The paper gives some properties of the space \(\text{BMO}_ *=\{w\in\text{BMO}: 1/\omega\in\text{BMO}\}\) such as Theorem 1. A real- valued function \(u\) is in \(\text{BMO }\Leftrightarrow\) there exists \(w\in \text{BMO}_ *\) such that \(u=w-1/w\) (a consequence of \(\text{Lip }\alpha\)-operatability on BMO); Theorem 3. \(\text{BMO}_ *\subset\bigcap_{p>1}A_ p\), a consequence of following Theorem 6. For \(1< p_ 0\leq 2\) we have \(w\), \(1/w\in A_{p_ 0}\Leftrightarrow L_ Q={1\over| Q|}\int_ Q | w-w_ Q|^{p_ 0'-1}\leq c<\infty\), for all cubes.