for a large class of functions on domains in Banach spaces, where \(P^*\varphi\) is the upper regularization of \(P\varphi\). If \(X=Image n\), then we prove that \(D^*\varphi\) is always equal to \(P^*\varphi\). [For part I, cf. Chirka, E. M. (ed.), Complex analysis in modern mathematics. On the 80th anniversary of the birth of Boris Vladimirovich Shabat. Moscow: Izdatel'stvo FAZIS, 47--55 (2001; Zbl 1056.30001).]
scientific article; zbMATH DE number 1278795
scientific article