Density of smooth functions in variable exponent Sobolev spaces

TL;DR

This paper establishes conditions under which smooth functions are dense in variable exponent Sobolev spaces, linking the density to properties of Riesz potentials and maximal operators, with implications for regularity assumptions.

Contribution

It provides new sufficient conditions for the density of smooth functions in variable exponent Sobolev spaces, including cases based on the bounds of the exponent function and properties of Riesz potentials.

Findings

Density holds if $p_- ext{ and }p_+$ satisfy certain bounds.

Riesz potentials' integrability characterizes density conditions.

Local boundedness of maximal operator implies density of smooth functions.

Abstract

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 $W^{1,p(\cdot)}(\mathbb R^n)$.