Weighted inequalities for the one-sided Hardy-Littlewood maximal functions.(English)Zbl 0627.42009

Let $$M^+f(x)=\sup_{h>0}(1/h)\int^{x+h}_{x}| f(t)| dt$$ denote the one-sided maximal function of Hardy and Littlewood. For $$w(x)\geq 0$$ and 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: $(A_ p^+)\quad [\frac{1}{h}\int^{a}_{a- h}w(x)dx][\frac{1}{h}\int^{a+h}_{a}w(x)^{-1/(p-1)}dx]^{p-1}\leq C$ for all real a and positive h. If in addition v(x)$$\geq 0$$ and $$\sigma =v^{-1/(p-1)}$$, then $$M^+$$ is bounded from $$L^ p(v)$$ to $$L^ p(w)$$ if and only if $$\int_{t}[M^+(\chi_ I\sigma)]^ pw\leq C\int_{I}\sigma <\infty$$ for all intervals $$I=(a,b)$$ such that $$\int^{a}_{-\infty}w>0$$. 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.

MSC:

 42B25 Maximal functions, Littlewood-Paley theory
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References:

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