Some kind of Bishop–Phelps–Bollobás property

In this paper we introduce two Bishop–Phelps–Bollobás type properties for bounded linear operators between two Banach spaces X and Y : property 1 and property 2. These properties are motivated by a Kim–Lee result which states, under our notation, that a Banach space X is uniformly convex if and only if the pair satisfies property 2. Positive results of pairs of Banach spaces satisfying property 1 are given and concrete pairs of Banach spaces failing both properties are exhibited. A complete characterization of property 1 for the pairs is also provided.     

Quantification of Pełczyński's property (V)

A Banach space X has Pełczyński's property (V) if for every Banach space Y every unconditionally converging operator is weakly compact. In 1962, Aleksander Pełczyński showed that spaces for a compact Hausdorff space K enjoy the property (V), and some generalizations of this theorem have been proved since then. We introduce several possibilities of quantifying the property (V). We prove some characterizations of the introduced quantitative versions of this property, which allow us to prove a quantitative version of Pelczynski's result about spaces and generalize it. Finally, we study the relationship of several properties of operators including weak compactness and unconditional convergence, and using the results obtained we establish a relation between quantitative versions of the property (V) and quantitative versions of other well known properties of Banach spaces.     

A new Brézis‐Gallouët‐Wainger inequality from the viewpoint of the real interpolation functors

The Brézis‐Gallouët‐Wainger inequality describes a subtle embedding property into . The relation between the Brézis‐Gallouët‐Wainger inequality and the real interpolation functor together with the sharpness of the results is discussed in the present paper. As our first main results shows, it turns out that there are two intermediate terms between and the logarithmic boundedness, which is supposed to be the right‐hand side of the Brézis‐Gallouët‐Wainger inequality. As the second result, the first result is extended to inequalities which reflect the meaning of the second index of Besov spaces and the interpolation theorem.     

Musielak–Orlicz–Bochner function spaces which are uniformly noncreasy

In this paper, the criteria for uniform noncreasiness of Musielak–Orlicz–Bochner function spaces are given. Moreover authors also prove that the space (resp ) is uniformly noncreasy if and only if the space (resp ) is uniformly convex or uniformly smooth. As a corollary, the criteria for uniform noncreasiness of Musielak–Orlicz function spaces are given.     

Characterization of Triebel–Lizorkin type spaces with variable exponents via maximal functions,local means and non‐smooth atomic decompositions

In this paper we study the maximal function and local means characterizations and the non‐smooth atomic decomposition of the Triebel–Lizorkin type spaces with variable exponents . These spaces were recently introduced by Yang et al. and cover the Triebel–Lizorkin spaces with variable exponents as well as the classical Triebel–Lizorkin spaces , even the case when . Moreover, covered by this scale are also the Triebel–Lizorkin‐type spaces with constant exponents which, in turn cover the Triebel–Lizorkin–Morrey spaces. As an application we obtain a pointwise multiplier assertion for those spaces.     

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.     

Nuclear embeddings in function spaces

Let be the usual Besov spaces in bounded Lipschitz domains Ω in (bounded intervals if ). The paper clarifies under which conditions the continuous embedding between two such spaces with is nuclear.     

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.     

Strict topologies on measure spaces

Let X be a measurable space, let be a family of measurable subsets of it, and let be a subspace of complex measures on X that is also closed under restrictions of measures. In this paper we introduce the ‐convergence topology and the ‐strict topology on . Among other results, we find necessary and sufficient conditions for Hausdorff‐ness and coincide‐ness of these topologies. Applications to Lebesgue spaces, and also examples in Hausdorff topological spaces and locally compact groups are given.     

Limit cases of reiteration theorems

We consider real interpolation methods defined by means of slowly varying functions and rearrangement invariant spaces, for which we present a collection of reiteration theorems for interpolation and extrapolation spaces. As an application we obtain interpolation formulas for Lorentz‐Karamata type spaces, for Zygmund spaces , and for the grand and small Lebesgue spaces.     

Caffarelli–Kohn–Nirenberg inequalities on Lie groups of polynomial growth

In the setting of a Lie group of polynomial volume growth, we derive inequalities of Caffarelli–Kohn–Nirenberg type, where the weights involved are powers of the Carnot–Caratheodory distance associated with a fixed system of vector fields which satisfy the Hörmander condition. The use of weak spaces is crucial in our proofs and we formulate these inequalities within the framework of Lorentz spaces (a scale of (quasi)‐Banach spaces which extend the more classical Lebesgue spaces) thereby obtaining a refinement of, for instance, Sobolev and Hardy–Sobolev inequalities.     

