# On a converse inequality for maximal functions in Orlicz spaces

Studia Mathematica (1996)

• Volume: 118, Issue: 1, page 1-10
• ISSN: 0039-3223

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## Abstract

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Let $\Phi \left(t\right)={ʃ}_{0}^{t}a\left(s\right)ds$ and $\Psi \left(t\right)={ʃ}_{0}^{t}b\left(s\right)ds$, where a(s) is a positive continuous function such that ${ʃ}_{1}^{\infty }a\left(s\right)/sds=\infty$ and b(s) is quasi-increasing and $li{m}_{s\to \infty }b\left(s\right)=\infty$. Then the following statements for the Hardy-Littlewood maximal function Mf(x) are equivalent: (j) there exist positive constants ${c}_{1}$ and ${s}_{0}$ such that ${ʃ}_{1}^{s}a\left(t\right)/tdt\ge {c}_{1}b\left({c}_{1}s\right)$ for all $s\ge {s}_{0}$; (jj) there exist positive constants ${c}_{2}$ and ${c}_{3}$ such that ${ʃ}_{0}^{2\pi }\Psi \left(\left({c}_{2}\right)/{\left(|⨍|}_{}\right)|⨍\left(x\right)|\right)dx\le {c}_{3}+{c}_{3}{ʃ}_{0}^{2\pi }{\Phi \left(1/\left(|⨍|}_{}\right)\right)Mf\left(x\right)dx$ for all $⨍\in {L}^{1}\left(\right)$.

## How to cite

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Kita, H.. "On a converse inequality for maximal functions in Orlicz spaces." Studia Mathematica 118.1 (1996): 1-10. <http://eudml.org/doc/216260>.

@article{Kita1996,
abstract = {Let $Φ(t) = ʃ_\{0\}^\{t\} a(s)ds$ and $Ψ(t) = ʃ_\{0\}^\{t\} b(s)ds$, where a(s) is a positive continuous function such that $ʃ_\{1\}^\{∞\} a(s)/s ds = ∞$ and b(s) is quasi-increasing and $lim_\{s→∞\}b(s) = ∞$. Then the following statements for the Hardy-Littlewood maximal function Mf(x) are equivalent: (j) there exist positive constants $c_1$ and $s_\{0\}$ such that $ʃ_\{1\}^\{s\} a(t)/t dt ≥ c_\{1\}b(c_\{1\}s)$ for all $s ≥ s_\{0\}$; (jj) there exist positive constants $c_2$ and $c_3$ such that $ʃ_\{0\}^\{2π\} Ψ((c_2)/(|⨍|_\{\}) |⨍(x)|) dx ≤ c_3 + c_\{3\} ʃ_\{0\}^\{2π\} Φ(1/(|⨍|_\{\})) Mf(x) dx$ for all $⨍ ∈ L^\{1\}()$.},
author = {Kita, H.},
journal = {Studia Mathematica},
keywords = {Hardy-Littlewood maximal function; Orlicz space; inequality},
language = {eng},
number = {1},
pages = {1-10},
title = {On a converse inequality for maximal functions in Orlicz spaces},
url = {http://eudml.org/doc/216260},
volume = {118},
year = {1996},
}

TY - JOUR
AU - Kita, H.
TI - On a converse inequality for maximal functions in Orlicz spaces
JO - Studia Mathematica
PY - 1996
VL - 118
IS - 1
SP - 1
EP - 10
AB - Let $Φ(t) = ʃ_{0}^{t} a(s)ds$ and $Ψ(t) = ʃ_{0}^{t} b(s)ds$, where a(s) is a positive continuous function such that $ʃ_{1}^{∞} a(s)/s ds = ∞$ and b(s) is quasi-increasing and $lim_{s→∞}b(s) = ∞$. Then the following statements for the Hardy-Littlewood maximal function Mf(x) are equivalent: (j) there exist positive constants $c_1$ and $s_{0}$ such that $ʃ_{1}^{s} a(t)/t dt ≥ c_{1}b(c_{1}s)$ for all $s ≥ s_{0}$; (jj) there exist positive constants $c_2$ and $c_3$ such that $ʃ_{0}^{2π} Ψ((c_2)/(|⨍|_{}) |⨍(x)|) dx ≤ c_3 + c_{3} ʃ_{0}^{2π} Φ(1/(|⨍|_{})) Mf(x) dx$ for all $⨍ ∈ L^{1}()$.
LA - eng
KW - Hardy-Littlewood maximal function; Orlicz space; inequality
UR - http://eudml.org/doc/216260
ER -

## References

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1. [1] M. de Guzmán, Differentiation of Integrals in ${ℝ}^{n}$, Lecture Notes in Math. 481, Springer, Berlin, 1975.
2. [2] H. Kita, On maximal functions in Orlicz spaces, submitted. Zbl0864.42007
3. [3] H. Kita and K. Yoneda, A treatment of Orlicz spaces as a ranked space, Math. Japon. 37 (1992), 775-802. Zbl0766.46008
4. [4] V. Kokilashvili and M. Krbec, Weighted Inequalities in Lorentz and Orlicz Spaces, World Sci., 1991. Zbl0751.46021
5. [5] G. Moscariello, On the integrability of the Jacobian in Orlicz spaces, Math. Japon. 40 (1994), 323-329. Zbl0805.46026
6. [6] M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Marcel Dekker, 1991.
7. [7] E. M. Stein, Note on the class L log L, Studia Math. 31 (1969), 305-310. Zbl0182.47803
8. [8] A. Torchinsky, Real-Variable Methods in Harmonic Analysis, Academic Press, New York, 1985.
9. [9] A. Zygmund, Trigonometric Series, Cambridge Univ. Press, Cambridge, 1959. Zbl0085.05601

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