On $H=W$ in Banach function spaces

Abstract

In this paper we prove ``$H=W$" in the context of a Banach function space $X(\hat{I}\text{c◯})$. Let $\hat{I}\text{c◯}$ be a subset of ${\mathbb{R}}^n$ and denote by $W_X^1(\hat{I}\text{c◯})$ the collection of all those $f\in X(\hat{I}\text{c◯})$ whose distributional derivatives ${\partial }_{j}f$ are contained in $X(\hat{I}\text{c◯})$. Our main result provides a small collection of ``universal" hypotheses on $X(\hat{I}\text{c◯})$ that ensure $W_X^1(\hat{I}\text{c◯})$ is equal to $H_X^1(\hat{I}\text{c◯})$, the formal closure of $Lip(\hat{I}\text{c◯})\cap W_X^1(\hat{I}\text{c◯})$ 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 $C_c^{\infty }({\mathbb{R}}^n)$ in $W_X^1({\mathbb{R}}^n)$.