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Transactions of the American Mathematical Society

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Maximal functions on classical Lorentz spaces and Hardy's inequality with weights for nonincreasing functions


Authors: Miguel A. Ariño and Benjamin Muckenhoupt
Journal: Trans. Amer. Math. Soc. 320 (1990), 727-735
MSC: Primary 42B25; Secondary 26D15, 46E30, 47B38
DOI: https://doi.org/10.1090/S0002-9947-1990-0989570-0
MathSciNet review: 989570
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Abstract: A characterization is given of a class of classical Lorentz spaces on which the Hardy Littlewood maximal operator is bounded. This is done by determining the weights for which Hardy's inequality holds for nonincreasing functions. An alternate characterization, valid for nondecreasing weights, is also derived.


References [Enhancements On Off] (What's this?)

  • [1] K. Andersen and B. Muckenhoupt, Weighted weak type inequalities with applications to Hilbert transforms and maximal functions, Studia Math. 72 (1982), 9-26. MR 665888 (83k:42018)
  • [2] R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241-250. MR 0358205 (50:10670)
  • [3] R. A. Hunt, On $ L(p,q)$ spaces, Enseign. Math. (2) 12 (1966), 249-276. MR 0223874 (36:6921)
  • [4] G. G. Lorentz, Some new functional spaces, Ann of Math. (2) 51 (1950), 37-55. MR 0033449 (11:442d)
  • [5] -, On the theory of spaces $ \Lambda $, Pacific J. Math. 1 (1951), 411-429. MR 0044740 (13:470c)
  • [6] B. Muckenhoupt, Hardy's inequality with weights, Studia Math. 44 (1972), 31-38. MR 0311856 (47:418)
  • [7] -, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207-226. MR 0293384 (45:2461)
  • [8] E. Sawyer, Weighted Lebesgue and Lorentz norm inequalities for the Hardy operator, Trans. Amer. Math. Soc. 281 (1984), 329-337. MR 719673 (85f:26013)
  • [9] E. Stein, Note on the class $ L\log L$, Studia Math. 32 (1969), 305-310. MR 0247534 (40:799)
  • [10] E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, Princeton, N.J., 1971. MR 0304972 (46:4102)
  • [11] J. O. Strömberg and A. Torchinsky, Weighted Hardy spaces, Lecture Notes in Math., vol. 1381, Springer-Verlag, New York, 1989. MR 1011673 (90j:42053)
  • [12] A. Zygmund, Trigonometric series, vol. I, Cambridge Univ. Press, London, New York, 1959. MR 0107776 (21:6498)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1990-0989570-0
Article copyright: © Copyright 1990 American Mathematical Society

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