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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.
MSC:
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.)
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