Weighted inequalities for the one-sided Hardy-Littlewood maximal functions
Author:
E. Sawyer
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
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: Let denote the one-sided maximal function of Hardy and Littlewood. For
on
and
, we show that
is bounded on
if and only if
satisfies the one-sided
condition:
![$\displaystyle \left( {A_p^ + } \right)\qquad \left[ {\frac{1} {h}\int_{a - h}^a... ...1} {h}\int_a^{a + h} {w{{(x)}^{ - 1/(p - 1)}}dx} } \right]^{p - 1}} \leqslant C$](images/img14.gif)







![$\displaystyle \int_I {{{[{M^ + }({\chi _I}\sigma )]}^p}w \leqslant C\int_I {\sigma < \infty } } $](images/img22.gif)





- [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. MR 665888, https://doi.org/10.4064/sm-72-1-9-26
- [2] Kenneth F. Andersen and Wo-Sang Young, On the reverse weak type inequality for the Hardy maximal function and the weighted classes 𝐿(𝑙𝑜𝑔𝐿)^{𝑘}, Pacific J. Math. 112 (1984), no. 2, 257–264. MR 743983
- [3] 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, https://doi.org/10.4064/sm-74-1-35-47
- [4] 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, https://doi.org/10.1090/S0002-9939-1983-0687639-3
- [5] G. H. Hardy and J. E. Littlewood, A maximal theorem with function-theoretic applications, Acta Math. 54 (1930), no. 1, 81–116. MR 1555303, https://doi.org/10.1007/BF02547518
- [6] 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
- [7] Björn Jawerth, Weighted inequalities for maximal operators: linearization, localization and factorization, Amer. J. Math. 108 (1986), no. 2, 361–414. MR 833361, https://doi.org/10.2307/2374677
- [8] Benjamin Muckenhoupt, Hardy’s inequality with weights, Studia Math. 44 (1972), 31–38. MR 311856, https://doi.org/10.4064/sm-44-1-31-38
- [9] Benjamin Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226. MR 293384, https://doi.org/10.1090/S0002-9947-1972-0293384-6
- [10] -, Weighted reverse weak type inequalities for the Hardy-Littlewood maximal function, preprint.
- [11] Eric T. Sawyer, A characterization of a two-weight norm inequality for maximal operators, Studia Math. 75 (1982), no. 1, 1–11. MR 676801, https://doi.org/10.4064/sm-75-1-1-11
- [12] Eric Sawyer, Weighted Lebesgue and Lorentz norm inequalities for the Hardy operator, Trans. Amer. Math. Soc. 281 (1984), no. 1, 329–337. MR 719673, https://doi.org/10.1090/S0002-9947-1984-0719673-4
Retrieve articles in Transactions of the American Mathematical Society with MSC: 42B25
Retrieve articles in all journals with MSC: 42B25
Additional Information
DOI:
https://doi.org/10.1090/S0002-9947-1986-0849466-0
Article copyright:
© Copyright 1986
American Mathematical Society