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On Korenblum convex functions

Ereú J., López L., Merentes N.

Commentationes Mathematicae

2017

Vol 57, No. 1

25--44

We introduce a new class of generalized convex functions called the K-convex functions, based on Korenblum’s concept of k-decreasing functions, where K is an entropy (distortion) function. We study continuity and differentiability properties of these functions, and we discuss a special subclass which is a counterpart of the class of so-called d.c. functions. We characterize this subclass in terms of the space of functions of bounded second k-variation, extending a result of F. Riesz. We also present a formal structural decomposition result for the K-convex functions.

The Nemitskij operator on Lipk-type and BVk-type spaces

Giménez J., López L., Merentes N.

Demonstratio Mathematica

2013

Vol. 46, nr 3

543--558

In this paper, we discuss and present various results about acting and boundedness conditions of the autonomous Nemitskij operator on certain function spaces related to the space of all real valued Lipschitz (of bounded variation, absolutely continuous) functions defined on a compact interval of R. We obtain a result concerning the integrability of products of the form (…) and a generalized version of the chain rule for functions a.e differentiable, in the sense of Lebesgue. As an application, we get a generalization of a theorem due to V. I. Burenkov for the case of functions of bounded Riesz-p-variation.