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On Lorentz spaces \(\Gamma_{p,w}\). (English) Zbl 1068.46019

This article studies Lorentz spaces \(\Gamma_{p,w}\), where \(0<p<\infty\) and \(w\) is a nonnegative measurable weight function. New formulas for the quasi-norm, duality, embeddings and Boyd indices are obtained. It is shown that, whenever \(\Gamma_{p,w}\) does not coincide with \(L^1+L^\infty\), it contains an order isomorphic and complemented coply of \(\ell^p\). Characterizations of order convexity and concavity as well as lower and upper estimates in \(\Gamma_{p,w}\) are obtained. The type and cotype of \(\Gamma_{p,w}\) for \(1\leq p<\infty\) are described.

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46B20 Geometry and structure of normed linear spaces
46B03 Isomorphic theory (including renorming) of Banach spaces
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