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Some remarks on level functions and their applications

Foralewski P., Leśnik K., Maligranda L.

Commentationes Mathematicae

2016

Vol 56, No. 1

55--86

A comparison of the level functions considered by Halperin and Sinnamon is discussed. Moreover, connections between Lorentz-type spaces, down spaces, Cesàro spaces, and Sawyer's duality formula are explained. Applying Sinnamon's ideas, we prove the duality theorem for Orlicz−Lorentz spaces which generalizes a recent result by Kamińska, Leśnik, and Raynaud (and Nakamura). Finally, some applications of the level functions to the geometry of Orlicz−Lorentz spaces are presented.

Asymptotically isometric and isometric copies of ℓ1 in some Banach function lattices

Kamińska A., Mastyło M.

Commentationes Mathematicae

2013

Vol. 53, No. 2

283--300

We identify the class of Calderón-Lozanovskii spaces that do not contain an asymptotically isometric copy of ℓ1, and consequently we obtain the corresponding characterizations in the classes of Orlicz-Lorentz and Orlicz spaces equipped with the Luxemburg norm. We also give a complete description of order continuous Orlicz-Lorentz spaces which contain (order) isometric copies of ℓ1(n) for each integer n≥2. As an application we provide necessary and sufficient conditions for order continuous Orlicz-Lorentz spaces to contain an (order) isometric copy of ℓ1. In particular we give criteria in Orlicz and Lorentz spaces for (order) isometric containment of ℓ1(n) and ℓ1. The results are applied to obtain the description of universal Orlicz-Lorentz spaces for all two-dimensional normed spaces.

Best constant approximants in Orlicz-Lorentz spaces

Levis F.E., Cuenya H.H., Priori A.N.

Commentationes Mathematicae

2008

Vol. 48, [Z] 1

59-73

The best constant approximant operator is extended from an Orlicz- Lorentz space [...]to the space[...] , where 0 is the derivative of φ. Monotonicity property of its extension is established.

Complex convexity of Orlicz-Lorentz spaces and its applications

Choi C., Kamińska A., Lee H. J.

Bulletin of the Polish Academy of Sciences. Mathematics

2004

Vol. 52, nr 1

19-38

We give sufficient and necessary conditions for complex extreme points of the unit ball of Orlicz-Lorentz spaces, as well as we find criteria for the complex rotundity and uniform complex rotundity of these spaces. As an application we show that the set of norm-attaining operators is dense in the space of bounded linear operators from d* (w, 1) into d(w, 1), where d*(w, 1) is a predual of a complex Lorentz sequence space d[w, 1), if and only if w [is an element of] c0 \ L2.