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.


42B25 Maximal functions, Littlewood-Paley theory
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