Volume 19, Issue 2

Abstract. In this paper we study the boundedness of the parametric Marcinkiewicz operator on generalized Morrey spaces . We find the sufficient conditions on the pair which ensure the boundedness of the operators from one generalized Morrey space to another , , and from the space to the weak space . As an application of the above result, the boundedness of the commutator of Marcinkiewicz operators on generalized Morrey spaces is also obtained. In the case , we find the sufficient conditions on the pair which ensure the boundedness of the operators from one generalized Morrey space to another , , and from the space to the weak space . In all the cases the conditions for boundedness are given in terms of Zygmund-type integral inequalities on which do not require any assumption on the monotonicity of .

Abstract. Let R be a ring and M a left R -module. An R -module N is called a cofinite extension of M in case and is finitely generated. We say that M has the property CE (resp. CEE) if M has a supplement (resp. ample supplements) in every cofinite extension. In this study we give various properties of modules with these properties. We show that a module M has the property CEE iff every submodule of M has the property CE. A ring R is semiperfect iff every left R -module has the property CE. We also study cofinitely injective modules, direct summands of every cofinite extension, as a generalization of injective modules.

Abstract. For a homogeneous system of differential equations of statics of the elastic mixture theory, we solve the basic boundary value problems using Papkovich–Neuber representations and Fourier integral transforms when the displacement vector or stress vector values are given on the boundary . The solution uniqueness theorem is proved for the problems posed. Solutions are obtained in quadratures.

Abstract. The notion of the classical Lorentz space is generalized to the grand Lorentz space . Some basic properties of the space are proved and the boundedness of the maximal operator M is characterized in . It is shown that is bounded if and only if the weight w belongs to the famous -class.

Abstract. The paper is concerned with measurable and nonmeasurable subsets of the unit square (unit cube) whose linear (planar) sections have constant linear (planar) Lebesgue measure. Some properties of such subsets are studied and their connections with the existence of iterated integrals are pointed out.

Abstract. Linear spaces are considered in the following four situations: a real space admits multiplication by complex scalars without changing the set itself; a real space is embedded into a wider set with multiplication by complex scalars; a complex space has an involution and thus has “real” and “imaginary” elements; a combination of the previous ones. We study how they manifest themselves when the initial space possesses additional structures such as topology, norm, inner product, and also the behavior of linear operators between such spaces.

Abstract. In the present paper, the continuous Hu cohomology with coefficients is defined in an arbitrary topological abelian group. The aim of this work is to find the connection between this cohomology and the Alexander–Spanier cohomology for different groups of coefficients.

Abstract. We provide a positive answer to a question raised by Al-Thagafi and Shahzad [Nonlinear Anal. 70 (2009), 3665–3671] about the existence of a best proximity point for a cyclic -contraction map in a reflexive Banach space by using appreciable generalized notions. In turn, this also gives a positive answer to a question raised by Eldred and Veeramani [J. Math. Anal. Appl. 323 (2006), 1001–1006]. To this end, we introduce a new concept of C -proximity point and a new class of maps, called cyclic -contractions, which contains cyclic contraction maps and cyclic -contraction maps as subclasses. Convergence and existence results of best C -proximity points for cyclic -contraction maps are also obtained.

Abstract. It is shown that a particular categorical axiomatisation of the category of locales is slice stable. This localic slice stability can be used to recover the fundamental theorem of topos theory. A categorical account of the ideal completion of a preorder is developed and is used to give a new proof of Joyal and Tierney's result on the slice stability of locales.

Abstract. Let X and Y be smooth, connected, projective complex curves and let . Let and be two integers. Let and be two partitions of d . In this paper we study equivalence classes of pairs satisfying the following: is a degree d covering of Y , f is unramified at b 0 and it is branched in points, n of which are points of simple branching, one is a special point whose local monodromy has cycle type and one is a special point whose local monodromy has cycle type . Moreover, is a bijection and the monodromy group of f is . We prove that the corresponding Hurwitz spaces are irreducible under the hypothesis where and denote, respectively, the genus of X and Y .

Abstract. Let be an analytic self-map of the open unit disk in the complex plane and be an analytic map on . Such maps induce a weighted composition operator followed by differentiation acting between the weighted Banach spaces of holomorphic functions. We characterize the boundedness and compactness of such operators in terms of the involved weights as well as the functions and .