A description of weights satisfying the $A_{\infty }$ condition of Muckenhoupt

Abstract:

A nonnegative weight $w$ on ${R^n}$ satisfies the ${A_\infty }$ condition iff \[ \sup \limits _{Q \in \mathcal {A}} \left ( {{{\left | Q \right |}^{ - 1}} \cdot \int _Q {wdx} } \right ) \cdot \exp \left \{ {\frac {1}{{\left | Q \right |}}\int _Q {\log \frac {1}{w}dx} } \right \} < \infty .\] Here $\mathcal {A}$ stands for a family of all cubes in ${R^n}$. Applications to BMO are considered.