Density of smooth functions in variable exponent Sobolev spaces
We show that if $p_-\geq 2$, then a sufficient condition for the density of smooth functions with compact support, in the variable exponent Sobolev space $W^{1,p(\cdot)}(\mathbb R^n)$, is that the Riesz potentials of compactly supported functions of $L^{p(\cdot)}(\mathbb R^n)$, are also elements of $L^{p(\cdot)}(\mathbb R^n)$. Using this result we then prove that the above density holds if (i) $p_-\geq n$ or if (ii) $2\leq p_-< n$ and $p_+<\frac{np_-}{n-p_-}$. Moreover our result allows us to give an alternative proof, for the case $p_-\geq 2$, that the local boundedness of the maximal operator and hence local log-H{\"o}lder continuity imply the density of smooth functions with compact support, in the variable exponent Sobolev space $W^{1,p(\cdot)}(\mathbb R^n)$.