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Weighted inequalities
for the maximal geometric mean operator


Authors: Xiangrong Yin and Benjamin Muckenhoupt
Journal: Proc. Amer. Math. Soc. 124 (1996), 75-81
MSC (1991): Primary 26D15, 42B25
DOI: https://doi.org/10.1090/S0002-9939-96-03252-2
MathSciNet review: 1307575
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Abstract: For nonnegative Borel measures $\mu$ on $R^1$ and for the maximal geometric mean operator $G_f$, we characterize the weight pairs $(w,v)$ for which $G_f$ is of weak type $(p,p)$ and of strong type $(p,p)$, $0<p<\infty$. No doubling conditions are needed. We also note that a previously published different characterization for the strong type inequality for $G_f$ has an incorrect proof.


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

  • 1. N. Bourbaki, Elements de mathematique, Livre VI Integration, Hermann, Paris, 1952. MR 14:960h
  • 2. S. V. Hruscev, A description of weights satisfying the $A_\infty$ condition of Muckenhoupt, Proc. Amer. Math. Soc. 90 (1984), 253--257. MR 85k:42049
  • 3. W. Hu, X. Shi, and Q. Sun $A_\infty$ condition characterized by maximal geometric mean operator, Lecture Notes in Math., vol. 1494, Springer-Verlag, Berlin and New York, 1991, pp. 68--72. MR 94k:26023
  • 4. B. Muckenhoupt, The equivalence of two conditions for weight functions, Studia Math. 49 (1974), 101--106. MR 50:2790
  • 5. ------, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207--226. MR 45:2461
  • 6. E. T. Sawyer, A characterization of a two-weight norm inequality for maximal operators, Studia Math. 75 (1982), 1--11. MR 84i:42032
  • 7. X. Shi, Two inequalities related to geometric mean operators, J. Zhejiang Teacher's College 1 (1980), 21--25.
  • 8. P. Sjogren, A remark on the maximal function for measures in $R^n$, Amer. J. Math. 105 (1983), 1231--1233. MR 86a:28003

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Additional Information

Xiangrong Yin
Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903

Benjamin Muckenhoupt
Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903

DOI: https://doi.org/10.1090/S0002-9939-96-03252-2
Received by editor(s): January 26, 1993
Received by editor(s) in revised form: January 31, 1994, and May 18, 1994
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1996 American Mathematical Society

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