Weighted Lorentz norm inequalities for general maximal operators associated with certain families of Borel measures. (English) Zbl 0917.46021
Summary: Given a certain family \({\mathcal F}\) of positive Borel measures and \(\gamma\in[0, 1)\), we define a general one-sided maximal operator \(M^+_{{\mathcal F},\gamma}\) and we study weighted inequalities in \(L_{p,q}\) spaces for these operators. Our results contain, as particular cases, the characterization of weighted Lorentz norm inequalities for some well-known one-sided maximal operators such as the one-sided Hardy-Littlewood maximal operator associated with a general measure \(M^+_\mu\), the one-sided fractional maximal operator \(M^+_\gamma\) and the maximal operator \(m^+_\alpha\) associated with the Cesàro-\(\alpha\) averages.
MSC:
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
Keywords:
weighted inequalities; weighted Lorentz norm inequalities; one-sided fractional maximal operator; Cesàro-\(\alpha\) averagesReferences:
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