Inequalities involving integrals of polar-conjugate concave functions

An inequality of K. Mahler, together with its case of equality, due to M. Meyer, are extended to integrals of powers of polar-conjugate concave functions. An application to estimation of the volume-product of certain convex bodies is given.     

A vanishing viscosity approach to a quasistatic evolution problem with nonconvex energy

We study a quasistatic evolution problem for a nonconvex elastic energy functional. Due to lack of convexity, the natural energetic formulation can be obtained only in the framework of Young measures. Since the energy functional may present multiple wells, an evolution driven by global minimizers may exhibit unnatural jumps from one well to another one, which overcome large potential barriers. To avoid this phenomenon, we study a notion of solution based on a viscous regularization. Finally we compare this solution with the one obtained with global minimization.     

Generalizations of some results concerning microperiodic mappings

It is well known that a microperiodic function that maps the set of reals into itself and is continuous at a point (Lebesgue measurable, respectively) must be constant (constant almost everywhere, resp.). We generalize those results in several directions. As a consequence we obtain conclusions concerning some systems of functional inequalities.     

Convergence in measure with respect to a matrix-valued measure and some matrix completion problems

For a matrix-valued measure M we introduce a notion of convergence in measure M, which generalizes the notion of convergence in measure with respect to a scalar measure and takes into account the matrix structure of M. Let S be a subset of the set of matrices of given size. It is easy to see that the set of S-valued measurable functions is closed under convergence in measure with respect to a matrix-valued measure if and only if S is a ρ-closed set, i.e. if and only if SP is closed for any orthoprojector P. We discuss the behaviour of ρ-closed sets under operations of linear algebra and the ρ-closedness of particular classes of matrices.     

A metric approach to asymptotic analysis

We introduce a notion of “firm” (or uniform) asymptotic cone to an unbounded subset of a normed space. We relate this notion to a concept of “firm” asymptotic function. We use these notions to study boundedness properties which can be applied to continuity questions for some operations on sets and functions. Such questions arise in stability analysis of Hamilton-Jacobi equations. We present some other applications such as an extension of a theorem of Dieudonné and existence results in optimization and fixed point theory.     

The convergence inL 1 of singular integrals in harmonic analysis and ergodic theory

We study the behavior of the ergodic singular integral associated to a nonsingular measurable flow {:t } on a finite measure space and a Calderón—Zygmund kernel with support in (0, ). We show that if the flow preserves the measure or, with more generality, if the flow is such that the semiflow {_t:t>-0} is Cesàrobounded,f and f are integrable functions, then the truncations of the singular integral converge to f not only in the a.e. sense but also in the L¹-norm. To obtain this result we study the problem for the singular integrals in the real line and in the setting of the weighted L¹-spaces.This research has been partially supported by a D.G.I.C.Y.T. grant (PB94-1496), a D.G.E.S. grant (PB97-1097) and Junta de Andalucía.     

Product measures and convergence theorems for product integrals in integration theory for semigroup valued functions

In [11] Sion has investigated measures and integrals having values in uniform semigroups. In this paper a result which gives sufficient conditions for the existence of a unique product of semigroup valued measures is established. Moreover a convergence theorem for product integrals of semigroup valued functions is proved.     

On several classes of sets II

Summary In this paper several classes of sets are stuided. It is shown that these classes are incomparable under the relation inclusion. If the setA is in one of our classes we examine whether it must follow thatA\K is in the same class, forK a finite or countable set. We also discuss which functions preserve the various classes under consideration.The research on this paper was supported by the Foundation for Scientific Work of the Republic of Bosnia and HerzegovinaDedicated to the memory of Alexander Ostrowski on the occasion of the 100th anniversary of his birth     

A unified approach to the topological centre problem for certain Banach algebras arising in abstract harmonic analysis

Let be a locally compact group. Consider the Banach algebra , equipped with the first Arens multiplication, as well as the algebra LUC , the dual of the space of bounded left uniformly continuous functions on , whose product extends the convolution in the measure algebra M . We present (for the most interesting case of a non-compact group) completely different - in particular, direct - proofs and even obtain sharpened versions of the results, first proved by Lau-Losert in [9] and Lau in [8], that the topological centres of the latter algebras precisely are and M , respectively. The special interest of our new approach lies in the fact that it shows a fairly general pattern of solving the topological centre problem for various kinds of Banach algebras; in particular, it avoids the use of any measure theoretical techniques. At the same time, deriving both results in perfect parallelity, our method reveals the nature of their close relation.Received: 1 January 2002     

A generalisation of the functional approach to compactness

A topological space X is compact iff the projection π:X×Y→Y is closed for any space Y. Taking this as a definition and then asking that π maps α-closed subspaces of X×Y onto β-closed subspaces of Y, for different closures α and β, extends the notion of compactness to include also examples of “asymmetric compactness” pursued in the article.Categorical closure operators and a so-called “functional approach to general topology” are employed to define and prove fundamental properties of compact objects and proper maps in this generalised setting.     

On strong differentiability of integrals along different directions

Theorems are proved as regards strong differentiability of integrals in different directions.     

一类粗糙核参数型Marcinkiewicz积分

In this paper,the L2-boundedness of a class of parametric Marcinkiewicz integral μρΩ,h with kernel function Ω in Bq0.0 (Sn-1) for some q＞ 1,and the radial function h (x)∈ l∞ (Ls) (R+) for 1＜s≤∞ are given. The Lp(Rn) (2≤p＜∞) boundedness of μ*Ω,ph,λ and μρΩ,h,s with Ω in Bq0,0(Sn-1) and h(|x|)∈l∞(Ls)(R+) in application are obtained. Here μ*Ω,p h,λ and μpΩ,h,s are parametric Marcinkiewicz integrals corresponding to the Littlewood-Paley gλ* function and the Lusin area function S,respectively.     

A symbolic operator approach to several summation formulas for power series II

Here expounded is a kind of symbolic operator method that can be used to construct many transformation formulas and summation formulas for various types of power series including some old ones and more new ones.     

Second order initial value problems with non-absolute integrals in ordered Banach spaces

In this paper we derive the existence and comparison results for second order initial value problems in ordered Banach spaces. The considered problems can be implicit, singular, functional, discontinuous and nonlocal. The main tools are fixed point results in ordered spaces and the theory of locally HL integrable vector-valued functions.     

Some results involving series representations of hypergeometric functions

In this paper, we prove some generalisations of several theorems given in [K.A. Driver, S.J. Johnston, An integral representation of some hypergeometric functions, Electron. Trans. Numer. Anal. 25 (2006) 115-120] and examine some special cases which correspond to a transformation given by Chaundy in [T.W. Chaundy, An extension of hypergeometric functions, I., Quart. J. Maths. Oxford Ser. 14 (1943) 55-78] and other transformations involving the Riemann zeta function and the beta function.     

A new proof for the estimates of Calderón-Zygmund type singular integrals

In this paper, a new proof for the estimates of Calderón-Zygmund type singular integrals will be presented. Received: 30 October 2005     

Box dimensions of Riemann-Liouville fractional integrals of continuous functions of bounded variation

If a continuous function f(x) has bounded variation on the unit interval [0,1], the box dimension of f(x) is 1. Furthermore, the box dimension of a Riemann-Liouville fractional integral of f(x) is still 1.