Weighted inequalities for the one-sided geometric maximal operators.

Abstract

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+ ∞.