# Moscow Mathematical Journal

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

On Denseness of *C*_{0}^{∞}(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
*C*_{0}^{∞}(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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