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Boundedness of classical operators on classical Lorentz spaces. (English) Zbl 0705.42014

Summary: The classical Lorentz space \(\Lambda_ p(v)\) consists of those measurable functions f on \({\mathbb{R}}^ n\) such that \((\int^{\infty}_{0}f^*(x)^ pv(x)dx)^{1/p}<\infty.\) We characterize when a variety of classical operators, including Hilbert and Riesz transforms, fractional integrals and maximal functions, are bounded from one Lorentz space, \(\Lambda_ p(v)\), to another, \(\Lambda_ q(w)\). In addition, we give a simple and explicit description of the dual of \(\Lambda_ p(v)\) and determine when \(\Lambda_ p(v)\) is a Banach space.

MSC:

42B25 Maximal functions, Littlewood-Paley theory
26D15 Inequalities for sums, series and integrals
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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