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Interpolation restricted to decreasing functions and Lorentz spaces. (English) Zbl 0932.46017

The aim of the paper is to find conditions on the weights under which real interpolation results of the type \((\Lambda^{p_0}(\omega_0), \Lambda^{p_1}(\omega_1))_{\theta,q}= \Lambda^q(\omega)\), with \(\omega= \omega^{1-\theta}_0 \omega^\theta_1\), hold for couples of classical Lorentz spaces, including the case of couples of weighted Lorentz spaces. Since they appear as the symmetrized of weighted \(L_p\)-spaces, the results are obtained after the study of some interpolation properties of general symmetric spaces and of cones of decreasing functions of \(L_p\)-spaces. Special attention is given to the case of monotone weights.

MSC:

46B70 Interpolation between normed linear spaces
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46M35 Abstract interpolation of topological vector spaces
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