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.
Citation:
R. A. Bandaliev, S. G. Hasanov, “On denseness of $C_0^\infty(\Omega)$ and compactness in $L_{p(x)}(\Omega)$ for $0<p(x)<1$”, Mosc. Math. J., 18:1 (2018), 1–13
\Bibitem{BanHas18}
\by R.~A.~Bandaliev, S.~G.~Hasanov
\paper On denseness of $C_0^\infty(\Omega)$ and compactness in $L_{p(x)}(\Omega)$ for~$0<p(x)<1$
\jour Mosc. Math.~J.
\yr 2018
\vol 18
\issue 1
\pages 1--13
\mathnet{http://mi.mathnet.ru/mmj660}
\crossref{https://doi.org/10.17323/1609-4514-2018-18-1-1-13}
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\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85044055532}
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V. I. Burenkov, T. V. Tararykova, “O neravenstve Geldera v lebegovykh prostranstvakh s peremennym poryadkom summiruemosti”, Posvyaschaetsya 70-letiyu prezidenta RUDN V.M. Filippova, SMFN, 67, no. 3, Rossiiskii universitet druzhby narodov, M., 2021, 472–482
R. A. Bandaliyev, P. Gorka, V. S. Guliyev, Y. Sawano, “Relatively compact sets in variable exponent Morrey spaces on metric spaces”, Mediterr. J. Math., 18:6 (2021), 232
J. Xu, “Precompact Sets in Bochner–Lebesgue Spaces
with Variable Exponen”, Math. Notes, 110:6 (2021), 932–941
Jaume Giné, Jaume Llibre, “Formal Weierstrass Nonintegrability Criterion for Some Classes of Polynomial Differential Systems in ℂ2”, Int. J. Bifurcation Chaos, 30:04 (2020), 2050064
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