Geometry of holomorphic mappings and Hölder continuity of the pluricomplex Green function

Abstract

We provide a solution to a long-standing open problem that lives in the interface of pluripotential theory and multivariate approximation theory. The problem is to characterize the holomorphic maps which preserve Hölder continuity of the pluricomplex Green function associated with a compact subset of C N . We also prove, under mild restrictions, that nondegenerate holomorphic maps preserve Markov's inequality for polynomials.Geometry of holomorphic mappings... Definition 1.2 (see [29]) We say that a compact set ∅ = K ⊂ C N has the HCP property if there exist , μ > 0 such that, for each z ∈ K (1) ,(1.4)In the above definition, and subsequently, we use the following notation: for each set ∅ = A ⊂ C N and each r > 0, we put A (r ) := {z ∈ C N : dist(z, A) ≤ r } .