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On Kantorovich's result on the symmetry of Dini derivatives

Martin Koc, Luděk Zajíček (2010)

Commentationes Mathematicae Universitatis Carolinae

For f : ( a , b ) → ℝ , let A f be the set of points at which f is Lipschitz from the left but not from the right. L.V. Kantorovich (1932) proved that, if f is continuous, then A f is a “( k d )-reducible set”. The proofs of L. Zajíček (1981) and B.S. Thomson (1985) give that A f is a σ -strongly right porous set for an arbitrary f . We discuss connections between these two results. The main motivation for the present note was the observation that Kantorovich’s result implies the existence of a σ -strongly right porous set A ⊂ ( a , b ) ...

On Kurzweil-Henstock equiintegrable sequences

Štefan Schwabik, Ivo Vrkoč (1996)

Mathematica Bohemica

For the Kurzweil-Henstock integral the equiintegrability of a pointwise convergent sequence of integrable functions implies the integrability of the limit function and the relation m abfm(s)s = abm fm(s)s. Conditions for the equiintegrability of a sequence of functions pointwise convergent to an integrable function are presented. These conditions are given in terms of convergence of some sequences of integrals.

On Kurzweil-Stieltjes integral in a Banach space

Giselle A. Monteiro, Milan Tvrdý (2012)

Mathematica Bohemica

In the paper we deal with the Kurzweil-Stieltjes integration of functions having values in a Banach space X . We extend results obtained by Štefan Schwabik and complete the theory so that it will be well applicable to prove results on the continuous dependence of solutions to generalized linear differential equations in a Banach space. By Schwabik, the integral ∫ a b d [ F ] g exists if F : [ a , b ] → L ( X ) has a bounded semi-variation on [ a , b ] and g : [ a , b ] → X is regulated on [ a , b ] . We prove that this integral has sense also if F is regulated on [ a , b ] ...

On local properties of functions and singular integrals in terms of the mean oscillation

Rahim Rzaev, Lala Aliyeva (2008)

Open Mathematics

This paper is devoted to research on local properties of functions and multidimensional singular integrals in terms of their mean oscillation. The conditions guaranteeing existence of a derivative in the L p-sense at a given point are found. Spaces which remain invariant under singular integral operators are considered.

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