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