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Weighted \(L_ \Phi\) integral inequalities for maximal operators. (English) Zbl 0859.42017
Summary: Sufficient (almost necessary) conditions are given on the weight functions \(u(\cdot)\), \(v(\cdot)\) for \[ \Phi^{-1}_2\Biggl[\int_{\mathbb{R}^n}\Phi_2(C_2(M_sf)(x)) u(x)dx\Biggr]\leq \Phi^{-1}_1\Biggl[C_1 \int_{\mathbb{R}^n} \Phi_1(|f(x)|)v(x)dx\Biggr] \] to hold when \(\Phi_1\), \(\Phi_2\) are \(\varphi\)-functions with subadditive \(\Phi_1\Phi^{-1}_2\) and \(M_s\) \((0\leq s<n)\) is the usual fractional maximal operator.
42B25 Maximal functions, Littlewood-Paley theory
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
[1] J. Musielak, Orlicz spaces and modular space.Lecture Notes in Math. 1034,Springer, Berlin, 1983. · Zbl 0557.46020 · doi:10.1007/BFb0072210
[2] V. Kokilashvili and M. Krbec, Weighted inequalities in Lorents and Orlicz Spaces.World Scientific, Singapore, 1991. · Zbl 0751.46021 · doi:10.1142/1367
[3] M. Schechter, Potential estimates in Orlicz spaces.Pacific J. Math. 133,(2)(1988), 381–389. · Zbl 0686.31002 · doi:10.2140/pjm.1988.133.381
[4] E. Sawyer, A characterization of a two weight norm inequality for maximal operators.Studia Math. 75(1982), 1–11. · Zbl 0508.42023
[5] L. Qinsheng, Two weight mixed \(\Phi\)-inequalities for the Hardy operator and the Hardy-Littlewood maximal operator.J. London Math. Soc. 46(2)(1992), 301–318. · Zbl 0758.42012
[6] Y. Rakotondratsimba, Inégalités à poids pour des opérateurs maximaux et des opérateurs de type potentiel.Thèse de Dortorat.Univ. Orléans, France, 1991.
[7] R. Kerman and A. Torchinsky, Integral inequality with weights for the Hardy maximal function.Studia Math. 71(1981–82), 272–284. · Zbl 0517.42030
[8] J. C. Chen, Weights and L\(\Phi\)-boundedness of the Poisson integral operator.Insrael J. Math. 81(1993), 193–202. · Zbl 0801.47036
[9] Q. Sun, Weighted norm inequalities on spaces of homogeneous type.Studia Math. 101(3)(1992), 241–251. · Zbl 0812.46018
[10] Y. Rakotondratsimba, Weighted strong inequalities for maximal functions in Orlicz spaces.Preprint Univ. Orléands, France, 1992.
[11] C. Perez, Two weighted inequalities for potential and fractional type maximal operators.Indiana Univ. Math. J. 43(1994), 31–45.
[12] Y. Rakotondratsimba, Weigheted weak inequalities for some maximal functions in Orlicz spaces.Preprint Univ. Orlénas, France, 1991.
[13] L. Pick, Two-weight weak. type maximal inequalities in Orlicz classes.Studia Math. 100(3) (1991), 207–218. · Zbl 0752.42012
[14] Y. Rakotondratsimba, Weighted weak inequalities for fractional integrals in Orlicz spaces.Preprint Univ. Orléans, France, 1991.
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