2013/03/07 by Mirosław Baran, Miroslaw Baran, Baran, Miroslaw +3
Mathematics · #32U35 #41A17 #Analytic Number Theory Research #Complex Variables (math.CV) #FOS: Mathematics #Geometry and complex manifolds #Mathematical functions and polynomials #math.CV #msc:32U35 #msc:41A17
paper · pdf · doi:10.48550/arxiv.1303.1841
arxiv created 2013/03/07 · openalex publication_date 2013/03/07 · arxiv updated 2013/03/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let VE be the pluricomplex Green function associated to a compact subset E of CN. The well known Hölder Continuity Property of E means that there exist constants B > 0, 0< c =< 1 such that VE(z) =< B dist(z,E)c. The main result of this paper says that this condition is equivalent to a Vladimir Markov type inequality, i.e. || DαP ||E =< M|α| (deg P)m|α| (|α|!)1-m ||P||E, where m,M>0 are independent of the polynomial P of N variables. We give some applications of this equivalence and we present its generalization related to a notion of a fit majorant. Moreover, as a consequence of the main result we obtain a criterion for the Hölder Continuity Property in several complex variables of the type of Siciak's L-regularity criterion.