For each positive $C$, define a set $$A_C=\left\{g\ge0: \frac{1}{|I|}\int_Ig\le C\inf_{x\in I}g(x) \text{ for any interval } I\right\}$$ In other words, elements in $A_C$ are non-negative functions whose average over any interval is comparable to its minimum on the same interval. A non-negative constant function is an example of such a function. For any function $f$, its Hardy-Littlewood maximal function $Mf$ does not belong to any $A_C$ as the counter example in an answer to this question suggests: average of maximal function is less than its infimum? However, that example does not work for small powers of $Mf$. So I have the following conjecture:
For any $\delta\in(0,1)$, there exists a C depending only on $\delta$ such that $(Mf)^{\delta}\in A_C$ for any $f$.
Any idea on proving this?
The condition on $g$ that you stated is usually expressed by saying that $g$ is in the Muckenhoupt class $A_1$.
The result you conjectured is true: it is Proposition 2 in the paper Another characterization of BMO by Coifman and Rochberg, Proc. Amer. Math. Soc. 79 (1980), 249-254. The paper is in free access, so I don't reproduce the proof here.