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In this paper, we prove a Hölder’s type inequality for fully measurable grand Lebesgue spaces, which involves the notion of fully measurable small Lebesgue spaces. It is proved that these spaces are non-reflexive rearrangement invariant Banach function spaces. Moreover, under certain continuity assumptions, along with several properties of fully measurable small Lebesgue spaces, we establish Levi’s theorem for monotone convergence and that grand and small spaces are associated to each other. Keywords: Banach function norm, Grand Lebesgue space, Associate space and Levi’s theorem

In the present paper we introduce and study some Orlicz–Lorentz sequence space over real n -normed space as a base space. We make an effort to study some algebraic and topological properties of the space. Some inclusion relations are also establish in the paper. Finally, we show that the space is complete and paranormed.

In this paper we consider some Banach spaces of analytic functions on the unit disk generated by the cone of analytic functions with monotone decreasing Taylor coefficients. We get that some of these spaces are Banach lattices with respect to this cone. Different ordered spaces of linear bounded operators acting between the previous spaces are also investigated, with emphasis on the so-called regular multipliers and Hankel operators.

It is well known that there exist hypercyclic composition operators on the Hardy spaces \(H^p\) for \(0 1\), all weighted Bergman spaces \(A^p_\beta \) with weight \(z\mapsto (1-|z|^2)^\beta \) (\(0 -1\)), and a class of weighted Dirichlet spaces, mimic the behavior of the Hardy spaces.

The embedding of the Sobolev spaces W p s (ℝ n ) in a Lizorkin-type space of locally summable functions of zero smoothness is established. This result is extended to the case of the embedding of Sobolev spaces on nonregular domains of n-dimensional Euclidean space. The formulation of the theorem depends on the geometric parameters of the domain of the functions.

The phase space of the closed string theory may be identified with the space of smooth loops. This reduces the problem of quantization of string theory to the quantization of the space of smooth loops. In this paper we describe the solution of the latter problem obtained in a series of papers. But the symplectic form of string theory is correctly defined not only on the space of smooth loops but also on its Hilbert completion coinciding with the Sobolev space of half-differentiable functions. So it is reasonable to consider this space as the phase manifold of non-smooth string theory. There is a natural group associated with this Sobolev space, namely the group of quasisymmetric homeomorphisms of the circle acting by change of variable. Unfortunately, this action is not smooth. However, we are able to quantize the Sobolev space of half-differentiable functions provided with the action of the group of quasisymmetric homeomorphisms using methods of noncommutative geometry.

The Gurari\u{\i} space is the unique separable Banach space $\mathbb{G}$ which is of almost universal disposition for finite-dimensional Banach spaces, which means that for every $\varepsilon>0$, for all finite-dimensional normed spaces $E \subseteq F$, for every isometric embedding ${e}\colon{E}\to{\mathbb{G}}$ there exists an $\varepsilon$-isometric embedding ${f}\colon{F}\to{\mathbb{G}}$ such that $f \restriction E = e$. We show that $\mathbb{G}^{\mathbb{N}}$ with a special sequence of semi-norms is of almost universal disposition for finite-dimensional graded Fr\'echet spaces. The construction relies heavily on the universal operator on the Gurari\u{\i} space, recently constructed by Garbuli\'nska-Wegrzyn and the third author. This yields in particular that $\mathbb{G}^{\mathbb{N}}$ is universal in the class of all separable Fr\'echet spaces., Comment: 15 pages

We describe a new class of spaces which contains the classical Morrey spaces. Using a new approach, we prove definitive versions of duality and interpolation theorems for these spaces. The results obtained here are also of interest for the classical Morrey spaces.

The aim of this paper is to obtain the description of the first and second complex interpolations of grand Lebesgue spaces. We also investigate the complex interpolation of closed subspaces satisfying the lattice property.

Let be the open unit disk in the complex plane and the class of all analytic functions on . Let be an analytic self-map of and . In this paper, the boundedness and compactness for product-type oper...

For , the unit ball of , we consider Bergman–Orlicz spaces of holomorphic functions in , which are generalizations of classical Bergman spaces. We characterize the dual space of large Bergman–Orlicz space, and bounded Hankel operators between some Bergman–Orlicz spaces and , where and are either convex or concave growth functions.

We study mappings inducing bounded composition operators on weighted Sobolev spaces on Carnot groups. We give an analytical description of such mappings in terms of the integrability of weighted distortion function. Bibliography: 17 titles.

We give an explicit formula to factorize functions in the Lorentz spaces L ( p , q ) . Some applications to centralizers, twisted sums and interpolation theory are included.

The semi-inner product, in the sense of Lumer, on weighted Hardy space which generate the norm is unique. Also we will discuss some properties of the numerical range of bounded linear operators on weighted Hardy spaces.

Let [Formula: see text] be a divergence form inhomogeneous higher order elliptic operator with complex bounded measurable coefficients. In this paper, for all [Formula: see text] and [Formula: see text] satisfying a weak ellipticity condition, the authors introduce the local Hardy spaces [Formula: see text] associated with [Formula: see text], which coincide with Goldberg’s local Hardy spaces [Formula: see text] for all [Formula: see text] when [Formula: see text] (the Laplace operator). The authors also establish a real-variable theory of [Formula: see text], which includes their characterizations in terms of the local molecules, the square functions or the maximal functions, the complex interpolation and dual spaces. These real-variable characterizations on the local Hardy spaces are new even when [Formula: see text] (the divergence form homogeneous second-order elliptic operator). Moreover, the authors show that [Formula: see text] coincides with the Hardy space [Formula: see text] associated with the operator [Formula: see text] for all [Formula: see text], where [Formula: see text] is some positive constant depending on the ellipticity and the off-diagonal estimates of [Formula: see text]. As an application, the authors establish some mapping properties for the local Riesz transforms [Formula: see text] on [Formula: see text], where [Formula: see text] and [Formula: see text].

