Moscow Mathematical Journal

Volume 18, Issue 1, January–March 2018 pp. 1–13.

On Denseness of C₀^∞(Ω) and Compactness in L_p(x)(Ω) for 0<p(x)<1

Authors: R. A. Bandaliev (1) and S. G. Hasanov (2)
Author institution:(1) Institute of Mathematics and Mechanics of ANAS, AZ 1141 Baku, Azerbaijan;
S.M. Nikolskii Institute of Mathematics at RUDN University, 117198 Moscow, Russia
(2) Institute of Mathematics and Mechanics of ANAS, AZ 1141 Baku, Azerbaijan;
Gandja State University, Gandja, Azerbaijan

Summary:

The main goal of this paper is to prove the denseness of C₀^∞(Ω) in L_p(x)(Ω) for 0<p(x)<1. We construct a family of potential type identity approximations and prove a modular inequality in L_p(x)(Ω) for 0<p(x)<1. As an application we prove an analogue of the Kolmogorov–Riesz type compactness theorem in L_p(x)(Ω) for 0<p(x)<1.

2010 Math. Subj. Class. Primary: 46E30, 46E35; Secondary: 26D15.

Keywords: L_p(x) spaces, denseness, potential type identity approximations, modular inequality, compactness.

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