We characterize the pairs of weights (u, v) such that the one-sided geometric maximal operator G+, defined for
functions f of one real variable by
G+ f(x) = sup
h>0
exp
1
h
x+h
x
log |f|
,
verifies the weak-type inequality
{x∈R:G+ f (x)>λ}
u ≤ C
λp
∞
0
|f|p v
or the strong type inequality
R
(G+ f)p u ≤ C
R
|f|p v
for 0 < p < ∞.
We also find two new conditions which are equivalent to A+
∞.
