Orlicz Function Spaces Research Articles

Inequalities with weights for maximal functions in orlicz spaces

Inequalities with weights for maximal functions in orlicz spaces

On a converse inequality for maximal functions in Orlicz spaces

Let $Φ(t) = ʃ_{0}^{t} a(s)ds$ and $Ψ(t) = ʃ_{0}^{t} b(s)ds$, where a(s) is a positive continuous function such that $ʃ_{1}^{∞} a(s)/s ds = ∞$ and b(s) is quasi-increasing and $lim_{s→∞}b(s) = ∞$. Then the following statements for the Hardy-Littlewood maxi

Disjointly strictly singular operators and interpolation

Interpolation properties of the class of disjointly strictly singular operators on Banach lattices are studied. We also give some applications to compare the lattice structure of two rearrangement invariant function spaces. In particular, we obtain suitable analytic characterisations of when the inclusion map between two Orlicz function spaces is disjointly strictly singular.

Strongly Extreme Points in Orlicz Function Spaces

For any Orlicz function Φ and any σ-finite atomless measure μ, a criterion for x from the unit sphere of the Orlicz space L Φ(μ), equipped with the Luxemburg norm, to be strongly extreme is given. Further, Orlicz spaces L Φ(μ) which are isometric to L ∞(μ) are characterized.

On quasi ϕ(L)*-a.e. convergence of Fourier series of functions in Orlicz spaces

On quasi ϕ(L)*-a.e. convergence of Fourier series of functions in Orlicz spaces

Universal classes of Orlicz function spaces

It is shown that for each 0 < p < q < oo the space 1/(0, oo) + L9(09 oo), defined as in Interpellation Theory, is universal for the class of all Orlicz function spaces Lψ with Boyd indices strictly between p and q (i.e. every Orlicz function space Lψ is orderisomorphically embedded into L p (0, oo) + Lq(0, oo)). The extreme case of spaces having Boyd indices equal to p or q is also studied. In particular every space Z/(0, oo) + Ls(0, oo) embeds isomorphically into the sum Lp(0, oo) +Lq(0, oo) for any 0</;<r<5<^<oo. 0. Introduction. It is a well-known fact from Interpolation Theory

On uniformly bounded sequences in Orlicz spaces

A useful result for uniformly bounded sequences of functions in Orlicz spaces is generalised by means of a recent extension of Komlós' theorem. The same generalisation can also be proven differently, by means of Young measures.

On (V*) sets and Pelczynski's property (V*)

The concept of (V*) set was introduced, as a dual companion of that of (V)-set, by Pelczynski in his important paper [14]. In the same paper, the so called properties (V) and (V*) are defined by the coincidence of the (V) or (V*) sets with the weakly relatively compact sets. Many important Banach space properties are (or can be) defined in the same way; that is, by the coincidence of two classes of bounded sets. In this paper, we are concerned with the study of the class of (V*) sets in a Banach space, and its relationship with other related classes. To this general study is devoted Section I. A (as far as we know) new Banach space property (we called it property weak (V*)) is defined, by imposing the coincidence of (V*) sets and weakly conditionally compact sets. In this way, property (V*) is decomposed into the conjunction of the weak (V*) property and the weak sequential completeness. In Section II, we specialize to the study of (V*) sets in Banach lattices. The main result in the section is that every order continuous Banach lattice has property weak (V*), which extends previous results of E. and P. Saab ([16]). Finally, Section III is devoted to the study of (V*) sets in spaces of Bochner integrable functions. We characterize a broad class of (V*) sets in L1(μ, E), obtaining similar results to those of Andrews [1], Bourgain [6] and Diestel [7] for other classes of subsets. Applications to the study of properties (V*) and weak (V*) are obtained. Extension of these results to vector valued Orlicz function spaces are also given.

