Density of smooth functions in variable exponent Sobolev spaces. (November 2015)

• Record Type: Journal Article
• Authors: Kostopoulos, Thanasis; Yannakakis, Nikos
• Is Part Of: Nonlinear analysis. Volume 127(2015)
• Page Start: 196
• Page End: 205
• Publication Date: 2015-11
• Subjects: 46E30 -- 46E35; Variable exponent Lebesgue space -- Density of smooth functions -- Variable exponent Sobolev space -- Maximal operator -- Riesz potential; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2015.07.007
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We show that if p − ≥ 2, then a sufficient condition for the density of smooth functions with compact support, in the variable exponent Sobolev space W 1, p ( ⋅ ) ( R n ), is that the Riesz potentials of compactly supported functions of L p ( ⋅ ) ( R n ), are also elements of L p ( ⋅ ) ( R n ) . Using this result we then prove that the above density holds if (i) p − ≥ n or if (ii) 2 ≤ p − < n and p + < n p − n − p − . Moreover our result allows us to give an alternative proof, for the case p − ≥ 2, that the local boundedness of the maximal operator and hence local log-Hölder continuity imply the density of smooth functions with compact support, in the variable exponent Sobolev space W 1, p ( ⋅ ) ( R n ) .