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Henryk Hudzik : vita et opera

Mastyło M., Musielak J.

Commentationes Mathematicae

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2015

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Vol 55, No. 2

45--78

EN

This article contains a short vita of Henryk Hudzik's as well as a non-exhaustive survey of his contribution to various areas of analysis. We focus on the theory of Orlicz−Sobolev spaces and the geometry of Banach spaces. We highlight criteria for some important geometric properties related to the metric fixed point theory in some classes of Banach lattices, including Orlicz and Orlicz−Lorentz spaces, but we do not forget Henryk Hudzik's contribution to nonlinear integral equations and partial differential equations.

2

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

Kamińska A., Mastyło M.

Commentationes Mathematicae

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2013

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Vol. 53, No. 2

283--300

EN

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.

3

The type and cotype of Calderon-Lozanovskij spaces

Cerda J., Mastyło M.

Bulletin of the Polish Academy of Sciences. Mathematics

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1999

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Vol. 47, no 2

105-117

EN

We prove certain vector-valued continuous inclusions for Calderon-Lozanovskij spaces and, by interpolation, we obtain results on the type and cotype of these spaces. We also give an extension of Kwapień's result which states that, for [1 is less than or equal to p is less than or equal to 2], every operator from l[sub 1] into l[sub p] is (r,1)-summing, if 1/r=3/2 - 1/p.