Uniform rotundity of Orlicz function spaces equipped with the ‐Amemiya norm

Necessary and sufficient conditions for uniform rotundity of Orlicz function spaces equipped with the p‐Amemiya norm are presented. The obtained results unify, complete and widen the characterization of uniform rotundity of Orlicz spaces. In the case of the ∞‐Amemiya (i.e. the Luxemburg) norm or the 1‐Amemiya (i.e. the Orlicz) norm, these results were known earlier. Some connections with the fixed point theory and the best approximation theory are presented.     

Locally uniform convexity in Musielak-Orlicz function spaces of Bochner type endowed with the Luxemburg norm

Criteria for locally uniform convexity of Musielak-Orlicz function spaces of Bochner type equipped with the Luxemburg norm are given. We also prove that, in Musielak-Orlicz function spaces generated by locally uniformly convex Banach space, locally uniform convexity and strict convexity are equivalent.     

Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces

We consider the Hardy-Littlewood maximal operator M on Musielak-Orlicz Spaces Lφ(Rd). We give a necessary condition for the continuity of M on Lφ(Rd) which generalizes the concept of Muckenhoupt classes. In the special case of generalized Lebesgue spaces Lp(⋅)(Rd) we show that this condition is also sufficient. Moreover, we show that the condition is “left-open” in the sense that not only M but also Mq is continuous for some q>1, where .     

Local rotundity structure of Calderón-Lozanovskii spaces

General results saying that a point x of the unit sphere S(E) of a Köthe space E is an extreme point (a strongly extreme point) [an SU-point] of B(E) if and only if ‖x‖ is an extreme point (a strongly extreme point) [an SU-point] of B(E⁺) and ‖x‖ is a UM-point (a ULUM-point) [nothing more] of E are proved. These results are applied to get criteria for extreme points and SU-points of the unit ball in Caldern-Lozanovski spaces which refer to problem XII from [5]. Strongly extreme points in these spaces are also discussed.     

Nearly strict convexity in Musielak-Orlicz-Bochner function spaces

Criteria for nearly strict convexity of Musielak-Orlicz-Bochner function spaces equipped with the Luxemburg norm are given. We also prove that, in Musielak-Orlicz-Bochner function spaces generated by strictly convex Banach space, nearly strict convexity and strict convexity are equivalent.     

Noncreasy and uniformly noncreasy Orlicz-Bochner function spaces

We point out that uniform noncreasiness is a super-property in Banach spaces. Moreover, we prove that Orlicz-Bochner function spaces are noncreasy if and only if they are rotund or smooth. Finally, we obtain that Orlicz-Bochner function spaces are uniformly noncreasy if and only if they are uniformly rotund or uniformly smooth.     

Local rotundity structure of generalized Orlicz–Lorentz sequence spaces

We give some criteria for extreme points and strong U-points in generalized Orlicz–Lorentz sequence spaces, which were introduced in [P. Foralewski, H. Hudzik, L. Szymaszkiewicz, On some geometric and topological properties of generalized Orlicz–Lorentz sequence spaces, Math. Nachr. (in press)] (cf. [G.G. Lorentz, An inequality for rearrangements, Amer. Math. Monthly 60 (1953) 176–179; M. Nawrocki, The Mackey topology of some F-spaces, Ph.D. Dissertation, Adam Mickiewicz University, Poznań, 1984 (in Polish)]). Some examples show that in these spaces the notion of the strong U-point is essentially stronger than the notion of the extreme point. This paper is related to the results from [A. Kamińska, Extreme points in Orlicz–Lorentz spaces, Arch. Math. 55 (1990) 173–180] (see Remark 1).     

Complex extreme points and complex convexity in Orlicz-Bochner function spaces

It is well known that extreme points which are connected with strict convexity of the whole spaces, are the most basic and important geometric points in geometric theory of Banach spaces. In this paper, criteria for complex extreme points, complex strict convexity and complex uniform convexity in Orlicz-Bochner function spaces are given.     

Capacity for potentials of functions in Musielak-Orlicz spaces

We define a capacity for potentials of functions in Musielak-Orlicz spaces. Basic properties of such capacity are studied. We also estimate the capacity of balls and give some applications of the estimates.     

Average locally uniform rotundity and a class of nonlinear maps

We consider some topological characterizations of dual Banach spaces that admit an equivalent dual average locally uniformly rotund norm and provide a criterion for such renorming which involves the class of σ-slicely continuous maps.     

Rotundity structure of local nature for generalized Orlicz-Lorentz function spaces

Criteria for extreme points and rotund points in generalized Orlicz-Lorentz function spaces, which were introduced in Foralewski (2011) [27] are given. Some examples show that in these spaces the notion of rotund point is essentially stronger than the notion of extreme point. Finally, the criteria obtained in this paper are interpreted in the case of classical Orlicz-Lorentz spaces. Results of this paper are related to the results from Carothers et al. (1992) [9], Kamińska (1990) [4], Foralewski et al. (2008) [26].     

Hardy type spaces associated with compact unitary groups

We investigate Hilbertian Hardy type spaces of complex analytic functions of infinite many variables, associated with compact unitary groups and the corresponding invariant Haar’s measures. For such analytic functions we establish a Cauchy type integral formula and describe natural domains. Also we show some relations between constructed spaces of analytic functions and the symmetric Fock space.     

Kadec-Klee property in Musielak-Orlicz function spaces equipped with the Orlicz norm

It is well-known that the Kadec-Klee property is an important property in the geometry of Banach spaces. It is closely connected with the approximation compactness and fixed point property of non-expansive mappings. In this paper, a criterion for Musielak-Orlicz function spaces equipped with the Orlicz norm to have the Kadec-Klee property are given. As a corollary, we obtain that a class of non-reflexive Musielak-Orlicz function spaces have the Fixed Point property.

     

Two-weighted estimates for some integral transforms in Lebesgue spaces with mixed norm and imbedding theorems

Two-weighted inequalities are proved for anisotropic potentials. These estimates are used to obtain refinements of the well-known imbedding theorems in the scale of weighted Lebesgue spaces.     

Musielak–Orlicz–Bochner function spaces which are uniformly noncreasy

In this paper, the criteria for uniform noncreasiness of Musielak–Orlicz–Bochner function spaces are given. Moreover authors also prove that the space (resp ) is uniformly noncreasy if and only if the space (resp ) is uniformly convex or uniformly smooth. As a corollary, the criteria for uniform noncreasiness of Musielak–Orlicz function spaces are given.     

Traces of functions in Hardy and Besov spaces on Lipschitz domains with applications to compensated compactness and the theory of Hardy and Bergman type spaces

In this paper we study conditions guaranteeing that functions defined on a Lipschitz domain Ω have boundary traces in Hardy and Besov spaces on ∂Ω. In turn these results are used to develop a new approach to the theory of compensated compactness and the theory of non-locally convex Hardy and Bergman type spaces.     

Weak compactness, transfinite Schauder bases and differentiability

We provide a characterization of the class of nonseparable Banach spaces that contain a nonseparable weakly compact set (respectively a relatively weakly compact transfinite basic sequence) in terms of differentiability properties of those spaces.