On $H=W$ in Banach function spaces

Abstract

In this paper we prove ``$H=W$" in the context of a Banach function space $X(Ω)$. Let $Ω$ be a subset of ${\mathbb R}^n$ and denote by $W^1_X(Ω)$ the collection of all those $f\in X(Ω)$ whose distributional derivatives $\partial_jf$ are contained in $X(Ω)$. Our main result provides a small collection of ``universal" hypotheses on $X(Ω)$ that ensure $W^1_X(Ω)$ is equal to $H^1_X(Ω)$, the formal closure of ${Lip}(Ω)\cap W^1_X(Ω)$ with respect to the norm \[\|f\|_{W^1_X(Ω)} = \|f\|_{X(Ω)} + \|\nabla f\|_{X(Ω)}.\] The main theorem has two corollaries. The first gives a slightly stronger set of hypotheses for ``$H=W$", and the second gives density of $C^\infty_c({\mathbb R}^n)$ in $W^1_X({\mathbb R}^n)$.