Smoothness of the Orlicz norm in Musielak–Orlicz function spaces

In this paper, we present a characterization of support functionals and smooth points in , the Musielak–Orlicz space equipped with the Orlicz norm. As a result, criterion for the smoothness of is also obtained. Some expressions involving the norms of functionals in , the topological dual of , are proved for arbitrary Musielak–Orlicz functions.     

Bilinear multipliers on Lorentz spaces

We give one sufficient and two necessary conditions for boundedness between Lebesgue or Lorentz spaces of several classes of bilinear multiplier operators closely connected with the bilinear Hilbert transform. The author has been partially supported by grants DGESIC PB98-1246 and BMF 2002-04013.     

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).     

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.     

A strict topology on Orlicz spaces

Let φ be a Young function, Ω be a locally compact space, and μ be a positive Radon measure on Ω. We consider a strict topology (in the sense of Sentilles‐Taylor) on the Orlicz function space and investigate various properties of this locally convex topology. We also study the Orlicz space of a locally compact group G with a left Haar measure under the strict topology and certain other natural locally convex topologies. Finally we present some results on various continuity properties of convolution operators on under the topology and other natural ones.     

Korn type inequalities in Orlicz spaces

We exhibit balance conditions between a Young function A and a Young function B   for a Korn type inequality to hold between the L^B$L^B$ norm of the gradient of vector-valued functions and the L^A$L^A$ norm of its symmetric part. In particular, we extend a standard form of the Korn inequality in L^p$L^p$, with 1<p<∞$1<p<\infty$, and an Orlicz version involving a Young function A   satisfying both the _Δ2${\mathrm{\Delta }}_2$ and the _∇2${\mathrm{\nabla }}_2$ condition.     

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.     

Some fundamental properties for duals of Orlicz spaces

In this paper some basic properties of Orlicz spaces are extended to their dual spaces, and finally, criteria for extreme points in these spaces are given.     

Orlicz capacities and Hausdorff measures on metric spaces

In the setting of doubling metric measure spaces with a 1-Poincaré inequality, we show that sets of Orlicz Φ-capacity zero have generalized Hausdorff h-measure zero provided thatwhere Θ⁻¹ is the inverse of the function Θ(t)=Φ(t)/t, and s is the “upper dimension” of the metric measure space. This condition is a generalization of a well known condition in Rⁿ. For spaces satisfying the weaker q-Poincaré inequality, we obtain a similar but slightly more restrictive condition. Several examples are also provided.     

On non-effective weights in Orlicz spaces

Given a weight w in Ω ⊂ ∝^N, |Ω| < ∞ and a Young function φ, we consider the weighted modular ∫_Ω ω(f(x))w(x)dx and the resulting weighted Orlicz space L_ω(w). For a Young function Ω ∉ Δ₂(∞) we present a necessary and sufficient conditions in order that L_ω(w) = L_ω(X_Ω) up to the equivalence of norms. We find a necessary and sufficient condition for ω in order that there exists an unbounded weight w such that the above equality of spaces holds. By way of applications we simplify criteria from [5] for continuity of the composition operator from L_ω into itself when ω Δ₂(∞) and obtain necessary and sufficient condition in order that the composition operator maps L_ω. continuously onto L_ω.     

Rearrangement transformations on general measure spaces

For a general set transformation R between two measure spaces, we define the rearrangement of a measurable function by means of the Layer's cake formula. We study some functional properties of the Lorentz spaces defined in terms of R, giving a unified approach to the classical rearrangement, Steiner's symmetrization, the multidimensional case, and the discrete setting of trees.     

Multiplication operators on vector measure Orlicz spaces

Let m be a countably additive vector measure with values in a real Banach space X, and let L¹(m) and L_w(m) be the spaces of functions which are, correspondingly, integrable and weakly integrable with respect to m. Given a Young's function Φ, we consider the vector measure Orlicz spaces L^Φ(m) and L^Φ_w(m) and establish that the Banach space of multiplication operators going from W = L^Φ(m) into Y = L¹ (m) is M = L^Ψ_w (m) with an equivalent norm; here Ψ is the conjugated Young's function for Φ. We also prove that when W = L^Φ_w(m), Y = L¹(m) we have M = L^Ψ_w (m), and when W = L^Φ_w(m), Y = L¹(m) we have M = L^Ψ (m).     

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.     

Criteria of weighted inequalities in Orlicz classes for maximal functions defined on homogeneous type spaces

The necessary and sufficient conditions are derived in order that a strong type weighted inequality be fulfilled in Orlicz classes for scalar and vector-valued maximal functions defined on homogeneous type spaces. A weak type problem with weights is solved for vector-valued maximal functions.     

Tractable embeddings of Besov spaces into small Lebesgue spaces

This paper deals with dimension‐controllable (tractable) embeddings of Besov spaces on n‐dimensional torus into small Lebesgue spaces. Our techniques rely on the approximation structure of Besov spaces, extrapolation properties of small Lebesgue spaces and interpolation.     

Generalized perfect spaces

Given two Banach function spaces X and Y related to a measure μ, the Y-dual space X^Y of X is defined as the space of the multipliers from X to Y. The space X^Y is a generalization of the classical Köthe dual space of X, which is obtained by taking Y = L^t(μ). Under minimal conditions, we can consider the Y-bidual space X^YY of X (i.e. the Y-dual of X^Y). As in the classical case, the containment X ⊂ X^YY always holds. We give conditions guaranteeing that X coincides with X^YY, in which case X is said to be Y-perfect. We also study when X is isometrically embedded in X^YY. Properties involving p-convexity, p-concavity and the order of X and Y, will have a special relevance.     

Fourier multipliers of classical modulation spaces

Based on the observation that translation invariant operators on modulation spaces are convolution operators we use techniques concerning pointwise multipliers for generalized Wiener amalgam spaces in order to give a complete characterization of the Fourier multipliers of modulation spaces. We deduce various applications, among them certain convolution relations between modulation spaces, as well as a short proof for a generalization of the main result of a recent paper by Bènyi et al., see [À. Bènyi, L. Grafakos, K. Gröchenig, K.A. Okoudjou, A class of Fourier multipliers for modulation spaces, Appl. Comput. Harmon. Anal. 19 (1) (2005) 131–139]. Finally, we show that any function with ([d/2]+1)-times bounded derivatives is a Fourier multiplier for all modulation spaces with p(1,∞) and q[1,∞].