Weighted inequalities for the one-sided Hardy-Littlewood maximal functions

Let $M^{+}f(x)=\underset{h>0}{sup}(1/h){\int }_x^{x+h}|f(t)|dt$ denote the one-sided maximal function of Hardy and Littlewood. For $w(x)⩾0$ on $R$ and $1>p>\mathrm{\infty }$ , we show that $M^{+}$ is bounded on $L^p(w)$ if and only if $w$ satisfies the one-sided $A_p$ condition: \[ $\left(A_p^{+}\right)\left[\frac{1}{h}{\int }_{a-h}^{a}w(x)dx\right]{\left[\frac{1}{h}{\int }_a^{a+h}w{(x)}^{-1/(p-1)}dx\right]}^{p-1}⩽C$ \] for all real $a$ and positive $h$ . If in addition $v(x)⩾0$ and $\sigma =v^{-1/(p-1)}$ ,then $M^{+}$ is bounded from $L^p(v)$ to $L^p(w)$ if and only if \[ ${\int }_I{[M^{+}({\chi }_I\sigma )]}^{p}w⩽C{\int }_I\sigma >\mathrm{\infty }$ \] for all intervals $I=(a,b)$ such that ${\int }_{-\mathrm{\infty }}^{a}w>0$ . The corresponding weak type inequality is also characterized. Further properties of $A_p^{+}$ weights, such as $A_p^{+}\Rightarrow A_{p-\epsilon }^{+}$ and $A_p^{+}=(A_1^{+})(A_1^{-}{)}^{1-p}$ , are established.