2010•Shuxue de shijian yu renshiRequires access

On Reverse Maximal Inequalities of(Ф_1,Ф_2)-Type for Martingales

Jin Yan-ming

Abstract

LetΦ_1,Φ_2 be nonnegative convex functions.In this paper,it is proved that thereverse maximal inequalities ||f||_(Φ_2)≤C||f~*||_(Φ_1) of the strong(Φ_1,Φ_2)-types for martingalesholds if and only ifΦ_1(?)Φ_2.The maximal and minimal inequalities ||M(f)||_(Φ_2)≤||m(f)||_(Φ_1)and ||m(f)||_(Φ_2)≤||M(f)||_(Φ_1) equateΦ_1(?)Φ_2,where M(f) = max{f~*,S(f)} and m(f) =min{f~*,S(f)} are the maximal and minimal functions of the maximal operator and squarefunction for martingale.

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Key concepts: Mathematics, Maximal operator, Maximal function, Combinatorics, Martingale (probability theory), Type (biology), Regular polygon, Convex function