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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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