This article is cited in 5 papers

On denseness of $C_0^\infty(\Omega)$ and compactness in $L_{p(x)}(\Omega)$ for $0<p(x)<1$

R. A. Bandaliev^ab, S. G. Hasanov<sup>ac

a</sup> Institute of Mathematics and Mechanics of ANAS, AZ 1141 Baku, Azerbaijan
^b S.M. Nikolskii Institute of Mathematics at RUDN University, 117198 Moscow, Russia
^c Gandja State University, Gandja, Azerbaijan

Abstract: The main goal of this paper is to prove the denseness of $C_0^\infty(\Omega)$ in $L_{p(x)}(\Omega)$ for $0<p(x)<1$. We construct a family of potential type identity approximations and prove a modular inequality in $L_{p(x)}(\Omega)$ for $0<p(x)<1$. As an application we prove an analogue of the Kolmogorov–Riesz type compactness theorem in $L_{p(x)}(\Omega)$ for $0<p(x)<1$.

Key words and phrases: $L_{p(x)}$ spaces, denseness, potential type identity approximations, modular inequality, compactness.

MSC: Primary 46E30, 46E35; Secondary 26D15

Language: English

DOI: 10.17323/1609-4514-2018-18-1-1-13

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