arXiv subjects
Publications and source records attributed to Miroslaw Baran.
We introduce an asymptotic Markov's exponent and show that it is equal to Markov's exponent for a wide class of norms. As a consequence we obtain a lower bound for the optimal exponent in Markov's inequality considered with the norms possessing Nikolskii type property.
Let V_E be the pluricomplex Green function associated to a compact subset E of C^N. The well known Hölder 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^αP ||_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.