Density of smooth functions in variable exponent Sobolev spaces

We show that if p−≥2p−≥2, then a sufficient condition for the density of smooth functions with compact support, in the variable exponent Sobolev space W1,p(⋅)(Rn)W1,p(⋅)(Rn), is that the Riesz potentials of compactly supported functions of Lp(⋅)(Rn)Lp(⋅)(Rn), are also elements of Lp(⋅)(Rn)Lp(⋅)(Rn). Using this result we then prove that the above density holds if (i) p−≥np−≥n or if (ii) 2≤p−