Abstract [11] In the present article it is presented a characterization of all those func- tions in the space of bounded mean oscillation functions, BM O, in terms of an appropriate wavelet, using an isomorphism between the aforemen- tioned space and the homogeneous space of Triebel-Lizorkin F˙ 0,2. In ad- dition, a new inequality that involves the vector inequality of the maximal function of Hardy-Littlewood is proved.