H\"older continuity of pluricomplex Green function and Markov brothers' inequality
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.
Keywords
Cite
@article{arxiv.1303.1841,
title = {H\"older continuity of pluricomplex Green function and Markov brothers' inequality},
author = {Miroslaw Baran and Leokadia Bialas-Ciez},
journal= {arXiv preprint arXiv:1303.1841},
year = {2013}
}