The locally uniformly non‐square points of Orlicz–Bochner sequence spaces

In this paper, we investigate the locally uniformly non‐square point of Orlicz–Bochner sequence spaces endowed with Luxemburg norm. Analysing and combining the generating function M and properties of the real Banach space X , we get sufficient and necessary conditions of locally uniformly non‐square point, which contributes to criteria for locally uniform non‐squareness in Orlicz–Bochner sequence spaces. The results generalize the corresponding results in the classical Orlicz sequence spaces.     

I‐convexity and Q‐convexity in Orlicz–Bochner function spaces endowed with the Orlicz norm

Some characterizations of I‐convexity and Q‐convexity of Banach space are obtained. Moreover, the criteria is shown for Orlicz–Bochner function spaces $L_M(\mu ,X)$ endowed with the Orlicz norm being I‐convex as well as being Q‐convex.     

Higher‐order Sobolev‐type embeddings on Carnot–Carathéodory spaces

A sufficient condition for higher‐order Sobolev‐type embeddings on bounded domains of Carnot–Carathéodory spaces is established for the class of rearrangement‐invariant function spaces. The condition takes form of a one‐dimensional inequality for suitable integral operators depending on the isoperimetric function relative to the Carnot–Carathéodory structure of the relevant sets. General results are then applied to particular Sobolev spaces built upon Lebesgue, Lorentz and Orlicz spaces on John domains in the Heisenberg group. In the case of the Heisenberg group, the condition is shown to be necessary as well.     

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.     

Some Inequalities in Classical Spaces with Mixed Norms

We consider some inequalities in such classical Banach Function Spaces as Lorentz, Marcinkiewicz, and Orlicz spaces. Our aim is to explore connections between the norm of a function of two variables on the product space and the mixed norm of the same function, where mixed norm is calculated in function spaces on coordinate spaces, first in one variable, then in the other. This issue is motivated by various problems of functional analysis and theory of functions. We will currently mention just geometry of spaces of vector-valued functions and embedding theorems for Sobolev and Besov spaces generated by metrics which differ from L ^p. Our main results are actually counterexamples for Lorentz spaces versus the natural intuition that arises from the easier case of Orlicz spaces (Section 2). In the Appendix we give a proof for the Kolmogorov–Nagumo theorem on change of order of mixed norm calculation in its most general form. This result shows that L ^p is the only space where it is possible to change this order.     

Gâteaux Differentiability of Bump Functions in Banach Spaces

Upper estimates for the order of Gâteaux smoothness of bump functions in Orlicz spaces ℓ_M(Γ) and Lorentz spaces d(w, p, Γ), Γ uncountable, are obtained.     

Duality theorem for B‐valued martingale Orlicz–Hardy spaces associated with concave functions

The dual space of B ‐valued martingale Orlicz–Hardy space with a concave function Φ, which is associated with the conditional p‐variation of B ‐valued martingale, is characterized. To obtain the results, a new type of Campanato spaces for B ‐valued martingales is introduced and the classical technique of atomic decompositions is improved. Some results obtained here are connected closely with the p‐uniform smoothness and q‐uniform convexity of the underlying Banach space.     

Sobolev‐Jawerth embedding of Triebel‐Lizorkin‐Morrey‐Lorentz spaces and fractional integral operator on Hardy type spaces

A Sobolev type embedding for Triebel‐Lizorkin‐Morrey‐Lorentz spaces is established in this paper. As an application of this result, the boundedness of the fractional integral operator on some generalizations of Hardy spaces such as Hardy‐Morrey spaces and Hardy‐Lorentz spaces are obtained.     

Complex interpolation of Orlicz spaces with respect to a vector measure

We apply the Calderón interpolation methods to Orlicz and weakly Orlicz function spaces with respect to a Banach‐space‐valued measure defined on a σ‐algebra. The results we obtain generalize those in the case of Banach lattices of p‐integrable and weakly p‐integrable functions with respect to such a vector measure.     

On the convexity coefficient of Musielak–Orlicz function spaces equipped with the Orlicz norm

In Hudzik and Landes, the convexity coefficient of Musielak–Orlicz function spaces over a non-atomic measure space equipped with the Luxemburg norm is computed whenever the Musielak–Orlicz functions are strictly convex see [6]. In this paper, we extend this result to the case of Musielak–Orlicz spaces equipped with the Orlicz norm. Also, a characterization of uniformly convex Musielak–Orlicz function spaces as well as k-uniformly convex Musielak–Orlicz spaces equipped with the Orlicz norm is given.     

