Weighted inequalities for one-sided maximal functions in Orlicz spaces

Ortega Salvador, Pedro

Studia Mathematica, Tome 129 (1998), p. 101-114 / Harvested from The Polish Digital Mathematics Library

Résumé

Let $M_{g}^{+}$ be the maximal operator defined by $M_{g}^{+} ⨍ (x) = s u p_{h > 0} (ʃ_{x}^{x + h} | ⨍ | g) / (ʃ_{x}^{x + h} g)$ , 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 $u (x \in ℝ | M_{g}^{+} ⨍ (x) > λ) \leq C / (Φ (λ)) ʃ_{ℝ} Φ (| ⨍ |) ω$ holds for every ⨍ in the Orlicz space $L_{Φ} (ω)$ . We also characterize the positive functions ω such that the integral inequality $ʃ_{ℝ} Φ (| M_{g}^{+} ⨍ |) ω \leq ʃ_{ℝ} Φ (| ⨍ |) ω$ holds for every $⨍ \in L_{Φ} (ω)$ . Our results include some already obtained for functions in $L^{p}$ and yield as consequences one-dimensional theorems due to Gallardo and Kerman-Torchinsky.

Publié le : 1998-01-01

Zbl 0922.42012

EUDML-ID : urn:eudml:doc:216567

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     author = {Pedro Ortega Salvador},
     title = {Weighted inequalities for one-sided maximal functions in Orlicz spaces},
     journal = {Studia Mathematica},
     volume = {129},
     year = {1998},
     pages = {101-114},
     zbl = {0922.42012},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv131i2p101bwm}
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Ortega Salvador, Pedro. Weighted inequalities for one-sided maximal functions in Orlicz spaces. Studia Mathematica, Tome 129 (1998) pp. 101-114. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv131i2p101bwm/

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