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.     

Calderón--Lozanovskii; construction for mixed norm spaces

We show that the Calderón--Lozanovskii; construction φ(.) commutes with arbitrary mixed norm spaces, that is, φ(E₀[F₀], E₁[F₁]) = φ(E₀, E₁) [φ(F₀, F₁)] if and only if φ is equivalent to a power function. This result we obtain by giving characterizations of the corresponding embeddings of φ(E₀[F₀], E₁[F₁]) into φ₀ (E₀, E₁)[φ₁ (F₀, F₁)] and vice versa in terms of the functions φ, φ₀, φ₁. As a particular case, we get embeddings of an Orlicz space with mixed norms into an Orlicz space on a product of measure spaces. Applications to classical operators between mixed norm Orlicz spaces are also discussed. This revised version was published online in June 2006 with corrections to the Cover Date.     

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.     

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)     

M‐constants in Orlicz–Lorentz function spaces

In this paper some lower and upper estimates of M‐constants for Orlicz–Lorentz function spaces for both, the Luxemburg and the Amemiya norms, are given. Since degenerated Orlicz functions φ and degenerated weighted sequences ω are also admitted, this investigations concern the most possible wide class of Orlicz–Lorentz function spaces. M‐constants were defined in 1969 by E. A. Lifshits, and used by many authors in the study of lattice structures on Banach spaces, as well as in the fixed point theory.     

Normkonvergenz von Fourierreihen in rearrangement invarianten Banachräumen

This paper studies rearrangement invariant Banach spaces of 2π-periodic functions with respect to norm convergence of Fourier series. The main result is that norm convergence takes place if and only if the space is an interpolation space of $\text{(}\text{L}{}^{\mathrm{p\prime }}\text{(T),}\text{L}{}^{\mathrm{p}}\text{(T)), 1 < p < 2,}\text{1}\text{p′}\text{+}\text{1}\text{p}\text{= 1}$, and L^p′(T) is dense in it (compare Satz 2.8). Since norm convergence and continuity of the conjugation operator are closely connected (compare Satz 2.2), this is achieved by a careful examination of this operator similar to that of D. W. Boyd for the Hilbert transform on the whole real axis. Finally, there are applications to Orlicz and Lorentz spaces.     

Best Local Approximation in Orlicz Spaces

We get results in Orlicz spaces L ^φ about best local approximation on non-balanced neighborhoods when φ satisfies a certain asymptotic condition. This fact generalizes known previous results in L ^p spaces.     

Complex rotundity of Orlicz sequence spaces equipped with the p-Amemiya norm

In this paper, two equivalent definitions of complex strongly extreme points in general complex Banach spaces are shown. It is proved that for any Orlicz sequence space equipped with the p-Amemiya norm (1?p<∞, p is odd), complex strongly extreme points of the unit ball coincide with complex extreme points of the unit ball. Moreover, criteria for them in Orlicz sequence spaces equipped with the p-Amemiya norm are given. Criteria for complex mid-point locally uniform rotundity and complex rotundity of Orlicz sequence spaces equipped with the p-Amemiya norm are also deduced.     

A minimizing property that characterizes Lp-spaces

It is shown that an approximative property with respect to Orlicz or Luxemburg norms in Orlicz spaces, useful for computing best approximants from some class of functions, is generally satisfied only for the L^p-norms, whenever the measure space contains at least four pairwise disjoint sets of finite and positive measure.     

Best approximation with wavelets in weighted Orlicz spaces

Democracy functions of wavelet admissible bases are computed for weighted Orlicz Spaces L ^??(w) in terms of the fundamental function of L ^??(w). In particular, we prove that these bases are greedy in L ^??(w) if and only if L ^??(w) =?L ^p (w), 1?<?p?<???. Also, sharp embeddings for the approximation spaces are given in terms of weighted discrete Lorentz spaces. For L ^p (w) the approximation spaces are identified with weighted Besov spaces.     

On interpolation of function spaces by methods defined by means of polygons

We describe the spaces obtained by applying the interpolation methods associated to polygons to N-tuples of weighted L^p-spaces, N-tuples of classical Lorentz spaces and some other N-tuples of function spaces.     

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.     

Second‐order elliptic and parabolic equations with BMO coefficients in the whole space

In this paper, we obtain the global regularity estimates in Orlicz spaces for second‐order divergence elliptic and parabolic equations with BMO coefficients in the whole space. In fact, the global result can follow from the local estimates. As a corollary we obtain L^p‐type regularity estimates for such equations. Copyright © 2011 John Wiley & Sons, Ltd.     

Minimal projections in spaces of functions of N variables

We will construct a minimal and co-minimal projection from L^p([0,1]ⁿ) onto L^p([0,1]^n₁)++L^p([0,1]^n_k), where n=n₁++n_k (see Theorem 2.9). This is a generalization of a result of Cheney, Halton and Light from (Approximation Theory in Tensor Product Spaces, Lecture Notes in Mathematics, Springer, Berlin, 1985; Math. Proc. Cambridge Philos. Soc. 97 (1985) 127; Math. Z. 191 (1986) 633) where they proved the minimality in the case n=2. We provide also some further generalizations (see Theorems 2.10 and 2.11 (Orlicz spaces) and Theorem 2.8). Also a discrete case (Theorem 2.2) is considered. Our approach differs from methods used in [8,13,20].     

Approximation by means of nonlinear Kantorovich sampling type operators in Orlicz spaces

In this paper we introduce a nonlinear version of the Kantorovich sampling type series in a nonuniform setting. By means of the above series we are able to reconstruct signals (functions) which are continuous or uniformly continuous. Moreover, we study the problem of the convergence in the setting of Orlicz spaces: this allows us to treat signals which are not necessarily continuous. Our theory applies to L^p-spaces, interpolation spaces, exponential spaces and many others. Several graphical examples are provided.     

On the Orlicz function spacesL M (0, ∞)

The isomorphic properties of the Orlicz function spacesL _M (0, ∞) are investigated. Especially we treat the question, whether theL _p-spaces are the only symmetric function spaces on (0, ∞), which are isomorphic to a symmetric function space on (0, 1). For the class of slowly varying Orlicz functions we answer this in the affirmative, and we also prove some results concerning the general case, which indicate, that it might be true there also.     

On orlicz sequence spaces III

It is proved that the set ofp's such thatl _p is isomorphic to a subspace of a given Orlicz spacel _Fforms an interval. Some examples and properties of minimal Orlicz sequence spaces are presented. It is proved that an Orlicz function space (different froml ₂) is not isomorphic to a subspace of an Orlicz sequence space. Finally it is shown (under a certain restriction) that if two Orlicz function spaces are isomorphic, then they are identical (i.e. consist of the same functions).     

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.     

On the embedding of BV space into Besov-Orlicz space

We give a sufficient (and, in the case of a compact domain, a necessary) condition for the embedding of Sobolev space of functions with integrable gradient into Besov-Orlicz spaces to be bounded. The condition has a form of a simple integral inequality involving Young and weight functions. We provide an example with Matuszewska-Orlicz indices of involved Orlicz norm equal to one. The main tool is the molecular decomposition of functions from a BV space.