Weighted inequalities for one-sided maximal functions

Abstract:

Let $M_g^ +$ be the maximal operator defined by \[ M_g^ + f(x) = \sup \limits _{h > 0} \left ( {\int _x^{x + h} {|f(t)|g(t)dt} } \right ){\left ( {\int _x^{x + h} {g(t)dt} } \right )^{ - 1}},\] where $g$ is a positive locally integrable function on ${\mathbf {R}}$. We characterize the pairs of nonnegative functions $(u,v)$ for which $M_g^ +$ applies ${L^p}(v)$ in ${L^p}(u)$ or in weak- ${L^p}(u)$. Our results generalize Sawyer’s (case $g = 1$) but our proofs are different and we do not use Hardy’s inequalities, which makes the proofs of the inequalities self-contained.