Finite representability ofl p(X) in Orlicz function spaces

We show that iflp(X),p ≠ 2, is finitely crudely representable in an Orlicz spaceLϕ (which does not containc0) then the Banach spaceX is isomorphic to a subspace ofLp. The same remains true forp = 2 whenLϕ is 2-concave or 2-convex, or ifX has local unconditional structure. We extend a theorem of Guerre and Levy to Orlicz function spaces.

On minimality and L-complemented subspaces of Orlicz function spaces

Several properties of Use class of minimal Orlicz flhnclion spaces L’~ are described. la panicular, an expliciily defined class of non-trivial mininial fiinctions is showed, which providesconcrete examples ofOrlicz spaces without cornplemented copies of P-spaces. A classicai tepic in Banach spaces is tite study of tite existence of l’-compiemented subspaces. It is weii-knewn titat frern tite existence of l~-subspaces in a Banach space E does not follow that E contains a complementad cepy of sorne l;-space (1 ). Recali titat a functien Fis minimal at co ([H-PJ) ifE;,= E~ for every function GcE~,, where E;, is tite cempact subset of tite continuous function space C[0,cc) defined by

Weighted inductive and projective limits of normed Köthe function spaces

We consider locally convex inductive and projective limits of weighted normed Köthe function spaces of.measurable functions on a σ-finite measure space (X, Σ,μ). This general framework includes inductive and projective limits of weighted Lp-spaces, 1 ≤ p ≤ ∞, Lorentz spaces L p,q 1 < p < ∞, 1 ≤ q ≤ ∞, and Orlicz function spaces. In the projective case we characterize the quasinormability of L p(A) in terms of the increasing sequence A = (a n)n∈ℕ of weights on X. For the most important normed Köthe function spaces Lp, like Lp-spaces or L p,q-spaces, this condition on A = (an)n∈ℕ (V = (v n = 1/an)n∈ℕ) allows a useful projective description of the inductive limit topology of k p(V) by an associated system ofweights. A characterization of the Montel spaces L P(A) in terms of A = (an)n∈ℕ is also given. As an application we obtain functional analytic results on weighted inductive limits of spaces of holomorphic and harmonic functions.

On approximation by polynomials and rational functions in Orlicz spaces

Для пространств Орли ча получен аналог изв естного неравенства С. Б. Стечк ина об оценке наименьших по линомиальных уклоне ний через модуль гладкости про извольного порядка. Например, еслиƒ∈L* Φ (I), то $$R_n (f,I)_\Phi \leqq E(f,I)_\Phi \leqq C(\Phi ,r)\omega _r \left( {\tfrac{1}{n},f,I} \right)_\Phi $$ при всех натуральныхr иn≧r (теорема I). Доказана неулучшаем ость этой теоремы, ее а налог для случая приближения т ригонометрическими полиномами и тригоно метрическими рацион альными функциями. Установлена связьΔ 2-условия на функциюΦ(u) со свойствами аппрокси мации соответствующ их классов функций (теор ема 3).

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).

Some properties of functions in Orlicz space

For functions in Orlicz space L M * , we study the behavior of $$\mathop \smallint \nolimits_0^\tau x^*$$ (t)dt, where x* (t) is non-increasing and equimeasurable with ¦ x(t) ¦. We establish the existence of unbounded functions in L M * that are not limits of bounded functions and for which $$\mathop \smallint \nolimits_0^\tau x^* (t)dt = o(\tau M^{ - 1} (1/\tau ))$$ . Moreover, we establish a new criterion for an N-function to belong to the class Δ2 and a sufficiency test for a function to belong to Orlicz space.

Approximation of periodic functions in Orlicz spaces

Approximation of periodic functions in Orlicz spaces

On extremum problems for Orlicz spaces of functions analytic in the unit circle

The abstracts (in two languages) can be found in the pdf file of the article.&#x0D; Original author name(s) and title in Russian and Lithuanian:&#x0D; В. М. Терпигорева. Об экстремальных задачах для классов Орлича аналитических функций в единичном круге&#x0D; V. Terpigoreva. Ekstremaliniai uždaviniai analizinių vienetiniame skritulyje funkcijų Orličo klasėse