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Two-weighted estimations for the
Hardy-Littlewood maximal function
in ideal Banach spaces

Author: E. I. Berezhnoi
Journal: Proc. Amer. Math. Soc. 127 (1999), 79-87
MSC (1991): Primary 42B20, 42B25
MathSciNet review: 1622773
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Abstract: We give conditions on a couple of ideal Banach spaces with weights which are both necessary and sufficient for the Hardy-Littlewood maximal function to satisfy the two-weighted estimations of weak type, and we consider a modification of the Hardy-Littlewood maximal function. We also give some conditions on weights in order for the Hardy-Littlewood maximal function and the modification under consideration to fulfil the two-weighted estimations of strong type.

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  • 1. E. I. Berezhnoĭ, Sharp estimates for operators on cones in ideal spaces, Trudy Mat. Inst. Steklov. 204 (1993), no. Issled. po Teor. Differ. Funktsii Mnogikh Peremen. i ee Prilozh. 16, 3–34 (Russian); English transl., Proc. Steklov Inst. Math. 3 (204) (1994), 1–25. MR 1320016
  • 2. Huann Ming Chung, Richard A. Hunt, and Douglas S. Kurtz, The Hardy-Littlewood maximal function on 𝐿(𝑝,𝑞) spaces with weights, Indiana Univ. Math. J. 31 (1982), no. 1, 109–120. MR 642621, 10.1512/iumj.1982.31.31012
  • 3. E. M. Dynkin and B. P. Osilenkier, Weighted estimations of singular operators and their applications, Itogi Nauki i Tekhniki: Mat. Anal., vol. 21, VINITI, Moskow, 1983, pp. 42-130 (Russian); English transl. in J. Soviet Math. 30 (1985), no. 3.
  • 4. José García-Cuerva and José L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, vol. 116, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 104. MR 807149
  • 5. Z. G. Gorgadze and V. I. Tarieladze, On geometry of Orlicz spaces, Probability theory on vector spaces, II (Proc. Second Internat. Conf., Błażejewko, 1979) Lecture Notes in Math., vol. 828, Springer, Berlin, 1980, pp. 47–51. MR 611709
  • 6. Björn Jawerth, Weighted inequalities for maximal operators: linearization, localization and factorization, Amer. J. Math. 108 (1986), no. 2, 361–414. MR 833361, 10.2307/2374677
  • 7. V. M. Kokilashvili, The maximal functions and the integrals of the potential type in the Lebesgue and Lorentz spaces, Trudy MIAN SSSR 172 (1985), 192-201 (Russian).
  • 8. Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin-New York, 1979. Function spaces. MR 540367
  • 9. Benjamin Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226. MR 0293384, 10.1090/S0002-9947-1972-0293384-6
  • 10. S. Ya. Novikov, Cotype and type of Lorentz function spaces, Mat. Zametki 32 (1982), no. 2, 213–221, 270 (Russian). MR 672752
  • 11. C. Perez, Two weighted norm inequalities for the Riesz potentials and uniform $L^p$-weighted Sobolev inequalities, Indiana Univ. Math. J. 39 (1990), 31-44.
  • 12. E. T. Sawyer, A characterization for two weight norm inequalities for maximal operators, Studia Math. 75 (1982), 1-11.

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Additional Information

E. I. Berezhnoi

Received by editor(s): December 11, 1991
Received by editor(s) in revised form: September 19, 1996
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1999 American Mathematical Society