Two-weight weak type maximal inequalities in Orlicz classes

Luboš Pick

Studia Mathematica (1991)

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

Abstract

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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 ) d μ , or ϱ ( M μ f > λ ) ≤ C ʃ X Φ ( C λ - 1 | f ( x ) | ) σ ( x ) d μ 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.

How to cite

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Pick, 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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  1. [1] A. Carbery, S.-Y. A. Chang and J. Garnett, Weights and LlogL, Pacific J. Math. 120 (1) (1985), 33-45. 
  2. [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. [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. [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. [5] M. A. Krasnosel'skiĭ and Ya. B. Rutitskiĭ, Convex Functions and Orlicz Spaces, Gos. Izdat. Fiz.-Mat. Liter., Moscow 1958 (in Russian). 
  6. [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. [7] W. A. J. Luxemburg, Banach Function Spaces, thesis, Delft 1955. Zbl0068.09204
  8. [8] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207-226. Zbl0236.26016
  9. [9] L. Pick, Two weights weak type inequality for the maximal function in L ( l o g + 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. [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. [11] J.-O. Strömberg and A. Torchinsky, Weighted Hardy Spaces, Lecture Notes in Math. 1381, Springer, Berlin 1989. 

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