Density of smooth functions in variable exponent Sobolev spaces

Abstract

We show that if $p_{-}\ge 2$, 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) $p_{-}\ge n$ or if (ii) $2\le p_{-}<n$ and $p_{+}<\frac{n{p}_{-}}{n-p_{-}}$. Moreover our result allows us to give an alternative proof, for the case $p_{-}\ge 2$, that the local boundedness of the maximal operator and hence local log-H{ö}lder continuity imply the density of smooth functions with compact support, in the variable exponent Sobolev space $W^{1,p(\cdot )}({\mathbb{R}}^n)$.

arXiv commentTo appear in Nonlinear Analysis TM&A