# Weighted inequalities for one-sided maximal functions in Orlicz spaces

Studia Mathematica (1998)

- Volume: 131, Issue: 2, page 101-114
- ISSN: 0039-3223

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topOrtega Salvador, Pedro. "Weighted inequalities for one-sided maximal functions in Orlicz spaces." Studia Mathematica 131.2 (1998): 101-114. <http://eudml.org/doc/216567>.

@article{OrtegaSalvador1998,

abstract = {Let $M_\{g\}^\{+\}$ be the maximal operator defined by $M_\{g\}^\{+\}⨍(x) = sup_\{h>0\} (ʃ_\{x\}^\{x+h\} |⨍|g)/(ʃ_\{x\}^\{x+h\} g)$, where g is a positive locally integrable function on ℝ. Let Φ be an N-function such that both Φ and its complementary N-function satisfy $Δ_2$. We characterize the pairs of positive functions (u,ω) such that the weak type inequality $u(\{x ∈ ℝ | M_\{g\}^\{+\}⨍(x) > λ\}) ≤ C/(Φ(λ)) ʃ_ℝ Φ(|⨍|)ω$ holds for every ⨍ in the Orlicz space $L_Φ(ω)$. We also characterize the positive functions ω such that the integral inequality $ʃ_ℝ Φ(|M_\{g\}^\{+\}⨍|)ω ≤ ʃ_ℝ Φ(|⨍|)ω$ holds for every $⨍ ∈ L_Φ(ω)$. Our results include some already obtained for functions in $L^p$ and yield as consequences one-dimensional theorems due to Gallardo and Kerman-Torchinsky.},

author = {Ortega Salvador, Pedro},

journal = {Studia Mathematica},

keywords = {one-sided maximal functions; weighted inequalities; weights; Orlicz spaces; Hardy-Littlewood maximal function; weak-type estimates},

language = {eng},

number = {2},

pages = {101-114},

title = {Weighted inequalities for one-sided maximal functions in Orlicz spaces},

url = {http://eudml.org/doc/216567},

volume = {131},

year = {1998},

}

TY - JOUR

AU - Ortega Salvador, Pedro

TI - Weighted inequalities for one-sided maximal functions in Orlicz spaces

JO - Studia Mathematica

PY - 1998

VL - 131

IS - 2

SP - 101

EP - 114

AB - Let $M_{g}^{+}$ be the maximal operator defined by $M_{g}^{+}⨍(x) = sup_{h>0} (ʃ_{x}^{x+h} |⨍|g)/(ʃ_{x}^{x+h} g)$, where g is a positive locally integrable function on ℝ. Let Φ be an N-function such that both Φ and its complementary N-function satisfy $Δ_2$. We characterize the pairs of positive functions (u,ω) such that the weak type inequality $u({x ∈ ℝ | M_{g}^{+}⨍(x) > λ}) ≤ C/(Φ(λ)) ʃ_ℝ Φ(|⨍|)ω$ holds for every ⨍ in the Orlicz space $L_Φ(ω)$. We also characterize the positive functions ω such that the integral inequality $ʃ_ℝ Φ(|M_{g}^{+}⨍|)ω ≤ ʃ_ℝ Φ(|⨍|)ω$ holds for every $⨍ ∈ L_Φ(ω)$. Our results include some already obtained for functions in $L^p$ and yield as consequences one-dimensional theorems due to Gallardo and Kerman-Torchinsky.

LA - eng

KW - one-sided maximal functions; weighted inequalities; weights; Orlicz spaces; Hardy-Littlewood maximal function; weak-type estimates

UR - http://eudml.org/doc/216567

ER -

## References

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- [KT] R. A. Kerman and A. Torchinsky, Integral inequalities with weights for the Hardy maximal function, Studia Math. 71 (1981), 277-284. Zbl0517.42030
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- [MR] F. J. Martín Reyes, New proofs of weighted inequalities for the one-sided Hardy-Littlewood maximal functions, Proc. Amer. Math. Soc. 117 (1993), 691-698. Zbl0771.42011
- [MOT] F. J. Martín Reyes, P. Ortega Salvador and A. de la Torre, Weighted inequalities for one-sided maximal functions, Trans. Amer. Math. Soc. 319 (1990), 517-534. Zbl0696.42013
- [M] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, ibid. 165 (1972), 207-226. Zbl0236.26016
- [Mu] J. Musielak, Orlicz Spaces and Modular Spaces, Springer, 1983. Zbl0557.46020
- [S] E. Sawyer, Weighted inequalities for the one-sided Hardy-Littlewood maximal functions, Trans. Amer. Math. Soc. 297 (1986), 53-61. Zbl0627.42009
- [SW] E. M. Stein and G. Weiss, Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, 1971.

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