Maximal and fractional integral operators on generalized Morrey spaces over metric measure spaces

We establish the boundedness and weak boundedness of the maximal operator and generalized fractional integral operators on generalized Morrey spaces over metric measure spaces without the assumption of the growth condition on μ. The results are generalization and improvement of some known results. We also give the vector‐valued boundedness. Moreover we prove the independence of the choice of the parameter in the definition of generalized Morrey spaces by using the geometrically doubling condition in the sense of Hytönen.     

Isomorphic classification of mixed sequence spaces and of Besov spaces over [0, 1]d

The aim of this paper is to establish the isomorphic classification of Besov spaces over [0, 1]^d . Using the identification of the Besov space with the ‐infinite direct sum of finite‐dimensional spaces (which holds independently of the dimension and of the smoothness degree of the space ) we show that , , is a family of mutually non‐isomorphic spaces. The only exception is the isomorphism between the spaces and , which follows from Pełczyński's isomorphism between and . We also tell apart the isomorphic classes of spaces from the isomorphic classes of Besov spaces over the Euclidean space .     

Reduction theorems for Sobolev embeddings into the spaces of Hölder,Morrey and Campanato type

Let X be a rearrangement‐invariant Banach function space on Q where Q is a cube in and let be the Sobolev space of real‐valued weakly differentiable functions f satisfying . We establish a reduction theorem for an embedding of the Sobolev space into spaces of Campanato, Morrey and Hölder type. As a result we obtain a new characterization of such embeddings in terms of boundedness of a certain one‐dimensional integral operator on representation spaces.     

Sobolev‐type inequalities for potentials in grand variable exponent Lebesgue spaces

We introduce a new scale of grand variable exponent Lebesgue spaces denoted by . These spaces unify two non‐standard classes of function spaces, namely, grand Lebesgue and variable exponent Lebesgue spaces. The boundedness of integral operators of Harmonic Analysis such as maximal, potential, Calderón–Zygmund operators and their commutators are established in these spaces. Among others, we prove Sobolev‐type theorems for fractional integrals in . The spaces and operators are defined, generally speaking, on quasi‐metric measure spaces with doubling measure. The results are new even for Euclidean spaces.     

Some characterizations of BLO space

For , a real‐valued function belongs to space if In this paper, we establish a version of John–Nirenberg inequality suitable for the space with . As a corollary, it is proved that spaces are independent of the scale in sense of norm. Also, we characterize the space through weighted Lebesgue spaces and variable Lebesgue spaces, respectively.     

How do the positive embeddings of Banach lattices depend on the αth derivatives of K?

Let K and S be locally compact Hausdorff spaces and X be an abstract space. Suppose that T is a positive Banach lattice isomorphism from into . Then for each ordinal α the cardinalities of the αth derivatives and satisfy the following inequality Moreover, if then is a continuous image of a subset of which can be taken closed when K is compact. The first statement of this result for is a vector‐valued extension of a Cengiz's theorem and the second one is vector‐valued version of a Holsztyński's theorem. A simple example shows that the number is sharp in these vector‐valued theorems.     

Commutativity of the Valdivia–Vogt table of representations of function spaces

In this article, we show that the Valdivia–Vogt structure table—containing the sequence space representations of the most used spaces of smooth functions appearing in the theory of distributions—can be interpreted as a commutative diagram, i.e., there is an isomorphism between the space of infinitely differentiable functions and the space , where s is the space of rapidly decreasing sequences, such that its restriction to the other function spaces in the structure table yields an isomorphism between these spaces of smooth functions and their sequence space representation. This result answers the corresponding question of Prof. Dietmar Vogt formulated on the conference “Functional Analysis: Applications to Complex Analysis and Partial Differential Equations” held in B?dlewo in May 2012.     

A mean value theorem for metric spaces

We present a form of the Mean Value Theorem (MVT) for a continuous function f between metric spaces, connecting it with the possibility to choose the relation of f in a homeomorphic way. We also compare our formulation of the MVT with the classic one when the metric spaces are open subsets of Banach spaces. As a consequence, we derive a version of the Mean Value Propriety for measure spaces that also possesses a compatible metric structure.     

Compact embeddings of Sobolev spaces with power weights perturbed by slowly varying functions

We study compact embeddings of weighted Sobolev spaces into Lebesgue spaces on the unit ball in . The weight is of slowly varyingly disturbed polynomial growth with a singularity at the origin. It extends 21 , 27 to a wider class of weights. Special attention is paid to the influence of the growth rate of the weight on the quality of compactness, measured in terms of entropy and approximation numbers. In case of Hilbert spaces, the results are related to the distribution of eigenvalues of some degenerate elliptic operators.