×

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
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Kenneth F. Andersen and Benjamin Muckenhoupt, Weighted weak type Hardy inequalities with applications to Hilbert transforms and maximal functions, Studia Math. 72 (1982), no. 1, 9 – 26. · Zbl 0501.47011
[3] E. Atencia and A. de la Torre, A dominated ergodic estimate for \?_{\?} spaces with weights, Studia Math. 74 (1982), no. 1, 35 – 47. · Zbl 0442.28021
[4] R. Coifman, Peter W. Jones, and José L. Rubio de Francia, Constructive decomposition of BMO functions and factorization of \?_{\?} weights, Proc. Amer. Math. Soc. 87 (1983), no. 4, 675 – 676. · Zbl 0514.30025
[5] G. H. Hardy and J. E. Littlewood, A maximal theorem with function-theoretic applications, Acta Math. 54 (1930), no. 1, 81 – 116. · JFM 56.0264.02
[6] R. A. Hunt, D. S. Kurtz, and C. J. Neugebauer, A note on the equivalence of \?_{\?} and Sawyer’s condition for equal weights, Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981) Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983, pp. 156 – 158.
[7] Björn Jawerth, Weighted inequalities for maximal operators: linearization, localization and factorization, Amer. J. Math. 108 (1986), no. 2, 361 – 414. · Zbl 0608.42012
[8] Benjamin Muckenhoupt, Hardy’s inequality with weights, Studia Math. 44 (1972), 31 – 38. Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, I. · Zbl 0236.26015
[9] Benjamin Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207 – 226. · Zbl 0236.26016
[10] -, Weighted reverse weak type inequalities for the Hardy-Littlewood maximal function, preprint. · Zbl 0578.42017
[11] Eric T. Sawyer, A characterization of a two-weight norm inequality for maximal operators, Studia Math. 75 (1982), no. 1, 1 – 11. · Zbl 0508.42023
[12] Eric Sawyer, Weighted Lebesgue and Lorentz norm inequalities for the Hardy operator, Trans. Amer. Math. Soc. 281 (1984), no. 1, 329 – 337. · Zbl 0538.47020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.