Weighted Orlicz space integral inequalities for the Hardy-Littlewood maximal operator

S. Bloom; R. Kerman

Studia Mathematica (1994)

  • Volume: 110, Issue: 2, page 149-167
  • ISSN: 0039-3223

Abstract

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Necessary and sufficient conditions are given for the Hardy-Littlewood maximal operator to be bounded on a weighted Orlicz space when the complementary Young function satisfies Δ 2 . Such a growth condition is shown to be necessary for any weighted integral inequality to occur. Weak-type conditions are also investigated.

How to cite

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Bloom, S., and Kerman, R.. "Weighted Orlicz space integral inequalities for the Hardy-Littlewood maximal operator." Studia Mathematica 110.2 (1994): 149-167. <http://eudml.org/doc/216106>.

@article{Bloom1994,
abstract = {Necessary and sufficient conditions are given for the Hardy-Littlewood maximal operator to be bounded on a weighted Orlicz space when the complementary Young function satisfies $Δ_2$. Such a growth condition is shown to be necessary for any weighted integral inequality to occur. Weak-type conditions are also investigated.},
author = {Bloom, S., Kerman, R.},
journal = {Studia Mathematica},
keywords = {integral inequalities; Hardy-Littlewood maximal operator; weighted inequalities; Orlicz spaces},
language = {eng},
number = {2},
pages = {149-167},
title = {Weighted Orlicz space integral inequalities for the Hardy-Littlewood maximal operator},
url = {http://eudml.org/doc/216106},
volume = {110},
year = {1994},
}

TY - JOUR
AU - Bloom, S.
AU - Kerman, R.
TI - Weighted Orlicz space integral inequalities for the Hardy-Littlewood maximal operator
JO - Studia Mathematica
PY - 1994
VL - 110
IS - 2
SP - 149
EP - 167
AB - Necessary and sufficient conditions are given for the Hardy-Littlewood maximal operator to be bounded on a weighted Orlicz space when the complementary Young function satisfies $Δ_2$. Such a growth condition is shown to be necessary for any weighted integral inequality to occur. Weak-type conditions are also investigated.
LA - eng
KW - integral inequalities; Hardy-Littlewood maximal operator; weighted inequalities; Orlicz spaces
UR - http://eudml.org/doc/216106
ER -

References

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  1. [1] R. Bagby, Weak bounds for the maximal function in weighted Orlicz spaces, Studia Math. 95 (1990), 195-204. Zbl0718.42018
  2. [2] N. K. Bari and S. B. Stečkin [S. B. Stechkin], Best approximation and differential properties of two conjugate functions, Trudy Moskov. Mat. Obshch. 5 (1956), 483-522 (in Russian). 
  3. [3] S. Bloom and R. Kerman, Weighted L Φ integral inequalities for operators of Hardy type, Studia Math. 110 (1994), 35-52. Zbl0823.42010
  4. [4] R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241-250. Zbl0291.44007
  5. [5] A. S. Gogatishvili, General weak-type estimates for the maximal operators and singular integrals, preprint. 
  6. [6] A. S. Gogatishvili and L. Pick, Weighted inequalities of weak and extra-weak type for the maximal operator and Hilbert transform, preprint. Zbl0798.42009
  7. [7] R. Kerman and A. Torchinsky, Integral inequalities with weights for the Hardy maximal function, Studia Math. 71 (1981/82), 277-284. Zbl0517.42030
  8. [8] V. Kokilashvili and M. Krbec, Weighted Inequalities in Lorentz and Orlicz Spaces, World Scientific, 1991. 
  9. [9] M. A. Krasnosel'skii [M. A. Krasnosel'skiǐ] and Ya. B. Rutickii [Ya. B. Rutitskiǐ], Convex Functions and Orlicz Spaces, Noordhoff, Groningen, 1961. Zbl0095.09103
  10. [10] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207-226. Zbl0236.26016
  11. [11] L. Pick, Weighted inequalities for the Hardy-Littlewood maximal operators in Orlicz spaces, preprint. 
  12. [12] L. Quinsheng, Two weight Φ-inequalities for the Hardy operator, Hardy-Littlewood maximal operator and fractional integrals, Proc. Amer. Math. Soc., to appear. Zbl0758.42012
  13. [13] M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Marcel Dekker, New York, 1991. Zbl0724.46032
  14. [14] T. Shimogaki, Hardy-Littlewood majorants in function spaces, J. Math. Soc. Japan 17 (1965), 365-373. Zbl0135.16304
  15. [15] A. Zygmund, Trigonometric Series, Vol. I, 2nd ed., Cambridge Univ. Press, Cambridge, 1959. Zbl0085.05601

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