Localization with respect to a class of maps II — Equivariant cellularization and its application

We present an example of a homotopical localization functor which is not a localization with respect to any set of maps. Our example arises from equivariant homotopy theory. The technique of equivariant cellularization is developed and applied to the proof of the main result.     

Random Fourier series with respect to a general orthonormal system inL ∞(μ) spaces

The aim of this work is to study Fourier series with respect to general orthonormal systems inL () spaces from the point of view of random unconditional convergence     

Unconditional convergence of Fourier series with respect to the haar system in the spaces Λ ω p

Criteria for a Haar system to be a basic system and an unconditional basic system in the spaces $$\Lambda _\omega ^p = \left\{ {f \in L^p :\omega _p \left( {\delta , f} \right) = O\left\{ {\omega \left( \delta \right)} \right\}} \right\}$$ , where 1ω is a modulus of continuity, are proved.     

On the conditional expectation with respect to a sequence of independent σ-fields

Summary In the paper we characterize those sequences of random variables which are conditional expectations of a p-integrable random variable with respect to a given sequence of independent -fields.     

Weak Hardy-type spaces associated with ball quasi-Banach function spaces I: Decompositions with applications to boundedness of Calderón-Zygmund operators

Let X be a ball quasi-Banach function space on R~n. In this article, we introduce the weak Hardytype space W H_X(R~n), associated with X, via the radial maximal function. Assuming that the powered HardyLittlewood maximal operator satisfies some Fefferman-Stein vector-valued maximal inequality on X as well as it is bounded on both the weak ball quasi-Banach function space W X and the associated space, we then establish several real-variable characterizations of W H_X(R~n), respectively, in terms of various maximal functions,atoms and molecules. As an application, we obtain the boundedness of Calderón-Zygmund operators from the Hardy space H_X(R~n) to W H_X(R~n), which includes the critical case. All these results are of wide applications.Particularly, when X := M_q~p(R~n)(the Morrey space), X := L~p(R~n)(the mixed-norm Lebesgue space) and X :=(E_Φ~q)_t(R~n)(the Orlicz-slice space), which are all ball quasi-Banach function spaces rather than quasiBanach function spaces, all these results are even new. Due to the generality, more applications of these results are predictable.     

On the extension of Hölder maps with values in spaces of continuous functions

We study the isometric extension problem for Hölder maps from subsets of any Banach space intoc ₀ or into a space of continuous functions. For a Banach spaceX, we prove that anyα-Hölder map, with 0<α ≤1, from a subset ofX intoc ₀ can be isometrically extended toX if and only ifX is finite dimensional. For a finite dimensional normed spaceX and for a compact metric spaceK, we prove that the set ofα’s for which allα-Hölder maps from a subset ofX intoC(K) can be extended isometrically is either (0, 1] or (0, 1) and we give examples of both occurrences. We also prove that for any metric spaceX, the above described set ofα’s does not depend onK, but only on finiteness ofK.     

On the relation between Fourier and Leont’ev coefficients with respect to smirnov spaces

Yu. Melnik showed that the Leontev coefficients _f () in the Dirichlet series 2}}$$ " align="middle" border="0"> of a function f E ^p (D), 1 < p < , are the Fourier coefficients of some function F L ^p , ([0, 2]) and that the first modulus of continuity of F can be estimated by the first moduli and majorants in f. In the present paper, we extend his results to moduli of arbitrary order.Published in Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 4, pp. 517–526, April, 2004.     

Associate space with respect to a locally $$\sigma $$-finite measure on a $$\delta $$δ-ring and applications to spaces of integrable functions defined by a vector measure

We show that for a locally \(\sigma \)-finite measure \(\mu \) defined on a \(\delta \)-ring, the associate space theory can be developed as in the \(\sigma \)-finite case, and corresponding properties are obtained. Given a saturated \(\sigma \)-order continuous \(\mu \)-Banach function space E, we prove that its dual space can be identified with the associate space \(E ^\times \) if, and only if, \(E^\times \) has the Fatou property. Applying the theory to the spaces \(L^p (\nu )\) and \(L_w^p (\nu )\), where \(\nu \) is a vector measure defined on a \(\delta \)-ring \(\mathcal {R}\) and \(1 \le p < \infty \), we establish results corresponding to those of the case when the vector measure is defined on a \(\sigma \)-algebra.     

AL- and AM-spaces of integrable scalar functions with respect to a Fréchet-valued measure

We study the question of determining conditions for the space of R-valued integrable functions with respect to a vector measure taking its values in a real Fréchet space to be an AL- or an AM-space.     

Differential as a harmonic morphism with respect to Cheeger–Gromoll-type metrics

We investigate horizontal conformality of a differential of a map between Riemannian manifolds, where the tangent bundles are equipped with Cheeger–Gromoll-type metrics. As a corollary, we characterize the differential of a map as a harmonic morphism.     

A characterization of tribes with respect to the Łukasiewicz t-norm

We give a complete characterization of tribes with respect to the ukasiewicz t-norm, i. e., of systems of fuzzy sets which are closed with respect to the complement of fuzzy sets and with respect to countably many applications of the ukasiewicz t-norm. We also characterize all operations with respect to which all such tribes are closed. This generalizes the characterizations obtained so far for other fundamental t-norms, e. g., for the product t-norm.     

On the Kluvánek construction of the Lebesgue integral with respect to a vector measure

The Kluvánek construction of the Lebesgue integral is extended in two directions. First, instead of a compact interval [a, b] in the real line an abstract non-empty set X is considered, instead of the ring generated by subintervals of [a, b] an arbitrary ring A of subsets of X. Secondly, instead of the length of intervals (λ([c, d]) = d?c) any vector measure λ: A→V is considered, where V is a Riesz space.     

Embeddings of Lorentz—Marcinkiewicz spaces with mixed norms

X(Y) f -:X(Y)={fM(×): f_{X(Y)=f(x,.)YX<} . =(0, ), M (×) — , ×, X, Y, Z— . X(Y) Z(×).     

Function spaces of Besov-type and Triebel-Lizorkin-type — a survey

This article is devoted to presenting a recapitulative introduction for the theory of Besov-type and Triebel-Lizorkin-type spaces developed in recent years.