Orlicz Classes Research Articles

Transfer of Fourier Multipliers into Schur Multipliers and Sumsets in a Discrete Group

Abstract We inspect the relationship between relative Fourier multipliers on noncommutative Lebesgue– Orlicz spaces of a discrete group and relative Toeplitz-Schur multipliers on Schatten–von- Neumann–Orlicz classes. Four applications are given: lacunary sets, unconditional Schauder bases for the subspace of a Lebesgue space determined by a given spectrum , the norm of the Hilbert transformand the Riesz projection on Schatten–von-Neumann classes with exponent a power of 2, and the norm of Toeplitz Schur multipliers on Schatten–von-Neumann classes with exponent less than 1.

The atomic decompositions of weak martingale Orlicz spaces

In this paper,the weak-type and strong-type atomic decomposition theorems in several weak mar-tingale Orlicz spaces are established and the boundedness of sublinear operators on these martingale spaces are investigated. The weak Orlicz space is a class of quasi-normed spaces,these results generalize some results for classical Orlicz spaces and weak-type Lorentz spaces.

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].

Approximation by rational functions in Smirnov–Orlicz classes

Let G be a doubly-connected domain bounded by Dini-smooth curves. In this work, the approximation properties of the Faber–Laurent rational series expansions in Smirnov–Orlicz classes E M ( G ) are studied.

On a Phase-Field Model for Electrowetting and Other Electrokinetic Phenomena

In three space dimensions, we present existence results for weak solutions to a novel two-phase model for various electrokinetic phenomena, including in particular dynamic electrowetting with electrolytes. The model is thermodynamically consistent. It combines Navier–Stokes- and Cahn–Hilliard-type phase-field equations with Nernst–Planck equations for ion density-evolution and with an elliptic transmission problem for the electrostatic potential. As physical energy estimates guarantee only boundedness of ion densities in the $L\log L$-Orlicz class uniformly with respect to time, an iteration method is proposed to establish higher regularity and integrability results of these quantities. In an appendix, the derivation of the model is sketched.

Nonsquareness in Musielak‐Orlicz‐Bochner Function Spaces

The criteria for nonsquareness in the classical Orlicz function spaces have been given already. However, because of the complication of Musielak‐Orlicz‐Bochner function spaces, at present the criteria for nonsquareness have not been discussed yet. In the paper, the criteria for nonsquareness of Musielak‐Orlicz‐Bochner function spaces are given. As a corollary, the criteria for nonsquareness of Musielak‐Orlicz function spaces are given.

On isometric copies of [formula omitted] and James constants in Cesàro–Orlicz sequence spaces

For a wide class of Orlicz functions not satisfying the growth condition δ 2 we show that the Cesàro–Orlicz sequence spaces ces φ equipped with the Luxemburg norm contain an order linearly isometric copy of ℓ ∞ . We also compute the n-th James constant in these spaces for any Orlicz function φ, under either the Luxemburg or Orlicz norm, showing that they are equal to n for any natural n ≥ 2 . In particular, we prove that the non-trivial spaces ces φ are not B-convex for any Orlicz function φ.

Weighted integral inequalities for the maximal geometric mean operator in martingales

Under appropriate conditions on Young's functions Φ 1 and Φ 2 , we give necessary and sufficient conditions in order that weighted integral inequalities hold for the maximal geometric mean operator G in martingale Orlicz classes. When Φ 1 = t p and Φ 2 = t p , the inequalities revert to the ones of strong or weak ( p , p ) -type in martingale spaces.

New Orlicz–Hardy spaces associated with divergence form elliptic operators

Let L be the divergence form elliptic operator with complex bounded measurable coefficients, ω the positive concave function on ( 0 , ∞ ) of strictly critical lower type p ω ∈ ( 0 , 1 ] and ρ ( t ) = t − 1 / ω − 1 ( t − 1 ) for t ∈ ( 0 , ∞ ) . In this paper, the authors study the Orlicz–Hardy space H ω , L ( R n ) and its dual space BMO ρ , L * ( R n ) , where L * denotes the adjoint operator of L in L 2 ( R n ) . Several characterizations of H ω , L ( R n ) , including the molecular characterization, the Lusin-area function characterization and the maximal function characterization, are established. The ρ-Carleson measure characterization and the John–Nirenberg inequality for the space BMO ρ , L ( R n ) are also given. As applications, the authors show that the Riesz transform ∇ L − 1 / 2 and the Littlewood–Paley g-function g L map H ω , L ( R n ) continuously into L ( ω ) . The authors further show that the Riesz transform ∇ L − 1 / 2 maps H ω , L ( R n ) into the classical Orlicz–Hardy space H ω ( R n ) for p ω ∈ ( n n + 1 , 1 ] and the corresponding fractional integral L − γ for certain γ > 0 maps H ω , L ( R n ) continuously into H ω ˜ , L ( R n ) , where ω ˜ is determined by ω and γ, and satisfies the same property as ω. All these results are new even when ω ( t ) = t p for all t ∈ ( 0 , ∞ ) and p ∈ ( 0 , 1 ) .

