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1991 | 100 | 3 | 207-218
Tytuł artykułu

Two-weight weak type maximal inequalities in Orlicz classes

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Necessary and sufficient conditions are shown in order that the inequalities of the form $ϱ({M_μ f > λ})Φ(λ) ≤ C ʃ_X Ψ(C|f(x)|) σ(x)dμ$, or $ϱ({M_μ f > λ}) ≤ C ʃ_X Φ(Cλ^{-1}|f(x)|) σ(x)dμ$ hold with some positive C independent of λ > 0 and a μ-measurable function f, where (X,μ) is a space with a complete doubling measure μ, $M_μ$ is the maximal operator with respect to μ, Φ, Ψ are arbitrary Young functions, and ϱ, σ are weights, not necessarily doubling.
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  • Mathematical Institute, Czechoslovak Academy of Sciences, Žitná 25, 115 67 Praha 1, Czechoslovakia
Bibliografia
  • [1] A. Carbery, S.-Y. A. Chang and J. Garnett, Weights and LlogL, Pacific J. Math. 120 (1) (1985), 33-45.
  • [2] D. Gallardo, Weighted weak type integral inequalities for the Hardy-Littlewood maximal operator, Israel J. Math. 67 (1) (1989), 95-108.
  • [3] A. Gogatishvili, V. Kokilashvili and M. Krbec, Maximal functions in ϕ(L) classes, Dokl. Akad. Nauk SSSR 314 (1) (1990), 534-536 (in Russian).
  • [4] R. A. Kerman and A. Torchinsky, Integral inequalities with weights for the Hardy maximal function, Studia Math. 71 (1982), 277-284.
  • [5] M. A. Krasnosel'skiĭ and Ya. B. Rutitskiĭ, Convex Functions and Orlicz Spaces, Gos. Izdat. Fiz.-Mat. Liter., Moscow 1958 (in Russian).
  • [6] M. Krbec, Two weights weak type inequalities for the maximal function in the Zygmund class, in: Function Spaces and Applications, Proc. Conf. Lund 1986, M.Cwikel et al. (eds.), Lecture Notes in Math. 1302, Springer, Berlin 1988, 317-320.
  • [7] W. A. J. Luxemburg, Banach Function Spaces, thesis, Delft 1955.
  • [8] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207-226.
  • [9] L. Pick, Two weights weak type inequality for the maximal function in $L(log^{+}L)^{K}$, in: Constructive Theory of Functions, Proc. Conf. Varna 1987, B. Sendov et al. (eds.), Publ. House Bulgar. Acad. Sci., Sofia 1988, 377-381.
  • [10] L. Pick, Weighted inequalities for the Hardy-Littlewood maximal operators in Orlicz spaces, preprint, Math. Inst. Czech. Acad. Sci. 46 (1989), 1-22.
  • [11] J.-O. Strömberg and A. Torchinsky, Weighted Hardy Spaces, Lecture Notes in Math. 1381, Springer, Berlin 1989.
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