H\"older continuity of pluricomplex Green function and Markov brothers' inequality

TL;DR

This paper establishes an equivalence between the H"older continuity of pluricomplex Green functions and a Vladimir Markov type inequality, providing new insights and criteria for regularity in several complex variables.

Contribution

It proves that H"older continuity of pluricomplex Green functions is equivalent to a specific polynomial inequality, extending the understanding of pluripotential regularity.

Findings

Equivalence between H"older continuity and Markov type inequality.

Application of the equivalence to regularity criteria in complex variables.

Generalization involving a notion of a fit majorant.

Abstract

Let V_E be the pluricomplex Green function associated to a compact subset E of C^N. The well known H\"older Continuity Property of E means that there exist constants B > 0, 0< c =< 1 such that V_E(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. || D^\alpha P ||_E =< M^{|\alpha |} (deg P)^{m|\alpha|} (|\alpha |!)^{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 H\"older Continuity Property in several complex variables of the type of Siciak's L-regularity criterion.