Abstract Lorentz spaces and Köthe duality

Abstract

Given a symmetric Banach function space E and a decreasing positive weight w on I=(0,a), 0<a≤∞, the generalized Lorentz space ΛE,w is defined as the symmetrization of the canonical copy Ew of E on the measure space associated with the weight. A class of functions ME,w is similarly defined in the spirit of Marcinkiewicz spaces as the symmetrization of the space wEw. Differently from the Lorentz space, which is a Banach function space, the class ME,w does not need to be even a linear space; but we show that if the weight w is regular then this class is normable. Let also QE,w be the smallest fully symmetric Banach function space containing ME,w. The Köthe duality of these classes is developed here. The Köthe dual of the class ME,w is identified as the Lorentz space ΛE′,w, while the Köthe dual of ΛE,w is QE′,w. Several characterizations of QE,w are obtained, one of them states that a function belongs to QE,w if and only if its level function in Halperin’s sense with respect to w, belongs to ME,w. The other characterizations are by optimization with respect to the Hardy–Littlewood submajorization order. These results are applied to a number of concrete Banach function spaces. In particular a new description of the Köthe dual space is provided for the classical Lorentz space Λp,w and for the Orlicz–Lorentz space Λφ,w, which correspond respectively to the cases E=Lp and E=Lφ.