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

Let ${M^ + }f(x) = {\sup _{h > 0}}(1/h)\int_x^{x + h} {\vert f(t)\vert\,dt}$ denote the one-sided maximal function of Hardy and Littlewood. For $w(x) \geqslant 0$ on $R$ and $1 < p < \infty$, we show that ${M^ + }$ is bounded on ${L^p}(w)$ if and only if $w$ satisfies the one-sided ${A_p}$ condition:

$$\left( {A_p^ + } \right)\qquad \left[ {\frac{1} {h}\int_{a - h}^a... ...1} {h}\int_a^{a + h} {w{{(x)}^{ - 1/(p - 1)}}dx} } \right]^{p - 1}} \leqslant C$$

for all real $a$ and positive $h$. If in addition $v(x) \geqslant 0$ and $\sigma = {v^{ - 1/(p - 1)}}$,then ${M^ + }$ is bounded from ${L^p}(v)$ to ${L^p}(w)$ if and only if

$$\int_I {{{[{M^ + }({\chi _I}\sigma )]}^p}w \leqslant C\int_I {\sigma < \infty } }$$

for all intervals $I = (a,b)$ such that $\int_{ - \infty }^a {w > 0}$. The corresponding weak type inequality is also characterized. Further properties of $A_p^ +$ weights, such as $A_p^ + \Rightarrow A_{p - \varepsilon }^ +$ and $A_p^ + = (A_1^ + ){(A_1^ - )^{1 - p}}$, are established.