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1998 | 131 | 2 | 101-114
Tytuł artykułu

Weighted inequalities for one-sided maximal functions in Orlicz spaces

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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.
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Bibliografia
  • [G] D. Gallardo, Weighted weak type integral inequalities for the Hardy-Littlewood maximal operator, Israel J. Math. 67 (1989), 95-108.
  • [KT] R. A. Kerman and A. Torchinsky, Integral inequalities with weights for the Hardy maximal function, Studia Math. 71 (1981), 277-284.
  • [KR] M. A. Krasnosel'skiĭ and V. B. Rutitskiĭ, Convex Functions and Orlicz Spaces, Noordhoff, Groningen, 1961.
  • [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.
  • [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.
  • [M] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, ibid. 165 (1972), 207-226.
  • [Mu] J. Musielak, Orlicz Spaces and Modular Spaces, Springer, 1983.
  • [S] E. Sawyer, Weighted inequalities for the one-sided Hardy-Littlewood maximal functions, Trans. Amer. Math. Soc. 297 (1986), 53-61.
  • [SW] E. M. Stein and G. Weiss, Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, 1971.
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bwmeta1.element.bwnjournal-article-smv131i2p101bwm
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