We proved the noncommutative analogue of Calderon’s result for fully symmetric spaces \(E_1\) and \(E_2\) on (0, 1) and for a finite von Neumann algebra \({{\mathcal {M}}}\). We also proved the noncommutative symmetric Hardy space’s analogue of Calderon’s result for fully symmetric spaces and for finite subdiagonal subalgebras.

We introduce the weak Hardy-Morrey spaces in this paper. We also obtain the atomic decompositions of the weak Hardy-Morrey spaces. By using these decompositions, we establish the Hardy inequalities on the weak Hardy-Morrey spaces.

Mitsuo Izuki was partially supported by Grand-in-Aid for Scientific Research (C), No. 15K04928, for Japan Society for the Promotion of Science. Yoshihiro Sawano was partially supported by Grand-in-Aid for Scientific Research (C), No. 16K05209, for Japan Society for the Promotion of Science.

We exhibit a new class of Banach spaces Y such that the pair ( c 0 , Y ) has the Bishop–Phelps–Bollobas property for operators. This class contains uniformly convex Banach spaces and spaces with the property β of Lindenstrauss. We also provide new examples of spaces in this class.

In this paper we show the Jordan decomposition for bounded variation functions withvalues in Riesz spaces. Through an equivalence relation, we prove that this decomposition is satisfiedfor functions valued in Hilbert spaces. This result is a generalization of the real case. Moreover, weprove that, in general, the Jordan decomposition is not satisfied for vector-valued functions.

In this paper, we propose an extension of quasi-equilibrium problems from the convex case to the nonconvex case and from Hilbert spaces to Banach spaces. The proposed problem is called quasi-variational problem. We study the convergence of some algorithms to solutions of the proposed nonconvex problems in Banach spaces.

In this study, we introduce the norm-attaining property for a dual pair of Banach spaces. We use this property to determine a quotient Banach space as a dual space. We also apply this property to obtain another proof of the James's characterization theorem regarding the reflexivity of a separable Banach space. In addition, the norm-attaining property of a Fourier space is studied.

R0 space in L-topological spaces are defined and studied. The authors give eight definitions of R0 space in L-topological spaces and discuss certain relationship among them. They showed that all of these satisfy ‘good extension’ property. Moreover, some of their other properties are obtained.Journal of Bangladesh Academy of Sciences, Vol. 40, No. 2, 117-124, 2016

We extend the extrapolation theory to Morrey spaces associated with Banach function spaces. Some applications by this theory such as the Fefferman–Stein vector-valued maximal inequalities are obtained. By using this extrapolation theory, we obtain the John-Nirenberg inequalities on Morrey spaces associated with Banach function spaces. In addition, by using the John-Nirenberg inequalities, we have the characterizations of the function spaces of bounded mean oscillation in terms of Morrey spaces associated with Banach function spaces.

This paper investigates the solution of fractional dynamical systems with the inverse operator method and Mittag-Leffler function. Controllability of linear fractional dynamical systems is studied by obtaining the Grammian operator. Sufficient conditions for both nonlinear and integrodifferential systems are established by the contraction principle. Some examples are provided to illustrate the theory.

We study Sobolev inequalities on doubling metric measure spaces. We investigate the relation between Sobolev embeddings and lower bound for measure. In particular, we prove that if the Sobolev inequality holds, then the measure μ satisfies the lower bound, i.e. there exists b such that μ(B(x,r))≥b r α for r∈(0,1] and any point x from metric space.

We characterize the dual spaces of weak martingale Hardy–Lorentz–Karamata spaces. As an application, we obtain a weak type John–Nirenberg theorem when the stochastic basis is regular. We also extend the boundedness of fractional integrals to the setting of martingale Hardy–Lorentz–Karamata spaces.

The aim of this paper is twofold. We first establish the Besov spaces on metric spaces endowed with a doubling measure, via the remarkable orthonormal wavelet basis constructed recently by T. Hytonen and O. Tapiola, and characterize the dual spaces of these Besov spaces. Second, we prove the T1 type theorem for the boundedness of Calderon–Zygmund operators on these Besov spaces. Finally, we introduce a new class of Lipschitz spaces and characterize these spaces via the Littlewood–Paley theory. Copyright © 2016 John Wiley & Sons, Ltd.

In this article, the well-known conjugate gradient (CG) method for linear systems in Hilbert spaces is extended to a reflexive Banach space setting. In this setting, the Riesz isomorphism has to be replaced by the duality mapping. Due to the nonlinearity of the duality mapping, the short term recursion and conjugacy of search directions cannot be maintained simultaneously. The well-posedness of the proposed iteration and its global convergence are shown under appropriate conditions. Error bounds and stopping criteria are presented as well. The results extend to a limited-memory variant of the algorithm. The behavior of the method is demonstrated by numerical examples.

In this paper, we characterize boundedness and compactness of difference of composition operators from the space of Cauchy integral transforms to Bloch-type spaces. Our characterizations are free from pseudo-hyperbolic metric, which is a common feature of all the characterizations of difference of composition operators acting between different spaces of holomorphic functions. Exact value of operator norm of difference of composition operators acting between these spaces is also computed.

Motivated by some recent results on the boundedness and dynamical properties of the differentiation and integration operators in weighted Banach spaces of holomorphic functions, we study conditions on weights that guarantee the compactness of these two operators in the corresponding weighted spaces.