Weighted inequalities for the one-sided Hardy-Littlewood maximal functions
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- by E. Sawyer
- Trans. Amer. Math. Soc. 297 (1986), 53-61
- DOI: https://doi.org/10.1090/S0002-9947-1986-0849466-0
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Abstract:
Let ${M^ + }f(x) = {\sup _{h > 0}}(1/h)\int _x^{x + h} {|f(t)| dt}$ 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: \[ \left ( {A_p^ + } \right )\qquad \left [ {\frac {1} {h}\int _{a - h}^a {w(x)dx} } \right ]{\left [ {\frac {1} {h}\int _a^{a + h} {w{{(x)}^{ - 1/(p - 1)}}dx} } \right ]^{p - 1}} \leqslant C\] 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.References
- 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. MR 665888, DOI 10.4064/sm-72-1-9-26
- Kenneth F. Andersen and Wo-Sang Young, On the reverse weak type inequality for the Hardy maximal function and the weighted classes $L(\textrm {log}\,L)^{k}$, Pacific J. Math. 112 (1984), no. 2, 257–264. MR 743983
- E. Atencia and A. de la Torre, A dominated ergodic estimate for $L_{p}$ spaces with weights, Studia Math. 74 (1982), no. 1, 35–47. MR 675431, DOI 10.4064/sm-74-1-35-47
- R. Coifman, Peter W. Jones, and José L. Rubio de Francia, Constructive decomposition of BMO functions and factorization of $A_{p}$ weights, Proc. Amer. Math. Soc. 87 (1983), no. 4, 675–676. MR 687639, DOI 10.1090/S0002-9939-1983-0687639-3
- G. H. Hardy and J. E. Littlewood, A maximal theorem with function-theoretic applications, Acta Math. 54 (1930), no. 1, 81–116. MR 1555303, DOI 10.1007/BF02547518
- R. A. Hunt, D. S. Kurtz, and C. J. Neugebauer, A note on the equivalence of $A_{p}$ 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. MR 730066
- Björn Jawerth, Weighted inequalities for maximal operators: linearization, localization and factorization, Amer. J. Math. 108 (1986), no. 2, 361–414. MR 833361, DOI 10.2307/2374677
- Benjamin Muckenhoupt, Hardy’s inequality with weights, Studia Math. 44 (1972), 31–38. MR 311856, DOI 10.4064/sm-44-1-31-38
- Benjamin Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226. MR 293384, DOI 10.1090/S0002-9947-1972-0293384-6 —, Weighted reverse weak type inequalities for the Hardy-Littlewood maximal function, preprint.
- Eric T. Sawyer, A characterization of a two-weight norm inequality for maximal operators, Studia Math. 75 (1982), no. 1, 1–11. MR 676801, DOI 10.4064/sm-75-1-1-11
- Eric Sawyer, Weighted Lebesgue and Lorentz norm inequalities for the Hardy operator, Trans. Amer. Math. Soc. 281 (1984), no. 1, 329–337. MR 719673, DOI 10.1090/S0002-9947-1984-0719673-4
Bibliographic Information
- © Copyright 1986 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 297 (1986), 53-61
- MSC: Primary 42B25
- DOI: https://doi.org/10.1090/S0002-9947-1986-0849466-0
- MathSciNet review: 849466