Relative types and extremal problems for plurisubharmonic functions

A type σ(u, ϕ) of a plurisubharmonic function u relative to a maximal plurisubharmonic weight ϕ with isolated singularity at ζ is defined as lim inf u(x)/ϕ(x) as x → ζ. We study properties of the relative types as functionals u ↦ σ(u, ϕ); it is shown that they give a general form for upper semicontinuous, positive homogeneous, and tropically additive functionals on plurisubharmonic singularities. We consider some extremal problems whose solutions are Green-like functions that give best possible bounds on u, given the values of its types relative to some of (or all) weights ϕ; in certain cases they coincide with known variants of pluricomplex Green functions. An analyticity theorem is proved for the upper level sets for the types with respect to exponentially Hölder continuous weights, which leads to a result on propagation of plurisubharmonic singularities.