Ergodic theorems for contractions in Orlicz-Kantorovich lattices

We obtain some versions of ergodic theorems for positive contractions in the Orlicz-Kantorovich lattices L M (m) associated with a measure m taking values in the algebra of measurable real functions. The proof is carried out by representing L M (m) as measurable bundles of classical Orlicz function spaces.

On a capacity for modular spaces

The purpose of this article is to define a capacity on certain topological measure spaces X with respect to certain function spaces V consisting of measurable functions. In this general theory we will not fix the space V but we emphasize that V can be the classical Sobolev space W 1 , p ( Ω ) , the classical Orlicz–Sobolev space W 1 , Φ ( Ω ) , the Hajłasz–Sobolev space M 1 , p ( Ω ) , the Musielak–Orlicz–Sobolev space (or generalized Orlicz–Sobolev space) and many other spaces. Of particular interest is the space V : = W ˜ 1 , p ( Ω ) given as the closure of W 1 , p ( Ω ) ∩ C c ( Ω ¯ ) in W 1 , p ( Ω ) . In this case every function u ∈ V (a priori defined only on Ω ) has a trace on the boundary ∂ Ω which is unique up to a Cap p , Ω -polar set.

INVERSE IMAGES OF SECTORS BY FUNCTIONS IN WEIGHTED BERGMAN–ORLICZ SPACES

AbstractFor ε>0, let Σε={z∈ℂ:∣arg z∣<ε}. It has been proved (D. E. Marshall and W. Smith,Rev. Mat. Iberoamericana15(1999), 93–116) that ∫f−1(Σε)∣f(z)∣ dA(z)≃∫𝔻∣f(z)∣ dA(z) for every ε>0, uniformly for every univalent functionfin the classical Bergman spaceA1that fixes the origin. In this paper, we extend this result to those conformal maps on 𝔻 belonging to weighted Bergman–Orlicz classes such thatf(0)=∣f′(0)∣−1=0.

Continuity properties of Riesz potentials of Orlicz functions

In this paper we are concerned with Sobolev type inequalities for Riesz potentials of functions in Orlicz classes. As an application, we study continuity properties of Riesz potentials.

$L^p$-integrability & weak type $L^{2}$-estimates for the gradient of harmonic mappings of $\mathbb D$

We give an elementary proof for the integrability properties of the gradient of the harmonic extension of a self homeomorphism of the circle giving explicit bounds for the $p$-norms, $p<2$, estimates in Orlicz classes and also an $L^2(\mathbb D)$-weak type estimate.

Vanishing exponential integrability for Riesz potentials of functions in Orlicz classes

Our aim in this paper is to show the vanishing exponential integrability for Riesz potentials of functions in Orlicz classes, as an improvement of continuity results of Sobolev functions. We also show the vanishing double exponential integrability.

Inequalities and properties of some generalized Orlicz classes and spaces

We discuss and complement the knowledge about generalized Orlicz classes \( \tilde X_\Phi \) and Orlicz spaces XΦ obtained by replacing the space L1 in the classical construction by an arbitrary Banach function space X. Our main aim is to focus on the task to study inequalities in such spaces. We prove a number of new inequalities and also natural generalizations of some classical ones (e.g., Minkowski’s, Holder’s and Young’s inequalities). Moreover, a number of other basic facts for further study of inequalities and function spaces are included.

Functions monotone close to boundary

Functions which are monotone close to boundary are defined. Some oscillation estimates are given for these functions in Orlicz classes. Criteria for monotonicity close to boundary are obtained.

Some notes on orlicz-bochner sequence spaces

In this paper, we discuss some basic properties of the Orlicz-Bochner sequence space l M (X) and its subspace h M (X). We present the equivalent definition of h M (X), the sufficient and necessary conditions under which l (M)(X) is complete, and l M (X) and h M (X) are separable respectively, and also give the sufficient condition that h M (X) has a basis. All these results generalize the results for the classical Orlicz sequence spaces.

On Asymptotics of Toeplitz Determinants with Symbols of Nonstandard Smoothness

We prove Szego’s strong limit theorem for Toeplitz determinants with a symbol having a nonstandard smoothness. We assume that the symbol belongs to the Wiener algebra and, moreover, the sequences of Fourier coefficients of the symbol with negative and nonnegative indices belong to weighted Orlicz classes generated by complementary N-functions both satisfying the Δ02-condition and by weight sequences satisfying some regularity and compatibility conditions.

THE CRITERIA OF WEAK GENERALIZED LOCALIZATION FOR MULTIPLE WALSH–FOURIER SERIES OF FUNCTIONS IN ORLICZ CLASSES

For the wide class of measurable sets [Formula: see text], [Formula: see text], N≥1, the criteria are found (in terms of structural and geometric characteristics of sets [Formula: see text] called [Formula: see text] and [Formula: see text] properties) for validity of the weak generalized localization almost everywhere (WGL) for multiple Walsh–Fourier series of functions equal zero on [Formula: see text], in the Orlicz classes Φ(L)(IN) "lying between" L1 and Lp, p&gt;1. In particular, it is found that in the class L( log +L)2WGL holds on the set [Formula: see text] iff [Formula: see text] has the [Formula: see text] property and in any class L( log + log +L)1-ε, 0&lt;ε&lt;1, WGL holds on [Formula: see text] iff [Formula: see text] has the [Formula: see text] property.