# Two-weight weak type maximal inequalities in Orlicz classes

Studia Mathematica (1991)

- Volume: 100, Issue: 3, page 207-218
- ISSN: 0039-3223

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topPick, Luboš. "Two-weight weak type maximal inequalities in Orlicz classes." Studia Mathematica 100.3 (1991): 207-218. <http://eudml.org/doc/215883>.

@article{Pick1991,

abstract = {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.},

author = {Pick, Luboš},

journal = {Studia Mathematica},

keywords = {two-weight weak type maximal inequalities in Orlicz classes; complete doubling measure; maximal operator; Young functions},

language = {eng},

number = {3},

pages = {207-218},

title = {Two-weight weak type maximal inequalities in Orlicz classes},

url = {http://eudml.org/doc/215883},

volume = {100},

year = {1991},

}

TY - JOUR

AU - Pick, Luboš

TI - Two-weight weak type maximal inequalities in Orlicz classes

JO - Studia Mathematica

PY - 1991

VL - 100

IS - 3

SP - 207

EP - 218

AB - 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.

LA - eng

KW - two-weight weak type maximal inequalities in Orlicz classes; complete doubling measure; maximal operator; Young functions

UR - http://eudml.org/doc/215883

ER -

## References

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- [2] D. Gallardo, Weighted weak type integral inequalities for the Hardy-Littlewood maximal operator, Israel J. Math. 67 (1) (1989), 95-108. Zbl0683.42021
- [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). Zbl0755.42011
- [4] R. A. Kerman and A. Torchinsky, Integral inequalities with weights for the Hardy maximal function, Studia Math. 71 (1982), 277-284. Zbl0517.42030
- [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. Zbl0068.09204
- [8] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207-226. Zbl0236.26016
- [9] L. Pick, Two weights weak type inequality for the maximal function in $L{\left(lo{g}^{+}L\right)}^{K}$, 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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