This paper presents noncommutative weak Orlicz space norm,then we obtain relevantinequalities in noncommutative weak LP space. in the last,we present weak averageinequality of Hardy-Littlewood maximal function inequalities ofτ-measurable operatorsand inequality of noncommutative weak Orlicz space norm.The three inequalities are asbelow:(a) If 1 < p <∞,we have‖MTiα(T)‖LPω≦A0‖Tiα‖LPω,Tiα∈LPω(M),(i=0,1)(b)supt>0Φ(t)λt(MT(|T|))≤CΦsupt>0Φ(t)λt(|T|), for all T∈LΦω(M).(c)IfΦis a strictly convex moderate young function, 1 < qΦ< pΦ<∞,then there isa constant CΦ> 0 such that‖MT(|T|)‖LΦω≤CΦ‖T‖LΦω,T∈LΦω(M).
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