Gelfand Numbers of Identity Operators Between Symmetric Sequence Spaces

Complementing and generalizing classical as well as recent results, we prove asymptotically optimal formulas for the Gelfand and approximation numbers of identities Eⁿ ↪ Fⁿ, where Eⁿ and Fⁿ denote the n-th sections of symmetric quasi-Banach sequence spaces E and F satisfying certain interpolation assumptions. We illustrate our results by considering classical spaces such as Lorentz and Orlicz sequence spaces. Supported by DFG grant Hi 584/2-2.     

Convexity and concavity constants in Lorentz and Marcinkiewicz spaces

We provide here the formulas for the q-convexity and q-concavity constants for function and sequence Lorentz spaces associated to either decreasing or increasing weights. It yields also the formula for the q-convexity constants in function and sequence Marcinkiewicz spaces. In this paper we extent and enhance the results from [G.J.O. Jameson, The q-concavity constants of Lorentz sequence spaces and related inequalities, Math. Z. 227 (1998) 129-142] and [A. Kamińska, A.M. Parrish, The q-concavity and q-convexity constants in Lorentz spaces, in: Banach Spaces and Their Applications in Analysis, Conference in Honor of Nigel Kalton, May 2006, Walter de Gruyter, Berlin, 2007, pp. 357-373].     

Boundedness and compactness of Hardy operators on Lorentz‐type spaces

Given non‐negative measurable functions on we study the high dimensional Hardy operator between Orlicz–Lorentz spaces , where f is a measurable function of and is the ball of radius t in . We give sufficient conditions of boundedness of and . We investigate also boundedness and compactness of between weighted and classical Lorentz spaces. The function spaces considered here do not need to be Banach spaces. Specifying the weights and the Orlicz functions we recover the existing results as well as we obtain new results in the new and old settings.     

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).     

Spaces of Bessel‐potential type and embeddings: the super‐limiting case

We consider Bessel‐potential spaces modelled upon Lorentz‐Karamata spaces and establish embedding theorems in the super‐limiting case. In addition, we refine a result due to Triebel, in the context of Bessel‐potential spaces, itself an improvement of the Brézis‐Wainger result (super‐limiting case) about the “almost Lipschitz continuity” of elements of H^1+n/p_p (?ⁿ). These results improve and extend results due to Edmunds, Gurka and Opic in the context of logarithmic Bessel potential spaces. We also give examples of embeddings of Besselpotential type spaces which are not of logarithmic type. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Best approximation properties in spaces of measurable functions

We research proximinality of μ‐sequentially compact sets and μ‐compact sets in measurable function spaces. Next we show a correspondence between the Kadec–Klee property for convergence in measure and μ‐compactness of the sets in Banach function spaces. Also the property S is investigated in Fréchet spaces and employed to provide the Kadec–Klee property for local convergence in measure. We discuss complete criteria for continuity of metric projection in Fréchet spaces with respect to the Hausdorff distance. Finally, we present the necessary and sufficient condition for continuous metric selection onto a one‐dimensional subspace in sequence Lorentz spaces .     

The Gustavsson–Peetre method for several Banach spaces

We extend the Gustavsson–Peetre method to the context of N ‐tuples of Banach spaces. We give estimates for the norm of the interpolated operator. The method is applied to tuples of weighted L _p ‐spaces and to tuples of Orlicz spaces identifying the outcoming spaces in both cases. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Cesaro mean convergence of martingale differences in rearrangement invariant spaces

We study the class of r.i. spaces in which Cesaro means of any weakly null martingale difference sequence is strongly null. This property is related to the Banach-Saks property. We show that in classical (separable) r.i. spaces (such as Orlicz, Lorentz and Marcinkiewicz spaces) these properties coincide but this is no longer true for general r.i. spaces. We locate also a class of r.i. spaces having this property where an analogue of the classical Dunford-Pettis characterization of relatively weakly compact subsets in L ₁ holds. Research was partially supported by the ARC and NSF grant DMS-0244515.     

赋Orlicz范数的Orlicz—Lorentz空间的严格凸性

本文对于赋Ｌｕｘｅｍｂｕｒｇ范数的Ｏｒｌｉｃｚ－Ｌｏｒｅｎｔｚ空间     

On the mutually non isomorphic spaces,II

This note is a companion to the article On the mutually non isomorphic spaces published in this journal, in which P. Cembranos and J. Mendoza showed that is a collection of mutually non isomorphic Banach spaces [5]. We now complete the picture by allowing the non‐locally convex relatives to be part of their natural family and see that, in fact, no two members of the extended class are isomorphic. Our approach is novel in the sense that we reach the isomorphism obstructions from the perspective of bases techniques and the different convexities of the spaces, both methods being intrinsic to quasi‐Banach spaces.